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LIUUXUV  or  THI:  I\IVI:KSITV  OF  C.\IJFOK.\I\. 


IMIVSIIS  DEPARTMEPTT. 

1,11   1     01 


WHIIINd. 

September.  1896. 
^7»/s>:  A 


NOTE. 

MACMIU,AN  &  Co.  take  pleasure  in  calling  your  attention 
to  the  new  edition  of  DANIELI/S  PHYSICS  in  the  hope  that  you 
may  find  it  possible  either  to  adopt  the  book  as  the  required 
text  in  your  general  course  in  physics  or  to  recommend  it 
as  valuable  for  collateral  reading  and  as  a  text-book  for 
reference.  The  copy  now  sent  you  is  one  of  a  small  edition 
printed  for  a  class  which  wished  to  make  immediate  use  of  the 
book.  A  few  typographical  errors  may  be  found  in  it  since 
the  author's  final  corrections  after  the  pages  were  electrotyped 
have  not  yet  been  made.  All  minor  defects  of  this  kind  will, 
however,  be  corrected  before  the  larger  edition  for  general 
use  is  printed.  The  index  is  still  in  the  author's  hands  for 
revision. 


UHI7IESIT7 


A  TEXT  BOOK 


OF  THE 


PRINCIPLES  OF  PHYSICS 


BY 

ALFRED  DANIELL,  M.A.,  LL.B.,  D.Sc.,  F.R.S.E. 

OF  THE  INNER  TEMPLE,   BARBISTER-AT-LAW 

MEMBER  OF  THE  FACULTY  OF  ADVOCATES  IN  SCOTLAND 

FORMERLY  LECTURER  ON   PHYSICS  IN  THE  SCHOOL  OF  MEDICINE,  EDINBURGH 


THIRD    EDITION 
(Sixth  Thousand) 


f  orfe 
MACMILLAN   AND    CO. 

AND     LONDON 
1894 

All  rights  reserved 


Entered  according  to  Act  of  Congress,  in  the  year  1894, 

BY  ALFEED  DANIELL, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


First  Edition,  Feb.  20,  1884.      1500  copies. 
Second  Edition,  Sept.  n,  1885.    4000  copies. 


Norfoooti 

J.  S.  Gushing  &  Co.  —  Berwick  &  Smith. 
Boston,  Mass.,  U.S.A. 


PREFACE  TO   THE    THIRD  EDITION. 

I  HAVE  tried,  in  revising  this  volume,  to  maintain  the 
characteristics  of  the  book,  to  improve  it  where  it  seemed 
to  need  improvement,  to  correct  any  errors  into  which 
I  had  fallen,  to  fill  up  any  gaps  which  seemed  impor- 
tant, and  to  keep  it  up  to  date.  I  have  considered 
every  equation  from  the  point  of  view  of  its  Dimensions, 
and  have  had  to  modify  some  of  the  equations  accord- 
ingly ;  I  have  tried  to  remove  some  difficulties  from  the 
student's  path  by  the  adoption  of  a  uniform  notation ; 
and  I  have  endeavoured  to  make  it  clear  to  him  what 
the  precise  physical  quantities  are  to  which  reference  is 
being  made  in  each  particular  case. 

Whatever  may  be  my  own  shortcomings  in  respect 
of  the  execution  of  this  revision,  I  hope  that  what  I 
have  been  able  to  do  may  at  any  rate  tend  in  the  direc- 
tion of  accuracy  and  precision ;  and  I  also  hope  that  the 
additional  labour  which  I  have  bestowed  upon  this  book 
may  be  accepted  as  an  earnest  acknowledgment  of  the 
uniform  kindness  with  which  the  work  has  been  received. 

8  NEW  COURT,  A.  D. 

LINCOLN'S  INN, 

LONDON,  W.C, 


JU7BRSIT 


PREFACE    TO    THE    FIRST    EDITION. 

IN  the  following  pages  I  have  endeavoured  to  give, 
in  terms  as  simple  as  the  nature  of  the  subject  will  per- 
mit, a  connected  account  of  the  leading  principles  of 
modern  physical  science. 

My  aim  has  not  been  to  build  up  a  mere  compendium 
of  physical  facts,  but  rather  to  put  the  reader  in  posses- 
sion of  such  principles  as  will  enable  him  with  small 
difficulty  to  apprehend  and  to  appreciate  those  facts. 

I  am  regretfully  aware  of  many  material  omissions. 
The  subject  of  Natural  Philosophy  is  so  vast  that  many 
things  which  in  themselves  are  by  no  means  devoid  of 
importance  —  but  to  which  different  writers  would  per- 
haps be  inclined  to  attribute  different  degrees  of  impor- 
tance—  must  necessarily  be  laid  aside  in  the  course  of 
the  preparation  of  a  text-book  of  limited  size.  One  of 
these  omissions,  which  my  own  love  of  the  develop- 
mental history  of  science  made  me  decide  upon  with 
extreme  unwillingness,  is  that  of  the  history  and  the 
personal  aspect  of  scientific  discovery.  As  a  general  rule, 
the  names  of  discoverers,  even  where  they  are  mentioned, 
play  a  very  subordinate  part  in  the  text. 

At  the  same  time,  I  trust  that  the  reader  of  this 
work  will  find  that,  after  assimilating  its  contents,  he 


yiii  PREFACE. 

is  in  some  measure  prepared  for  the  reception  of  further 
information  in  the  course  of  that  wider  reading  and 
practical  study  to  which  I  hope  the  following  pages  will 
be  found  fitted  to  serve  as  an  elementary  introduction. 

It  is  wholly  beyond  question  that  to  him  who  desires 
to  become  a  physicist,  Practical  Laboratory  Work  is  abso- 
lutely essential.  Thorough  knowledge  must  be  drunk  in 
by  the  eyes  and  the  ears,  and  absorbed  by  the  finger- 
tips ;  and  the  true  use  of  a  book  of  this  kind  is,  I  take 
it,  not  to  replace  practical  work  but  to  economise  the 
labours  of  the  student.  This  it  may  do  by  so  furnish- 
ing his  mind  with  a  store  of  general  principles,  that 
when  he  comes  to  enter  a  physical  laboratory  he  may 
there  find  around  him,  in  the  concrete  form,  a  collection 
of  pieces  of  apparatus,  the  construction  and  the  action 
of  which  he  is  able,  by  the  application  of  the  principles 
already  familiar  to  him,  promptly  and  intelligently  to 
comprehend.  Bearing  the  necessary  limitations  of  the 
usefulness  of  any  mere  book  steadily  before  me,  I  have 
endeavoured,  as  far  as  possible,  to  simplify  and  gener- 
alise all  descriptions  of  apparatus,  and  in  the  same  way 
to  simplify  and  generalise  the  accompanying  diagrams ; 
and  thus  I  have  tried  not  only  to  adapt  the  work  to  the 
requirements  of  those  who  may  use  it  as  a  stepping-stone 
to  further  attainments,  but  also  to  render  it  a  suitable 
text-book  for  that  larger  circle  of  readers  who,  having 
no  distinct  desire  to  follow  out  the  special  study  of  phys- 
ics, may  yet  wish  to  possess  an  elementary  acquaintance 
with  the  modern  aspect  of  natural  philosophy. 

This  book  was  primarily  designed  as  a  contribution 
to  Medical  Education,  and  as  such  I  hope  it  may  be 


PREFACE.  ix 

found  useful.  That  arrangement,  which  still  prevails  in 
some  of  our  Universities,  under  which  a  student  of  medi- 
cine may  even  proceed  to  the  degree  of  M.D.  without 
any  adequate  knowledge  of  physics,  is  self -evidently 
opposed  to  common  sense,  and  to  the  exigencies  of 
physiological  study  and  of  medical  practice.  Such  an 
anomaly  cannot,  it  may  be  anticipated,*  endure  much 
longer.  Before  many  years  are  over  it  will  be  univer- 
sally acknowledged  in  practice,  as  it  already  is  in  theory, 
that  knowledge  of  natural  philosophy  is  an  essential  part 
of  the  mental  equipment  of  the  medical  student  and 
of  the  properly-trained  medical  man.  The  needs  of  the 
intelligent  student  of  physiology  have  been  kept  con- 
stantly before  my  mind,  as  I  hope  those  of  my  readers 
who  are  already  physiologists  will  recognise ;  but  I  have 
been  careful  not  to  make  the  book  one  suited  for  admis- 
sion only  into  a  medical  class-room ;  my  aim  has  been  to 
produce  a  work  useful  at  once  to  the  Student  of  Medicine, 
the  Student  of  Science,  and  the  General  Reader. 

The  plan  of  the  work  is  that  of  a  gradual  progression 
from  the  simplest  to  the  more  complex.  No  preliminary 
knowledge  of  physical  principles  is  assumed,  and  every 
effort  has  been  made  to  attain  to  absolute  lucidity  of 
expression,  even  though  this  be  found  occasionally  to 
necessitate  the  frequent  repetition  of  a  single  word  in  the 
course  of  a  single  sentence.  While  the  reader  is  expected 
as  he  reads  each  page  to  remember  the  contents  of  the 
preceding  pages,  I  trust  that  I  have  sufficiently  carried 


*  It  is  gratifying  that  this  anomaly  has  ceased  to  exist,  in  the  United  Kingdom, 
since  the  year  1892,  in  which  year  the  new  regulations  of  the  General  Medical  Coun- 
cil came  into  force. 


x  PKEFACE. 

out  my  intention  of  nowhere  setting  before  him  anything 
of  the  nature  of  an  unsolved  riddle,  so  far  as  that  could 
be  guarded  against  by  my  own  efforts  on  his  behalf. 

I  have  endeavoured  to  secure  intelligible  continuity 
throughout  the  paragraphs  printed  in  larger  type,  and 
thereby  to  enable  the  reader  on  his  first  perusal  to  confine 
his  attention  to  the  more  prominent  portions  of  the  text. 

However  imperfectly  its  design  may  have  been  exe- 
cuted, I  shall  be  glad  if  this  work  be  found  to  contribute 
in  any  degree  to  the  extension  of  that  mode  of  teaching 
Natural  Philosophy  for  the  establishment  of  which  we 
have  come,  directly  and  indirectly,  to  owe  so  much  to  the 
advocacy  and  example  of  Professors  Thomson  [now  Lord 
Kelvin]  and  Tait  —  a  mode  of  teaching  under  which  the 
whole  of  Natural  Philosophy  is  regarded  as  substantially 
a  single  science,  in  which  scattered  facts  are  connected 
and  co-ordinated  by  reference  to  the  principles  of  Dynam- 
ics and  the  great  experimental  Law  of  the  Conservation 
of  Energy. 

20th  February  1884. 


UFI7BESITT 


INTRODUCTORY 


PAGE 
1 


CHAPTER  I. 

TIME,  SPACE,  AND  MASS. 


Standards  of  Time 

Standards  and  Dimensions  of  Space 

Standards  of  Mass 

The  C.G.S.  System  . 


9 
9 

12 
13 


CHAPTER  IL 

NOTIONS  DERIVED  FROM  THE  PRECEDING. 

Motion      .           .           .            .            .           .           .           .            .  .14 

Velocity   ..........      14 

Digression  on  Mathematical  Formulae  and  the  Theory  of  Dimensions  .      15 

Acceleration         .  .         .           '.            .            .            .        .    .<.          .  .18 

Momentum           »           .            .            .        .    .           >  ,     ,..,           .  .      19 

Force        .           .           .           .            .            .  „        «           .           .  .      19 

Weight        *.           .           .           .           .   f       ...          ;          .  .21 

Stress,  p.  23 ;  Pressure,  p.  24 ;  Tension    .           .           •    :       .  .25 


CHAPTER  TIL 

MEASUREMENTS . 

Measurement  of  Length  —  Line  Measurement,  p.  27  ;  End  Measurement  . 
Surface  .  .  .          ... 

Volume        .  .  . 

Time    . 


Force  . 


30 
32 
33 
34 
35 
35 


Work,  p.  40  ;  Activity  . 
Energy 

Potential  Energy 


CHAPTER  IV. 
WORK  AND  ENERGY. 


42 
42 
43 


CONTENTS. 

PAGE 

Kinetic  Energy        .  .  .  .  .  .  .  .46 

Conservation  of  Energy      .......      47 

Transformations  of  Energy  ,  .  .  .  .47 

Availability  of  Energy        .  .  .  .  .50 

Graphic  representation  of  Energy  .  .  .  .  .53 

The  Indicator  Diagram  .  .  .  .  .  .65 


CHAPTER  V. 

KINEMATICS. 

GENERAL  PROPOSITIONS  .  .  .        .  .  .          v .  .  .57 

Direction,  p.  57  ;   Velocity,  p.  58  ;   Dimensions,  p.  59  ;  Simultaneous 
Motions,  p.  60  ;  Parallelogram  of  Velocities,  p.  61 ;  Triangle  of  Veloc- 
ities, p.  63  ;  Resolution  of  Velocity  into  Components,  p.  63  ;  Compo- 
sition of  more  than  two  velocities,  p.  65  ;  The  Polygon  of  Velocities, 
p.  66  ;  Reference  to  Axes,  p.  66  j  Velocities  not  in  one  plane,  p.  67  ; 
Change  of  Velocity  .  .  .      68 

Parallelograms,  etc.,  of  Accelerations,  p.  68 ;  Accelerated  Motion,  p.  69 ; 

Composition  of  Uniform  with  Accelerated  Motion      .  .  .71 

Degrees  of  Freedom  of  a  Particle,  p.  72  ;  Translation,  p.  73  ;  Rigid 
Body,  p.  73  ;  Centre  of  Figure,  p.  73  ;  Rotation,  p.  74  ;  Composi- 
tion of  Rotations,  p.  74 ;  Precession  and  Nutation,  p.  75  ;  Degrees 
of  Freedom  of  a  Rigid  Body      ......      76 

Strain,  p.  77  ;  Shear  .  .  .  .  .  .78 

Circular  Motion,  p.  78 ;  Curvature  .  .  .  .79 

SIMPLE  HARMONIC  MOTION  AND  WAVE-MOTION         .  .  .    •        .80 

Simple  Harmonic  Motion      .  .  .  .  .  .  .80 

Acceleration  in  S.H.M.  proportional  to  Displacement,  p.  83  ;  Isochro- 
nous S.H.M.'s,  p.  83  ;  Frequency,  p.  83  ;  Projection  of  a  S.H.M., 
p.  84  ;  Harmonic  Curve    .  .  .  .  ....       85 

Composition  of  S.H.M.'s       .  .  .  .    -        .  .  .86 

Conversion  of  circular  into  reciprocating  motion,  p.  88  ;  of  reciprocat- 
ing into  circular    .  .  »         .  .  .          • .   •.  •        ,  .89 
Composition  of  S.H.M.'s  at  right  angles  —  of  the  same  period,  p.  89; 
of  different  period,  p.  91  ;  of  different  phase,  p.  92  ;   of  non-com- 
mensurable period           „>•'•.            .            .            .            .  .93 

Resolution  of  S.H.M.  into  rectangular  components         .  .  .      96 

Composition  of  S.H.M.  with  uniform  motion      .  .  .'  .96 

Composition  of  S.H.M.'s  in  the  same  line  .  .        ".    '        .97 

Beats V.     100 

Rotation  of  plane  of  S.H.M.          .  .  .  .  .101 

Composition  of  several  S.H.M.'s,  p.  101  ;  Fourier's  Theorem,  p.  101 ; 
Tide  calculating  machine  .  .  ...  .  .103 

Oscillatory  Movement  of  Systems  of  Particles         .  .-•         .  .     103 

Waves,  p.  104  ;  Wave-length,  p.  104  ;  Velocity  of  propagation  .     105 

Transversal  vibrations  of  a  cord,  p.  106  ;  Longitudinal  vibrations  of 
a  cord        .  .  .  .  .  ...  .  .    110 

Waves  on  a  surface,  p.  112  ;  in  a  tridimensional  substance,  p.  114  ; 
Concentric  Waves,  p.  114;  Direction  of  the  Wave,  p.  115;  Flat 
Wave-front,  p.  115  ;  Wave  passing  through  an  aperture  .  .  116 


CONTENTS.  xiii 

PAGE 

Reflexion  —  of  linear  waves,  p.  117  ;  of  a  plane  wave-front  at  a  plane 
surface,  p.  119  ;  of  a  curved  wave-front  at  a  plane  surface,  p.  120 ; 
General  Construction,  p.  121  ;  Problems  ....  122 

Transmission  of  a  linear  wave  into  a  denser  medium,  p.  124 ;  into  a 
rarer  medium  .  ...  .  .  .  .  125 

Refraction  of  a  plane  wave  at  a  plane  surface,  p.  126 ;  General  construc- 
tion for  refraction,  p.  127  ;  Kefraction  of  a  spherical  wave  at  a  plane 
surface,  p.  128 ;  Caustic  by  refraction,  p.  129  ;  Approximate  foci, 
p.  129 ;  Refraction  of  a  plane  wave  at  a  spherical  surface,  p.  130  ; 
Refraction  of  a  spherical  wave  at  a  spherical  surface  .  .  .130 

Rays  as  used  in  geometrical  construction,   p.   131 ;   Ptolemy's  Law, 

p.  133 ;  Fermat's  Law      .  .  .  .  .  .  .133 

Superposition  of  Wave-motions  —  on  an  indefinite  cord,  p.  133 ;   nodes 

and  loops,  p.  134;  Stationary  vibrations  .  .        -:- ••';•>  .    134 

Vibrations  of  a  Cord  whose  extremities  are  fixed,  p.  135 ;  transversal, 
p.  135 ;  longitudinal,  p.  135  ;  whose  extremities  are  free,  p.  136 ;  a 
rod  of  which  one  extremity  is  free  .  .  .  .  .136 

Vibrations  of  a  membrane  .......     137 

Interference      .  ...  .  .  .  .  .137 

Propagation  of  waves  along  normals,  p.  138  ;  Effects  of  a  screen, 
p.  139 ;  Diffraction  round  a  screen,  p.  139 ;  Wave  traversing  an 
aperture,  p.  140;  Relation  of  Wave-length  to  Diffraction-Fringes, 
p.  141 ;  Broken  wave-front,  p.  141 ;  Diffraction  .  .  .  141 

Energy  of  S.H.M.,  p.  141;  Energy  of  Conical  Pendulum,  p.  142; 
Energy  of  wave-motion,  p.  142 ;  Rate  of  propagation  of  groups  of 
waves  142 


CHAPTER  VI. 

KINETICS. 

GENERAL  PROPOSITIONS  relating  to  Force  parallel  to  those  relating  to 

Velocity     .,'.., 143 

Resolution  of  Forces       ..  .  .  .  .  .  .  .  .    143 

Experimental  proof  of  the  Parallelogram  of  Forces       .  .  .    144 

The  Equilibrium  of  Forces .  .  .  .  .  .  .145 

Centre  of  Figure      .  .  .  .  .  .  .  .146 

Inertia  of  Matter,  p.  146  ;  Coefficient  of  Inertia,  p.  147  ;  Examples     .     147 
Momentum,  p.  149 ;  Impact  of  Inelastic  Bodies,  p.  150  ;   Apparent  loss 
of  Energy  on  Impact,  p.  150  ;  Impact  of  Elastic  Bodies,  p.  151 ;  Ob- 
lique Jmpact,  p.  152  ;  Energy  in  Impact  of  Elastic  Bodies     .  .     152 
Accelerated  Motion,  p.  152 ;  Problems          .            .            .            .  .     153 

The  Principle  of  Moments     .  .  , '  '        .  .  .  .     155 

Torque,  p.  158;   Force  constant  in   direction,   Rotational  Component 

varies         .  .  .    f       .  .  .  .  .  .    158 

Couples,  p.  158 ;  Moment  of  a  Couple,  p.  159 ;   Examples  of  Couples, 

p.  160  ;  Equilibrium  of  Couples ......    160 

Rotation,  p.  161 ;  Moment  of  Inertia,  p.  162  ;  Radii  of  Inertia  and 
Moments  of  Inertia  in  particular  cases,  p.  162  ;  Angular  Momentum, 
p.  163 ;  Energy  of  a  Rotating  Body,  p.  163 ;  Minimum  Angular 
Velocity,  p.  164 ;  Centres  of  Oscillation  and  Percussion,  p.  164 ; 
Table  166 


CONTENTS. 

PAGE 

"  Centrifugal  Force"  .  .  .  .  .  .  .165 

The  Mechanical  Powers,  p.  169 ;  The  Lever,  p.  169 ;  The  Wheel  and 
Axle,  p.  171  ;  Wheelwork,  p.  172 ;  The  Inclined  Plane,  p.  172  ; 
The  Screw,  p.  173 ;  The  Wedge,  p.  174 ;  Pulleys,  p.  174 ;  The  Bell- 
crank,  p.  175  ;  The  Knee  .  ....     175 

FEICTION,  p.  176 ;  Statical  Friction  between  Solids,  p.  176 ;  The  Limiting 
Angle,  p.  177  ;  The  Angle  of  Repose,  p.  178  ;  Hope  round  Post,  p.  178  ; 
Kinetical  Friction  between  Solids,  p.  179 ;  Influence  of  Duration 
of  Contact,  p.  179 ;  Transformation  of  Energy  by  Friction,  p.  180  ; 
Negative  Acceleration,  p.  180 ;  Brakes,  p.  180 ;  Critical  Angle  in 
Kinetic  Friction,  p.  180;   The  Mechanical  Powers,  p.  180;   Work 
done  against  Friction,  p.  181 ;  Resistance  to  Traction,  p.  181 ;  Roll- 
ing Friction,  p.  182  ;  Belting,  p.  182 ;  Activity  in  Belting,  p.  183 ; 
Friction-Dynamometers,  p.  184  ;  Variations  in  Kinetic  Friction        .     184 
Friction  of  Solids  against  Liquids,  p.  184  ;  Friction  on  a  raindrop          .    185 
Viscosity-Resistances,  p.  185  ;  Friction  in  S.H.M.    .       •.  V          .  .    185 


CHAPTER  VII. 
ATTRACTION  AND  POTENTIAL. 

Attraction,  p.  187 ;  Particular  cases      .  .  .  ,  .    188 

Convention  as  to  Attraction  —  and  Repulsion  -f  .  .  ,  .    189 

Potential  Energy  in  the  case  of  Repulsion  .  .  /        .  .    190 

Work  done  by  Repulsion    .  .  .  _ . , .  .  .     190 

Potential  Energy  unexhausted  at  any  given  distance      .  .  .    191 

Direction  of  Movement       .  .  .  .  .  .  .191 

Potential  .  .  .  .  .  .  .  .  .  .    192 

Potential  a  condition  at  a  point  in  space   .....    192 

Gravitation-Potential  ....  .  .  .192 

Absolute  Zero  of  Potential,  p.  192  ;  Fields  of  Space  in  opposite  condi- 
tions, p.  192  ;  Continuity  of  Potential  through  Zero  value,  p.  193  ; 
Arbitrary  Zero  of  Potential,  p.  193  ;  Analogy  of  Sea-level,  p.  193 ; 
Equipotential  Surfaces,  p.  193  ;  Motion  parallel  to  these  surfaces, 
p.  194 ;  across  them,  p.  194 ;  Work  done  in  crossing  them,  p.  194 ; 
Distances  between  successive  Equipotential  Surfaces,  p.  195  ;  Equi- 
potential Surfaces  of  complex  form,  p.  196  ;  Free  movement  at  right 
angles  to  them,  p.  196  ;  Lines  of  Force,  p.  196 ;  Tubes  of  Force, 
p.  197  ;  Number  of  Lines  of  Force,  p.  197 ;  Systems  of  Surfaces  and 
Lines,  p.  198;  Variations  in  Differences  of  Potential,  p.  199; 
Theorem,  p.  199;  Potential  of  a  Double  Sheet,  p.  200;  Isody- 
namic  Surfaces  and  Lines  of  Slope  .....  200 

CHAPTER  VIII. 
GRAVITATION  AND  THE  PENDULUM. 

Law  of  Gravitation,  p.  201  ;  Cavendish's  Experiment         '  .        ;..'  .  202 

Accelerated  Motion  under  Gravity  .  .        -    .  .  .  202 

Path  of  a  Projectile,  p.  203  ;  Kepler's  Laws        .        '  •.  .  .  203 

Universal  Gravitation,  p.  203  ;   Variations  of  g  on  the  earth's  surface, 

p.  205 ;  Measurement  of  g          .  .  .  .  .  .  206 

Centre  of  Gravity,  p.  206  ;  of  Two  Masses,  p.  207 ;  of  a  System  of  Masses  .  207 


CONTENTS.  XV 

PAGE 

Overturning  a  body,  p.  207  ;  Work  done         .....  209 

Equilibrium,  stable,  unstable,  and  neutral       .                        ...  210 

Pendulum             .........  210 

Simple  Pendulum,  p.  210 ;   Harmonic  Motions,  p.  210 ;  Isochronous 
Oscillations,  p.  212  ;   Work  done,  p.  213  ;   Compound  Pendulum, 

p.  213 ;  Ballistic  Pendulum,  p.  214  ;  Bifilar  Suspension          .           .  215 


CHAPTER  IX. 

MATTER. 

THE  PROPERTIES  OF  MATTER      .            «            .            .            .            .  .216 

Essential  properties—  Quantity,   p.  216;   Quality  — Elements,  p.  217; 

Extension,  p.  218  ;  Impenetrability,  p.  218  ;  Indestructibility  .  .    219 
General  properties  —  Weight,  p.  219  ;  Divisibility,  p.  220  ;  Porosity        .    220 

Contingent  properties              .            .            .            .            .            .  .    220 

Density,  p.  220  ;  Specific  Densities,  p.  221 ;  Measurement  of     .  .    221 
THE  STATES  OF  MATTER,  p.  225  ;    Solid  and  Fluid,  p.  225  ;  Rigid  Solid, 

p.  225  ;   Soft  Solid,  p.  226  ;   Viscous  and  Mobile  Liquid,  p.  226  ; 

Viscosity    .           .  '                     .            .            .            .            .  .227 

Gas,  p.  229  ;  Vapour,  p.  231  ;  The  Critical  State,  p.  232  ;  Radiant 

Matter       .            .            .        ;V         .    !       .            .            .  .234 

The  Ether,  p.  234 ;  Vacuum                     ".    '•                    .            .  .    234 

Change  of  State      ^  .  •         ,           .           .  •         .            .            .  .235 

THE  CONSTITUTION  OF  MATTER             .            .-;           .            .             .  .    238 

Chemical  Views        .            ,            .            .            .            .            .  .     238 

Physical  Views,  p.  245  ;  Atoms,  p.  246  ;  Vortex- Atoms             .  .    246 

The  Kinetic  Theory     .           .           .            .            .           .           .  .247 

MOLECULAR  FORCES        .        ;;tJ-J            .            .•           .            .            .  .     253 

In  gases,  p.  253  ;  in  liquids,  p.  254  ;  in  solids      .            .            .  .255 


CHAPTER  X. 
SOLIDS. 

CONTINGENT  PROPERTIES  OF  SOLIDS    ......    257 

Cohesion,  p.  257  ;  Hardness  —  Softness,  p.  258  ;  Hardness  —  Fragility, 
p.  258 ;  Malleability,  p.  259 ;  Plasticity,  p.  259 ;  Resistance  to 
Deformation,  p.  259  ;  Cubical  Compressibility,  p.  259  ;  Shearability, 
p.  260  ;  Extensibility  —  Inextensibility,  p.  260  ;  Young's  Modulus, 
p.  261  ;  Linear  Compressibility,  p.  262  ;  Flexibility,  p.  262  ;  Tough- 
ness and  Brittleness,  p.  262  ;  Torsibility  .  .  .  .263 

Elasticity          .........    263 

Resistance  and  Power  of  Restitution,  p.  264 ;  RestHution-Pres- 
sure,  p.  264  ;  Coefficient  of  Restitution,  p.  264  ;  Perfect  and  Imper- 
fect Elasticity,  p.  264  ;  Deferred  Restitution,  p.  266  ;  Vibrations  due 
to  Elasticity,  p.  266  ;  Viscosity  of  Elastic  Solids,  p.  267  ;  Fatigue  of 
Elasticity,  p.  267  ;  Effect  of  Repeated  Variations  of  Stress,  p.  267  ; 
Velocity  of  Propagation  of  a  Displacement,  p.  267  ;  Physiological 
Examples  of  Elasticity,  p.  268 ;  Its  Mechanical  Advantages  .  .  268 

Strength  of  Structures  as  depending  on  their  form  .  .    269 


CONTENTS. 

CHAPTER  XI. 
LIQUIDS. 

1.  MOLECULAR  ACTIONS  .  .  .  .  .  .  .    271 

Cohesion,  p.  271 ;  Cohesive  Forces,  p.  271 :  Surface-Tension,  p.  272  ; 

its  measurement,  p.  275 ;  Capillarity    .                   ">,.  -        .            .  276 

Superficial  Viscosity ..278 

Cohesion  Figures     .                       ......  279 

Solubility  of  Solids  in  Liquids,  p.  279  ;  Dissociated  Molecules  in  Solu- 
tions, p.  280 ;  Ions,  p.  281  ;  Supersaturation,  p.  281 ;  Miscibility  of 

Liquids .            .            .  282 

Imbibition 282 

Diffusion,  p.  282  ;  Colloids  and  Crystalloids         ....  284 

Osmosis,  p.  285  ;  Osmotic  Pressure            .....  288 

2.  THE  STATICS  OF  LIQUID  MASSES     .            .            .            .            .            .  288 

Dilatancy      ......                                   .  288 

Pascal's  Principle,  p.  289  ;  Hydrostatic  Pressure,  p.  290  ;  Hydraulic 

Press 290 

Heavy  Liquids,  pressure  within,  p.  291  ;  Communicating  Vases           .  293 

Archimedes'  Principle,  p.  294  ;  Equilibrium        ....  294 

Measurement  of  Fluid  Pressure    .           .           .            .            .            .  295 

Superficial,  p.  295  ;  interior,  p.  296  ;  variable          .  .  .298 

3.  THE  KINETICS  OF  LIQUID  MASSES  ......  299 

Streams            ....                       ....  299 

Law  of  Continuity,  p.  300  ;  Forces  producing  flow,  p.  300  ;  Head  of 
Liquid,  p.  302  ;  Torricelli's  Law,  p.  302  ;  Velocity  of  Jet,  p.  303 ; 
Energy  of  Jet,  p.  303  ;   The  Vena  Contracta,  p.  304  ;  Ajutages, 
p.  305 ;  Recoil       .  .  .  .  .  .306 

Eesistances      .  .  .  .  .  .  .  .  .    306 

Surface  Adhesion,  p.  306  ;  Surface  Friction,  p.  306  ;  Eddies,  p.  306 ; 
Viscosity   .  .  .  :.  .  .  .  .    307 

Lateral  Diminution  of  Pressure         .  .  .  .  .  .309 

Flow  through  uniform  rigid  tubes,  p.  309  ;  Constant  flow,  p.  309  ;  Vari- 
able flow,  p.  312 ;  Interrupted  flow        .  .  .  .  .312 

Flow  through  bent  tubes         .  .  .  .  .  .313 

Flow  in  tubes  not  of  uniform  diameter         ....    313 

Flow  in  branched  rigid  tubes  .  .  .  .  .  .314 

Flow  through  capillary  tubes  .  .  ,.'..-  .  .    315 

Measurement  of  the  pressure  in  a  stream  ,  ,  .  .317 

Measurement  of  the  velocity  of  a  stream  .  .  .  .317 

Work  done  in  keeping  up  a  stream  .  .         _ . .        .  .  .319 

Streams  in  elastic  tubes       .  ,  .  ,  .    320 

Primary  waves  in  elastic  tubes,  p.  321 ;  The  form  of  a  simple  pulse- 
wave,  p.  322  ;  Secondary  waves  in  elastic  tubes,  p.  322 ;  The  form 
of  the  physiological  pulse-wave,  p.  322  ;  Reflected  waves  in  elastic 
tubes,  p.  322  ;  Amount  of  outflow  from  distensible  elastic  tubes  .  323 

CHAPTER    XII. 
GASES. 

PROPERTIES  OF  GASES  —  Density,  p.  324  ;  Elasticity,  p.   324 ;  Pressure 

Hydrostatic,  p.  325 ;  Compressibility    .  .  .  .    325 


CONTENTS. 


XVll 


PAGE 

Tendency  to  indefinite  expansion,  p.  326  ;  Air-pumps    .         *  .  .     326 

Absorption  of  gases  by  solids,  p.  327;  by  liquids  .  /  .     328 

Diffusion  of  Gases,  p.  330 ;  Effusion,  p.  330 ;  Transpiration,  p.  331  ; 
Membrane-Diffusion         .  .  .  .  .  .  .332 

Diffusion  of  Gases  from  Liquids    .  .   •        .   •         .  .  .     332 

THE  STATICS  OF  GASES  .   •         .   •        . .   •         .  .  .  .333 

STREAMS  OP  GAS  .....  .  .  .  .     334 

Recoil,  p.  334  ;  Viscosity,  p.  334  ;  Measurement  of  Flow          ....-        .    336 

THE  PRESSURE  OF  THE  ATMOSPHERE  .  .  ;          .  .  .  .     336 

General  propositions  .  .  .  .  .  ;-.  .    336 

Suction,  so-called,  p.  339 ;  examples         ...          .  .        ;    .  .    340 

Liquid  columns  supported  by  atmospheric  pressure,  p.  341 ;  Barometric 

Height .  .342 

Torricellian  Vacuum,  p.  342  ;  Suspended  Loops  .        >...'•        .    344 

Siphon,  p.  345  ;  Pump  and  force-pump,  p.  345  ;  Valves          ......         .     346 

Measurement  of  Atmospheric  Pressure    .  .  .  ,  .    347 

Variations  in  the  Barometric  Pressure      ..  '        .  .  .    .        .    348 

Standard  Atmospheric  Pressure     .  .  .  .  .  .     349 

Gases  passed  into  the  Torricellian  Vacuum         ....    349 


CHAPTER  Xin. 

HEAT. 

HEAT  A  FORM  OF  ENERGY        .            .            .            .            .           .  .    350 

First  Law  of  Thermodynamics      .            .            .            .            .  .353 

the  lowest  form  of  Energy,  p.  353  ;  Conclusions  .            .  .     354 
Change  of  state  a  cause  of  absorption  or  liberation  of  Energy  in  the 

form  of  Heat        .            .            .            .            .            .            .  .355 

Change  of  state  effected  adiabatically      .           .            .            .  .358 

EFFECTS  OF  HEAT           .            .            .             .             .             .            .  .     359 

Internal  and  External  Work,  p.  359  ;  Latent  Heat         .            .  .361 

Increase  of  kinetic  energy  of  molecules   .            .            .            .  .361 

The  Kadiometer,  p.  361  ;  The  Spheroidal  State        .            .  .     363 

Increase  of  Temperature     .            .            .            .            .            .  .    364 

Absolute  Zero   .            .    '                    .            .            .            .  .364 

Specific  Heat  and  Thermal  Capacity  .            .            .            .  .365 

Atomic  and  molecular  heat ......    366 

Thermal  Capacity  at  constant  Volume       .            .            .  .    367 

Thermal  Capacity  under  constant  Pressure"          .            .  .    367 

The  ratio  of  the  two  specific  heats,  p.  368  ;  Differences  therein  .    369 

Work  done  in  heating  a  gas           '.            .            .            .  .    369 

Variations  in  specific  heat    .            .                        .            .  .371 

The  six  thermal  capacities  .                       .            .            .  .    371 

Internal  work       .  '•  .    *        .            .            .    '        .            .            .  .    372 

Expansion       -•'.'-..           .    '        .            .            .            .  .    373 

Proof  that  there  is  no  perfect  gas  (Joule's  experiments,  etc.)  .    374 

Van  der  Waals'  Law             .            .            .            .  .    375 

The  Latent  Heat  of  Expansion       .            .            .            .  .377 

The  Coefficient  of  Expansion,  p.  378  ;  Linear,  p.  378  ;  Cubical  .    379 
Examples  of  expansion  by  heat      .....    379 


CONTENTS. 

PAGE 

Applications,  p.  380 ;  Measurement        .           »  :        >           .  380 
Fusion    .            .            .          ,  .            •            .            »        '- '.            .  384 
Prof.  James  Thomson's  proposition  as  to  melting  ice   .            .  384 
Kegelation,  p.  385 ;  Freezing  Point  and  Molecules  in  Solu- 
tion     .            .            .            .  •         .         .'.            -.            .  386 

Sublimation       .            .                        ...           y  .        .  386 

Boiling  and  Evaporation          .            .•           .            .            .            .  386 

The  boiling  point  at  different  pressures  .            ...            .  387 

Vapour-Pressure  of  a  Solution           .            .            .            .            .  387 

Saturation  pressure           .            .            .            .            .            .  390 

Measurement  of  Vapour  Density            ....  391 

Dewpoint,  p.  392  ;  Hygrometer,  p.  392  ;  Dew    .  .  .393 

TRANSFORMATIONS  OF  HEAT      .            .                                                  .            .  393 

Work  into  Heat       .            .            .                        .                                    .  393 

Heat  into  Work        .  .  .  .  .  .  .393 

The  nature  of  a  Cycle,  p.  394  ;  Adiabatic  Equation,   p.  395 ; 
Entropy         .  .  .  .  .  .  .395 

Carnot's  Cycle,  p.  395 ;  its  steps         .            .            .            .            .  395 

The  Reversibility  of  Carnot's  Cycle  .                       ...  397 

Carnot's  principle         .......  397 

Carnot's  function          .......  397 

Efficiency  of  Carnot's  reversible  engine  a  maximum           .            .  398 

Second  Law  of  Thermodynamics       .....  398 

Its  various  forms  .            .            .            .            .            .            .  398 

Degradation  of  Energy            ......  399 

MEASUREMENT  OF  HEAT  .  .  .  .  .  .  .    399 

Temperature  :  methods  of  measuring  it   .  .  .  .  .    399 

Air  thermometer  .  .  .  .  .  .  .    401 

Mercurial  thermometer,   p.   401  ;    its  construction,  p.  401  ;    its 
graduation,  p.  402  ;    its  sensitiveness,  p.  402  ;  testing,  p.  403  ; 
different  forms  .  ...  .  .  .    403 

Pyrometry          .  .  .  .   -.        ...  .  .404 

Calorimetry  .  .  .  .  .  .  .  .    404 

The  Method  of    Mixtures,   p.  404;   Water-Equivalent,   p.  405; 
Latent-Heat  Methods  .  .  .  .  .  .406 

TRANSFERENCE  OF  HEAT  .  ..  .  .  '-.  .     406 

Conduction   .  .  .  ....  .  .    406 

Conductivity,  p.  406  ;  Dynamical,  Calorimetric,  and  Thermometric 
Coefficients  of  Conductivity  .  .  .  .  ^         .407 

Steady  Flow  of  Heat,  p.  408  ;  in  bars  .  .  . '          .408 

Flow  of  Heat  and  flow  of  Temperature,  p.  409 ;  Waves  of  Tem- 
perature         .  .  .  .  .  .  .    409 

Relative  conductivities  .  .  .  .  .  .    409 

Convection  currents  .          •  .    •        ,  .  .  .  .    410 

Radiation      .  .  .  •         V  ...  .  '         .411 

Transport  of  Heat   .  .  .  .  .  .411 

CHAPTER  XIV. 

SOUND. 

NATURE  OF  SOUND          .  .  .  .  »  *  .  .    412 

Sound,  p.  412  ;  Sounding  bodies,  p.  412  ;  Sound-waves  .  .  ;.  413 


CONTENTS. 


XIX 


PAGE 

Characteristics  of  Sounds :  Pitch,  Loudness,  Quality      .  .  .    414 

Noise 417 

Pitch,  p.  418  ;  Physical  Determination,  p.  418  ;  Musical  Specification  .    420 
Musical  Intervals,  p.  422  ;  Transition  .  .  .  .423 

Loudness       .  .  .  .  .  .  .  .    425 

Quality  of  Sound     .  ....     428 

Analysis  of  Sound,  p.  429 ;  Resonators         .  .  .  .    430 

Synthesis  of  Sound       .  .  .  .  .  .  .432 

Complex  Sound- Waves,  p.  433  ;  The  Phonograph     .  .  .433 

LAWS  OF  VIBRATION  OF  SOUNDING  BODIES    .....     434 

Transverse  Vibrations  of   Strings,  p.  434  ;   The  Monochord,  p.  437  ; 
Experiments  with,  p.  437  ;  Melde's  Experiments,  p.  439  ;  Longitu- 
dinal Vibrations  of   Strings,  p.  440 ;  of  Rods,  p.  441  ;   Transverse 
Vibrations  of  Rods,  p.  441  ;  Torsional  Vibrations  of  Rods,  p.  443.; 
Vibration   of   Discs  or  Plates,  p.  443;   of  Membranes,  p.  444;  of 
Bells,  p.  444 ;  Effect  of  Loading  .    :  .         <.,  .    445 

Free  Vibrations        .  ...  .  .  .  .  .    445 

Resonance     .  ,  .  .  .  :          .  .  .  ..    446 

Forced  Vibrations   ........    447 

MUSICAL  INSTRUMENTS   .  .  .  .-•''.  .  .  .     449 

Singing  and  Sensitive  Flames,  p.  454  ;  Trevelyan's  Rocker,  p.  455 ; 
Radiophony  ....  .  .  .  .  .  .  455 

PROPAGATION  OF  SOUND  .  .  .  .  .  .  .     456 

In  Solids,  p.  456  ;  in  Liquids,  p.  457  ;  in  Gases,  p.  457  ;  Beats,  p.  458  ; 
Diffraction,  p.  458  ;  Reflexion,  p.  459 ;  Refraction,  p.  461  ;  Inter- 
ference, p.  461 ;  Velocity  of  Sound,  p.  461 ;  Propagation  according 
to  the  Kinetic  Theory  of  Gases,  p.  464  ;  Doppler's  principle  .  .  465 

THE  HUMAN  EAR  .    •     •  .  .  .  .  .  .  .465 

HARMONY  AND  DISSONANCE        .  .  .  .  .  .     471 

Differential  Tones,  p.  473 ;  Summational  Tones  .  .  .  .474 

VOICE  —  VOWELS         •    .-'.••          .  .  .  .  .  .     475 

TRANSFORMATIONS  of  the  Energy  of  Sound     .  .  .  .  .    476 


CHAPTER  XV. 

OF   ETHER-WAVES. 

PRELIMINARY       .            ,            .            .            ,            .            .            .            .  478 

NATURE  OF  RADIATION  .  .-  -  .  .  .  .  "  .  .  .  478 

Limits  of  Frequency,  p.  480  ;  Velocity  and  Wave-lengths  .  .  480 

Kinds  of  Radiation  so-called  :  Heat,  Light,  Actinic  "  Rays"  .  .  480 
Colour,  p.  483  ;  White  Light,  p.  484  ;  The  Spectrum  .  .  .486 

Compound  Coloured  Light,  p.  486  ;  Complementary  Colours  .  .  487 

RADIATIONS  OF  A  HOT  BODY    .            .            .....  488 

Exchange  of  Radiations,  p.  489;  Prevost's  Law,  p.  491  ;  Stokes's  Law  .  492 

Spectrum  Analysis,  p.  494 ;  Linear  Spectrum,  p.  495 ;  Band  Speot-f  um  .  496 

TRANSMISSION,  REFLEXION,  AND  ABSORPTION  .  .  .  .  497 

Absorption-bands,  p.  499 ;  Colour  -.  501 

Blue  colour  of  opalescent  bodies  ......  503 


xx  CONTENTS. 

PAGE 

FLUORESCENCE,  PHOSPHORESCENCE,  CALORESCENCE  .            .   '      V.  -         .  504 

SOURCES  OF  ETHER- WAVES       .            ..,"...         .            .            .            .  506 

Vibrations  of  Molecules,  p.  506  ;  communicated  to  the  Ether  .            .  507 

PROPAGATION  OF  WAVES  THROUGH  THE  ETHER         .    •        .    -        .            .  508 

Ether- Vibrations  transverse            ......  509 

The  Velocity  of  propagation,  p.  510  ;  Methods  of  measurement           .  512 

Intensity  at  a  place  .            .                        .    -        .                        .    "        .  513 

Direction  of  Propagation     .......  514 

Plane-polarised  Light,  p.  514  ;  Circularly-,  Elliptically-polarised  Light, 

p.  514  ;  Common  Light    .             .            .            .            ...  515 

Polariser,  p.  516 ;  Partially-polarised  Light          .  .          .  .  .516 

Rotatory  Polarisation                       ...            .            .            .            .  517 

KEFLEXION  AND  REFRACTION   .            .            .            .             ...            .  517 

Fresnel's  Laws :  for  light  whose  vibrations  are  at  right  angles  to  the 
plane  of  incidence,  p.  517  ;  parallel  to  that  plane,  p.  519  ;  Mixed 

Light,  p.  520  ;  Neumann  and  MacCullagh  .  .  .  .  521 

Plane  of  Polarisation,  p.  521 ;  Modification  of  Character  of  Light  by 

Reflexion  and  Refraction  .  •  .  .  .  .  .  522 

Mirrors         .            .            .            .            .            .            .            ...  523 

Prisms,  p.  528  ;  Monochromatic  Light,  p.  528  ;  Mixed  Light,  p.  530  ; 
Spectrum,  p.  530  ;  Deviation  without  Dispersion,  p.  531  ;  Dispersion 

without  Deviation,  p.  532  ;  Abnormal  Dispersion  f  ._  .  532 

Lenses,  p.  533  ;  Gauss's  Lens-Method,  p.  539  ;  Chromatic  Aberration  .  541 

INTERFERENCE     .            .            .            .             .           ;.            .            .            .  542 

Bent  Mirror,  p.  542  ;  Biprism,  p.  542  ;  Fringes,  p.  543  ;  Measurement 

,  of  wave-length     .            .            .         „            .        •  -  .    '                     .  544 

Colours  of  thin  films           .......  545 

Iridescence   .........  547 

Shadow,  p.  547  ;  Camera  Obscura             ..."       .            .           .           .  548 

Diffraction,  p.  548  ;  Diffraction-grating    .....  549 

Twinkling  of  Stars  .  .  .  ....  .  .550 

DOUBLE  REFRACTION      .            .            .          ••".•            .        .'   ;   .        '-.'••         .  551 

Uniaxial  Crystals,  p.  551  ;  Nicol's  Prism  .....  555 

Binaxial  Crystals,  p.  556  ;  Conical  Refraction     .            .            .            .  557 

Interposed  Lamina  between  two  prisms            f    .  •          .            .            .  559 

Circularly-  and  Elliptically-polarised  Light          ....  560 

Determination  of  the  character  of  a  Beam  of  Light       .            ...   '        .  563 

Colours  produced  by  interposed  film         .            .            .            ,            .  563 

ROTATORY  POLARISATION            .            .            .            .    ,       v           ..            .  566 
SoleiPs  Saccharimeter         .            .        ,    .        '  \            .            .            .568 

TRANSFORMATIONS  of  the  Energy  of  Ether- Waves      .            .            .            .  569 

OPTICAL  INSTRUMENTS    .            .            .            .            ....            .  570 

The  Eye,  p.  570  ;  The  Microscope,  p.  571  ;  The  Telescope,  p.  572  ; 
The  Opera  Glass,  p.  572 ;  The  Ophthalmoscope  .  .  .572 

VISUAL  PERCEPTION        .            .-          .            .            .            .            .            .  573 

Perception  of  Colour,  p.  573  ;  Mixture  of  Colours,  p.  574 ;   Comple- 
mentary Colours,  p.  574  ;  Primary  Colours      .           -.   '        .  ,         .  575 
Perception  of  Form,  p.  576  ;  Lustre,  p.  576  ;  Corresponding  Points     .  576 


CONTENTS.  xxi 

CHAPTER  XYI. 
ELECTRICITY  AND  MAGNETISM. 

PAGE 

GENERAL  PHENOMENA  of  Electricity  may  be  explained  as  those  of  Ether- 
stress         .            .            .            .            .            .            .          ,  .'           .  577 

Quantity  of  Electricity,  p.  578  ;  Unit  Quantity,  p.  578 ;  Vitreous  and 

Resinous,  p.  578  ;  their  mutual  action  .            .            .        '.'..'.           .  578 

Density          .        '    V        "-.'          .            .            .            .         ".            .  579 

Complementary  Distribution          .           ,.  '          .            .            .            .  580 

Imaginary  Electric  Matter .            .            .            .            .          '.*'/.  581 

Electric  Force,  p.  581  ;  Field  of  Force     .  .  .  .  .582 

Potential :  Absolute  Electric  Potential,  p.  584  ;  Difference  of  Potential     .  584 
Potential-Gradient   .            .            .            .            .            .            .            .585 

Travelling  of  the  electric  condition  of  a  body,  p.  585  ;  Current            .  585 

Displacement-Current     •    .            .            .            .            .            .            .  586 

Electromotive  difference  of  potential,  Voltage     ....  587 

Potential  of  Earth,  p.  588  ;  The  Potential  of  a  body            .            .  588 

Conductors  and  Non-  Conductors :  Dielectric.            ....  588 

Insulation,  p.  589  ;  Kinds  of  Conductors              ....  690 

Electrostatic  and  Electrokinetic-          '..,'"          .            .            .            .  591 

"Free"  and  "Bound"  Charges  ..         . '' •        .  .  .  .591 

Division  of  Charge        ...            ...            .            .            .            .  592 

Capacity,  p.  592  ;  Work  done  in  charging  a  conductor  .            .            .  593 

Electrostatic  Induction    .         <.»         :  .            .            .            .            .            .  594 

Lines  of  Force  and  Lines  of  Induction     , .  ~;                     .            .            .  595 
Condenser,  p.  596  ;  Specific  Inductive  Capacity,  p.  597  ;  Induction  in 
other  Media  than  Air,  p.  598  ;  Ley  den  Jar,  p.  599 ;   Ley  den  Bat- 
teries, p.  600 ;  Coefficients  of  Mutual  Induction,  p.  600  ;  Effects  of 

Induction,  p.  601  ;  Electric  Screens       .....  601 

Thermal  and  Fluid  Analogies          .  .  .  .  .602 

Electric  Stress  across  Dielectric,  p.  602  ;  Energy  therein      .  .  .603 

Dimensions  of  Electrostatic  Measures              .....  603 

Relations  of  Electrostatic  Quantities    .            .            .            .            .            .  604 

i 

OBSERVATION  OF  DIFFERENCES  OF  POTENTIAL            .             .                         .  604 
Gold-leaf  Electroscope,  p.  604 ;  Peltier's  Electroscope,  p.  605  ;  Bohnen- 
berger's,  p.   605  ;  Lord  Kelvin's  Quadrant  Electrometer,  p.  606  ; 

Coulomb's  Torsion  Balance       ".        t    .  ^        .            .            .            .  607 

Measurement  of  Difference  of  Potential  .....  608 

PRODUCTION  OF  DIFFERENCE  OF  POTENTIAL  .....  609 
Contact,  p.  609 ;  of  non-aonductors,  p.  610 ;  Frictional  machines, 

p.  610  ;  Contact  of  Metals,  p.  611 ;  Open  and  Closed  Circuit             .  613 

Chemical  Action      .            .            .  .          .  .         .            .            .            .  614 

One-fluid  cells  and  batteries,  p.  618  ;  Two-fluid,  p.  620  ;  others  .  622 
Friction  of  water  against  steam  or  air,  p.  623 ;  Evaporation,  p.  623 ; 

Pressure,  p.  623  ;  Heat,  p.  624  ;  Electro-capillarity  .  .  .  624 
Thermo-electricity,  p.  624  ;  Thermo-electric  Diagram,  p.  626  ;  Neutral 

point,   p.    626 ;    "  Gaugain's    Curves,"   p.    627 ;    Temperature    of 

Reversal     .            .            .            .            .            .            .                        .  627 

Atmospheric  Electricity       .  .  .  .  .  .  .629 

The  Electrophorus,  p.  630  ;  Lord  Kelvin's  Replenisher,  p.  631 ;  his 

Water-Gravity  Electric  Machine             .....  631 


CONTENTS. 

PAGE 

STEADY  ELECTRICAL  CURRENTS            .                        .                        .  632 

Discharge  of  a  Condenser,  p.  632  ;  of  a  Cell        .            .            .            .  632 

Current-intensity  and  Resistance  .                                    ...  633 

Current-intensity,  p.  633;  Resistance,  p.  633;   Current-Density, 
p.  633  ;   Ohm's  Law,  p.  633  ;   Conductance,  p.  634 ;   Conduc- 
tivity, p.  634;   Resistivity,  p.  634;  Table,  p.   635;   Value   of 
Ohm,  p.  634 ;  Variable  Conductivity,  p.  636 ;  Reduced  Resist- 
ance,   p.    637  ;    Steady    Current    equal    throughout,    p.    637  ; 

Dimensions    ....                        ,  638 

Falls  of  Potential  and  Resistances  in  Conductors            .            .            .  638 

Flow  through  large  Conductors      .            .            .            .                        .  643 

Simultaneous  Currents      .  .  '..'".  '.        .            .            .            .  .         .  644 

Derived  Currents     .  .  .  .644 

Kirchhoff's  Laws,  p.  644  ;  Shunts,  p.  645  ;  Wheatstone's  Bridge  .  645 

Measurement  of  the  constants  of  a  battery    ....  646 

The  Energy  of  a  Steady  Current  ...            .            .            .            .  647 

Energy  in  a  Condenser,  p.  648  ;  stored  in  the  Ether       .            .            .  648 

Transmission  of  Energy  during  a  Steady  Current          .  .         .            .  648 

EFFECTS  OF  A  STEADY  CURRENT         ...  .          .           .,            .             .            .  649 

Production  of  Heat,  p.  649 ;  Peltier's  effect,  649 ;  Thomson's  effect, 
p.  650 ;  Derived  Currents  .  .         .  .  .  . '          .653 

Production  of  Light            .            .            .            .            .:.'-.  653 

Geissler's  Tubes,  p.  656 ;  Electrification  of  Radiant  Matter  .  656 
Electrolysis,  p.  657  :  Ions,  p.  658 ;  Faraday's  Laws  of  Electrolysis, 
p.  659  ;  Electrochemical  Equivalents,  p.  661  ;  Calculation  of  E.M.D.P. 
of  a  Cell,  p.  661 ;  Electrolysis  in  Gases,  p.  662  ;  Nobili's  and  Gueb- 
hard's  Equipotential  Surfaces,  p.  663 ;  Polarisation  of  Electrodes, 
p.  664  ;  Secondary  Cells  and  Batteries,  p.  665 ;  Electrical  Storage 

of  Energy,  p.  666  ;  Equalisation  of  a  Current  .            .            »    •       '.  667 

THE  DYNAMICAL  PROPERTIES  OF  A  STEADY  CURRENT          .            .    ,        .  667 
Electromagnetic  Field  of  Force      .            .            .        . .- „          ...            .  667 
Lines  of  Force,  p.  667 ;  their  Direction          ..-_,            .          '.            .  668 
Mutual  action  of  Currents,  p.  669  ;  Ampere's;  formula,  p.  670  ;  Differ- 
ences between  Electromagnetic  and  Electrostatic  Field  of  Force  .  671 
Field  of  Force  round  a  Closed  Circuit    .            .            .            .            .672 
Solenoidal  system  of  currents,  p.  673  ;  the  surrounding  Field  of  Force  673 

MAGNETISM  :  General  Phenomena,  p.  674  ;  Terrestrial  Magnetism  .            .  678 

Equivalence  of  Shell  and  Circuit  .            .            .            .            .            .  682 

Magnetic  Induction,  p.  684  ;  Permeability,  p.  685  ;  Diamagnets            .  687 

Electromagnets,  p.  689  ;  Hysteresis,  p.  690  ;  Astatic  Arrangements      .  691 

Magnetic  Circuit      ........  691 

Nature  of  Magnetism          .  ,         .            .            .            .            .            .  692 

Dimensions        .  j ':       .                                                  ...  693 

Magnetic  Rotatory  Polarisation  of  Light,  p.  694  ;  Hall's  Experiment, 

p.  694 ;  Kerr's  Experiment          .            .            .         '    .            .            .  695 

THE  VARIABLE  PERIOD.             ....        "'/           .            .  695 

Arrival- Curve,  p.  697 ;  Deep-sea  Cables  .           .                   .    ../         .  699 

»        State  of  the  Field  during  this  period          .            /",       .»        ',            .  699 

ELECTROMAGNETIC  CURRENT-INDUCTION           .                         .            »"           .  699 

Phenomena  during  the  Variable  Period   .                       ,          -V  <        .  699 


CONTENTS.  xxiii 

PAGE 

Equivalence  of  Circuits  and  Magnets        .  .  .  .  700 

Secondary  Currents,  p.  700 ;  Lenz's  Law .  .  .  .  701 

Mutual  Attraction  and  Eepulsion  of  Currents      ....  702 

Amount  of  Induction  .......  703 

Self -Induction,  p.  704  ;  Coefficient  of  Mutual  Induction,  p.  704  ;  do.  of 

Self-induction,  p.  704  ;  Extra-Currents,  p.  705  ;  Measurements        .  706 

Induction  Coils        .  .*  .  .  .  .          .  .    '        .  706 

MAGNETIC  OR  ELECTROMAGNETIC  MEASURE    .            .             .            .            .  707 

Electromagnetic  Unit  of  Current-Intensity  or  Strength  .          '-»...          .  707 

Measurement  of  V,  p.  708  ;  V  considered,  p.  709  ;  Dimensions            .  709 
Practical  Units,  p.  711 ;  Table      .            .            .            .        ,.           .714 

Measurement  of  Current- Intensity  or  Strength     .            .           .           .  712 

Galvanometers,  p.  712  ;  Galvanometer-Constant,  p.  713  ;  Ballistic 

Galvanometers,  p.  713  ;  Dead-Beat  — ,  p.  716  j  Differential  —  .  716 
Electrodynamometers,   p.    716 ;    Ammeters,   p.   717 ;    Langley's 

Bolometer      .            .            ...            .            .            .  717 

Magnetic  Measurement  of  Resistance,  p.  718  ;  The  Standard  Ohm      .  718 

Measurement  of  Capacity    .            .            .            .            .            .            .  718 

OSCILLATING  OR  ALTERNATING  CURRENTS      .....     721 
Their  properties,  p.  721 ;  Tesla's  Experiments,  p.  723  ;  Transformers  .    724 

PRODUCTION  or  ALTERNATING  CURRENTS        .            .            .  .  .  725 

Ley  den- jar  Methods             .            .            . '"]       .            .  .  .  725 

Dynamo  Methods     .            *           ...            .  .  .  .727 

Dynamos  in  general     .            .            .  -        .            .  .  .  727 

Alternators,  p.  729 ;  multipolar,  p.  729 ;  multiphase  .  .  730 

Armature,  p.  730 ;  Commutator      .....  730 

Direct-Current  Dynamos      .            .            .            .  .  .  731 

ELECTRIC  TRANSMISSION  OP  ENERGY  TO  A  DISTANCE           .            .            .     733 
Telegraphy 733 

Signalling  by  Alternating  Currents,  p.  735  ;  Phonophore,  p.  735  ;  Har- 
monic Telegraph,  p.  736  ;  Telephone     .....     736 

Electromotors,  p.  737  ;  Efficiency,  p.  738  ;  Voltage          .  .  .739 

Alternating-Current  Motors        ......     740 

OSCILLATORY  ELECTROMAGNETIC  DISTURBANCES  IN  FREE  ETHER   .  .     741 

Herz's  Experiments,  p.  741  ;  Character  of  Ether- Waves  produced        .     742 
Maxwell's  Theory  of  Light .          • .  .  .  .  .  .744 

THE  ETHER          .  .  .  '.  .  .  .  .  .745 

Maxwell"1  s  Theory,  p.  745 ;  Question  of  the  Inertia  of  Ether,  p.  745 ; 
Relation  of  the  Ether  to  Matter  ..^        .  .  .  .746 

APPENDIX  .  .  *         v  ..  v^        .  747 

BIBLIOGRAPHY     .  .  .        -  '.'  ».,,.-        .  .  .  .    751 

INDEX  ,    757 


INDEX   OF   SYMBOLS. 


A  Area. 

B  Varying  coefficient  of  Kinetic  Fric- 
tion. 

C  Electrostatic  Capacity  or  Permit- 
tance. 

D    Conductance,  in  electrostatic  units. 

E'  Potential-Difference  between  two 
points,  in  electrostatic  units. 

F    A  Force  or  Stress,  generally. 

G  The  Weight  of  a  Body,  Gravita- 
tional Attraction. 

H  Heat  (in  ergs)  :  Driving  Head  of  a 
Liquid  or  Gas ;  H,,  and  Hp,  p.  310. 

I  Current-Intensity  or  Strength,  in 
electrostatic  measure. 

K  Specific  Inductive  Capacity,  or  Per- 
mittivity. 

L  Coefficient  of  Self-Induction,  or 
Inductance ;  Latent  Heat  of 
Expansion,  Coefficient  of. 

M  Coefficient  of  Mutual  Magnetic  In- 
duction ;  Moment  of  a  Couple. 

N   Moment  of  Inertia. 

P    A  Total  Pressure,  generally. 

Q  Quantity  of  Electricity,  in  electro- 
static measure. 

R  Resistance  of  a  Conductor,  in  elec- 
trostatic measure. 

T  Period  of  S.H.M.  or  of  a  complete 
oscillation. 

U  Resistance  to  Fluid  Flow,  in  cm. 
of  liquid  column. 

V   Electric  Potential,  in  e.-s.  units. 

W  Work :  Energy. 

A  Number  of  Amperes. 

E  Linear  Extension  on  Stretching. 

F  Statical  Frictional  Resistance. 

Gr  Temperature-Gradient. 

H  Height  of  Barometer-Column. 

L  Coefficient  of  Linear  Expansion  by 
Heat. 

jR  Kinetical  Frictional  Resistance. 

S   Energy  of  Surface-Film  per  sq.  cm. 

T  Superficial  Tension  of  Surface- 
Film,  per  linear  centimetre. 

V  Number  of  Volts. 

B   Total  Electromagnetic  Induction. 
F  Total    Force,   in    some   particular 
direction. 


H  Total  Magnetic  Force,  across   an 

Area. 
I    Total  Electrostatic  Induction  across 

an  Area. 
P   Total  Pressure,  directed,  across  an 

Area. 
T   Total  Tension,   directed,  across  a 

cross- section. 
V  The    Electromagnetic-Electrostatic 

Ratio  ;  the  Velocity  of  Light  etc. 

31    Intensity  of  Magnetisation. 

i!H  Magnetic  Moment. 

3&  Thermodynamic  Constant,  =k— c, 

a    Amplitude  in  Periodic  Motion. 

b    Coefficient  of  Kinetic  Friction. 

c  Thermal  Capacity  at  constant  Vol- 
ume, in  ergs  per  gramme. 

ca  Calories  (gramme-calories). 

d    Distance  :  Diameter  of  Tube. 

e  Potential-Difference  in  magnetic 
measure  ;  Epoch  in  S.H.M. 

/  Force  or  Stress  per  sq.  cm.,  in 
general ;  Focal  Distance. 

g  Acceleration  due  to  Gravitation : 
Intensity  of  Gravity. 

h  Vertical  Height ;  Height  of  Liquid 
Column. 

i  Current-Intensity  or  Strength  in 
magnetic  measure. 

k  Thermal  Capacity  at  constant  Pres- 
sure, in  ergs  per  gramme ;  Co- 
efficient of  Transpiration ;  Pois- 
euille's  Coefficient;  generally,  a 
Coefficient. 

I     A  Length. 

m  Mass :  Coefficient  of  Inertia. 

m   Average  Mass  of  a  Molecule. 

n  Frequency  :  Number  of  Turns  in  a 
Coil. 

o    Area  of  Cross-Section. 

p  Pressure  per  sq.  cm.,  general;  or 
Hydrostatic  Pressure  per  sq.  cm. 

po  "Electric  Tension"  per  sq.  cm. 

p  Small  increment  of  Pressure,  dynes 
per  sq.  cm. 

q  Quantity  of  Electricity,  in  mag- 
netic units. 

r  Radius  :  Resistance  of  a  Conductor 
in  magnetic  units. 


INDEX  OF   SYMBOLS. 


XXV 


s  Distance  traversed. 

t  Time  :  Temperature  in  °C. 

v  Velocity  or  Speed  in  general. 

w  Activity,  ergs  per  second. 

c  Curvature. 

k  Viscosity  log.  dec. 

n  Coefficient  in  S.H.M. 


Magnetic  Induction  per  sq.  cm. 
Force  in  a  stated  direction,  per  sq. 

cm.  of  Area. 

Magnetic  Field-Intensity,  per  sq.  cm. 
Electrostatic  Induction  per  sq.  cm. 
Pressure  per  sq.  cm.,  in  a  stated 

direction. 
Displacement  in  a  given  straight 

line. 

Tension  per  sq.  cm.  of  cross-section. 
Displacement  along  axis  of  x. 
Velocity  in  a  given  straight  line. 
Displacement  along  axis  of  y. 


Electric  Force  in  magnetic  units. 
Horizontal  component  of  Earth's 

Magnetic  Field  (per  sq.  cm.). 
Coefficient  of  Resistance  to  Com- 

pression. 

1     Length  of  the  Simple  Pendulum. 
tn   Quantity  of  Magnetism. 
n    Coefficient  of  Rigidity  to  Shear. 
r     Impedance. 

t     Coefficient  of  Rigidity  to  Twist. 
ij    Volume. 

ft     Small  increment  of  Volume. 
g    Coefficient  of  Rigidity  to  Stretching 
(Young's  Modulus). 

D    Conductivity. 

N    No.  of  grooves  per  cm.  in  Diffrac- 

tion-grating. 
R    Resistivity. 

n    Atmospheric  Pressure,  per  sq.  cm. 
\>    Thermometric  Coefficient  of  Ther- 
mal Conductivity. 

a    Acceleration  in  general. 

/3    Index   of   Refraction;    (3^,  do.  do. 

when  wave-length  X  =  <x>. 
7    Gravitation-Constant. 

5  An  Angle  of  Deviation. 
5x  A  small  increment  of  x. 
e     2-718281... 

6  Angular   Displacement:    Angle  of 

Shear  ;  Critical  Temperature. 
#    Calorimetric  Coefficient  of  Thermal 
Conductivity. 


17     Coefficient  of  Viscosity. 

i  Radius  of  Inertia:  Angle  of  Inci- 
dence. 

to    Rad.  of  Inertia  round  Cent.  Gravity. 

£     An  Angle. 

K     Magnetic  Susceptibility. 

X  Wave-Length :  Latitude :  Coeffi- 
cient of  Restitution :  Latent 
Heat  of  Evaporation. 

H  Magnetic  Permeability  :  Coefficient 
of  Statical  Friction. 

v    Velocity  of  Wave-Propagation. 

TT    3-1416... 

w    Critical  Pressure. 

p  Density  of  a  Mass :  Angle  of  Re- 
flexion. 

g     Angle  of  Refraction. 

<r  Electric  Surf  ace- Density  :  Specific 
Heat  in  ca  per  gramme. 

s     Magnetic  Surf  ace- Density. 

T    Temperature  on  the  Absolute  scale. 

f     Small  increment  of  Temperature. 

Cp  Strength  of  Magnetic  Shell:  En- 
tropy. 

0    Critical  Volume. 

X    Critical  Angle  in  Statical  Friction. 

•fy    Critical  Angle  in  Kinetical  Friction. 

w  Angular  Velocity  :  a  Solid  Angle  : 
=  "0hms." 

a    Acceleration  in  a  given  Direction. 

<r  Mass  per  unit  surface  of  a  Film  or 
Shell. 

4>  Electric  Force  (per  sq.  cm. ^Poten- 
tial-Slope or  Gradient. 

T  Galvanometer- Constant. 

A  Current-Density. 

0  Dynamical  Coefficient  of  Thermal 

Conductivity. 

A  Coefficient  of  Extensibility. 

II  Total  Atmospheric  Pressure. 

S  "Sum  of  all  the  £'s"=S(a;). 

li  Magnetic  (scalar)  Potential. 

/    "divided  by." 

oo    "Infinity." 

oc  "varies  as;"  i.e.,  "=some  con- 
stant x ." 

~  "Numerical  difference  between:" 
always  positive. 

109    -  1000,000000  ;  nine  ciphers. 

10-9  =  1  --  109  =  0-000,OCfO,001  :  eight 
ciphers  after  the  point. 

a*  =  V«;  a*=Vo»;  o~*=l/a*. 

at  "  Value  of  a  at  end  of  time  t." 


UII7BRSITT 


INTRODUCTORY. 

NATURAL  PHILOSOPHY  or  PHYSICS  may  be  briefly  defined  as  the 
Science  of  Matter  and  Energy.  This  definition  is  one  which  is 
obviously  comprehensive  enough  to  include  within  its  range  the 
whole  of  Chemistry  and  of  Biology  as  well  as  of  Chemical  and 
Physiological  Physics.  Chemistry  is  in  truth  but  a  colony  of 
facts  closely  related  to  one  another,  and  classified  by  us  on 
principles  which  depend  almost  entirely  upon  our  ignorance  of 
the  fundamental  nature  of  the  relation  between  those  apparently 
different  Forms  of  Matter  which  we  know  as  the  various  Chemi- 
cal Elements ;  and  the  consummation  of  Chemistry,  a  full  and 
accurate  knowledge  of  the  inner  mechanism  of  all  chemical 
reactions,  would  probably  result  in  the  absorption  of  all  Chemistry 
in  the  wider  science  of  Molecular  Physics.  In  the  meantime 
the  fundamental  unity  of  the  two  nominally  distinct  sciences, 
Chemistry  and  Physics,  is  shown  by  the  extent  to  which  they 
overlap  one  another  in  the  field  of  Chemical  Physics. 

Physiology,  again,  or  in  a  wider  sense  Biology,  is  con- 
cerned with  the  matter  and  the  energy  of  living  beings  ;  and  if  it 
ever  come  to  attain  its  highest  ideal,  even  Biology  must  thereupon 
necessarily  merge  in  Natural  Philosophy.  Already  we  see  that 
while  physiological  research  is  steadily  conquering  the  unknown, 
that  which  it  succeeds  in  thoroughly  explaining  falls  out  of  its 
grasp  and  comes  to  form  a  part  of  ordinary  physical  or,  it  may 
in  the  meantime  be,  of  ordinary  chemical  knowledge. 

We  may  more  amply  define  Natural  Philosophy  or  Physics 
as  the  systematic  exposition  of  the  Phenomena  and  Properties  of 
Matter  and  Energy,  in  so  far  as  these  phenomena  and  properties 
can  be  stated  in  terms  of  definite  Measurement  and  summarised 
by  the  formulation  of  mechanical  principles  or  Laws. 

Here,  again,  we  must  admit  that  our  definition  is,  in  the 
present  state  of  our  knowledge,  too  ideal.  A  perfect  and  accu- 

B 


2  PRINCIPLES   OF  PHYSICS. 

rate  knowledge  even  of  the  simplest  actual  phenomenon  would 
imply  absolute  omniscience.  Often  we  find  that  we  can  measure 
but  cannot  systematise  the  phenomena  of  Nature ;  and  we  find 
at  the  very  outset  of  our  exposition  that  we  are  compelled  to 
confess  entire  ignorance  as  to  the  very  nature  of  our  subject- 
matter  ;  for  we  do  not  know  what  Matter  is. 

To  us  the  question,  What  is  Matter?  —  What  is,  assum- 
ing it  to  have  a  real  existence  outside  ourselves,  the  essential 
basis  of  the  phenomena  with  which  we  may  as  physicists  make 
ourselves  acquainted?  —  appears  absolutely  insoluble.  Even  if 
we  became  perfectly  and  certainly  acquainted  with  the  intimate 
structure  of  what  we  call  Matter,  we  would  but  have  made  a 
further  step  in  the  study  of  its  properties ;  and  as  physicists  we 
are  forced  to  say  that  while  somewhat  has  been  learned  as  to  the 
properties  of  Matter,  its  essential  nature  is  quite  unknown  to  us. 

As  little  can  we  give  any  full  and  satisfactory  answer  to 
the  question,  What  is  Energy?  As  a  provisional  statement 
we  may  say  that  Energy  is  the  Power  of  doing  Work ;  a  rifle- 
bullet  in  motion,  a  coiled  watch-spring,  possesses  the  power  of 
doing  Work  upon  other  bodies  suitably  arranged ;  but  plainly 
this  power  depends  upon  the  relation  into  which  the  matter 
which  is  said  to  possess  it  is  brought  with  reference  to  other 
matter,  and  it  ultimately  depends  upon  the  position  of  one  set 
of  particles  of  matter  with  reference  to  other  sets.  Since 
Energy  depends,  then,  upon  the  relative  position  of  particles 
of  Matter,  we  are  not  able  to  explain  its  own  essential  nature, 
though  we  may  acquire  a  considerable  amount  of  information 
as  to  its  very  remarkable  properties. 

These  properties  of  Energy,  those  of  Matter,  their  mutual 
relations,  and  the  laws  of  these  properties  and  relations,  con- 
stitute the  subject-matter  of  Natural  Philosophy;  and  these 
have  been  ascertained  by  observation,  by  measurement,  and  by 
judicious  reasoning  upon  the  data  supplied  by  investigation. 

In  the  investigations  upon  which  Natural  Philosophy  is 
founded,  the  guiding  principle  is  a  belief,  based  on  the  recorded 
experience  of  the  human  race,  in  the  Constancy  of  the  order  of 
Nature.  This  does  not  mean  that  things  are  to  continue  for 
ever  as  they  are  at  present.  If  a  closed  boiler  containing  water 
be  heated  to  a  certain  temperature,  the  Constancy  of  Nature 
would  not  be  interfered  with  by  the  consequent  explosion  of  the 
boiler ;  it  would  be  seriously  infringed  if  the  boiler  did  not 
burst.  So,  again,  if  volcanic  eruptions  thrust  up  mountain 


INTRODUCTORY.  3 

ranges  through  a  flat  plain,  as  in  the  case  of  the  Rocky  Moun- 
tains, or  if  a  crack  in  the  earth's  crust  allow  a  flood  of  lava  to 
flow  over  a  wide  region,  as  in  the  geological  history  of  Idaho, 
such  a  cataclysm  would  appear  to  be  an  awful  break  in  the 
uniformity  of  Nature ;  yet  if  the  earth's  crust  be  so  pressed 
upwards  that  it  can  resist  no  further  pressure,  the  Constancy  of 
Nature  is  confirmed  by  its  giving  way.  On  this  belief  in  the 
Constancy  of  Nature  are  based  all  rational  calculations  of 
eventualities,  and  all  our  arrangements  from  day  to  day,  which 
are  subject  to  the  transpiry  of  facts  unknown  or  unforeseen  at 
the  time  when  these  arrangements  were  made. 

This  belief  finds  formulated  expression  in  the  Law  of 
Causality,  which  affirms  that  every  effect  has  a  sufficient  cause. 
If  we  observe  any  given  phenomenon,  we  conceive  ourselves 
entitled  as  the  result  of  all  experience  to  enquire  into  its  cause, 
and  conversely,  to  affirm  that  if  there  be  no  cause  tending  to 
produce  change  in  any  particular  respect  in  the  present  condition 
of  things,  there  will  in  that  respect  be  no  change.  It  is  scarcely 
necessary  here  to  investigate  the  meaning  of  the  word  Cause 
itself;  it  will  be  quite  sufficient  to  point  out  that  for  us  the 
relation  of  Cause  and  Effect  is  one  of  Sequence,  found  to  be 
invariable  if  not  interfered  with  by  the  intervention  of  circum- 
stances which  render  cases  dissimilar.  In  similar  cases,  the 
same  causes  are  observed  to  be  followed  by  the  same  effects. 
It  is  plain,  however,  that  the  same  effects  are  not  always  and 
necessarily  the  results  of  the  same  causes  ;  and  when  different 
causes  are  found  to  produce  the  same  effects  they  are  e  q  u  i  v  a- 
lent  in  effectiveness  to,  and  may  be  substituted  for,  one  another. 

Again,  the  principle  may  be  stated  that  the  cause  is  equiva- 
lent and  in  proper  terms  of  measurement  numerically  equal  to 
the  effect  produced  by  it.  Apparent  exceptions  to  this  state- 
ment arise  only  when  the  problem  is  not  of  the  extremely  simple 
form  in  which  one  cause,  and  one  cause  alone,  is  brought  into 
play.  It  is  not,  except' in  a  loose  popular  sense,  the  heat  of  the 
spark  which  causes  the  explosion  of  a  magazine  and  consequent 
destruction  of  property ;  it  is  not  drawing  the  trigger  which  is 
the  cause  of  the  bullet's  leaving  the  gun.  The  heat  of  the  spark, 
the  drawing  of  the  trigger,  is  necessary  as  one  cause  out  of 
several ;  but  the  problem  is  not  here  so  simple  that  th^se  can  be 
adduced  as  cases  in  which  the  effect  is  greater  than'  the  cause. 
They  only  point  out  an  extended  statement,  that  the  total  effect 
produced  is  equivalent  to  the  effective  sum  of  the  causes 


4  PRINCi^ES  OF  PHYSICS. 

acting;  and  when  one  of  the  causes  acting  is  an  arrangement 
of  matter  which  is  explosive  or  in  unstable  equilibrium,  ready 
to  topple  over  so  as  to  assume  a  stable  position,  the  effect 
produced,  though  apparently  greater  than  the  small  disturbance 
which  disarranged  the  unstably-balanced  matter,  must  be  traced 
not  to  it  only,  but  to  all  the  conditions  and  circumstances 
involved,  including  the  unstable  equilibrium,  the  antecedent 
cause  of  which  may  itself  be  sought  for. 

If  several  causes  act  simultaneously,  each  produces  only  a 
part  of  the  aggregate  effect,  and  the  total  effect  is  equal  to  the 
sum  of  the  acting  causes.  Under  the  name  of  Galileo's  princi- 
ple this  is  one  of  the  fundamental  truths  of  physics,  and  is  thus 
enunciated: — If  a  body  be  acted  on  by  two  or  more  Forces 
(force  being  meanwhile  defined  as  any  cause  of  motion),  each 
of  these  forces  acts  independently,  and  produces  its  own  effect 
without  reference  to  the  others,  the  total  effect  produced  being 
ascertained  by  finding,  in  any  appropriate  way,  the  sum  of  the 
effects  due  to  the  several  forces.  A  cannon-ball,  for  instance, 
fired  from  a  height,  is,  as  it  passes  through  the  air,  executing 
movement  due  to  at  least  two  forces  or  causes  of  motion :  the 
force  exerted  upon  the  ball  during  the  explosion  sends  the  ball 
forwards,  that  of  gravity  continuously  draws  it  downwards. 
If,  for  the  sake  of  convenience,  we  neglect  the  resistance  of  the 
air,  and  enquire  what  would  be  the  path  pursued  by  a  shot 
travelling  in  vacuo,  we  would  find  by  making  use  of  this  princi- 
ple that  the  position  of  the  shot  at  any  moment  would  be  found 
by  enquiring  (1)  How  far  outwards  the  shot  would  have  been 
projected  had  there  been  no  tendency  to  fall ;  and  (2)  How 
far  the  ball  would  have  fallen  if  gravity  had  alone  acted  on  it. 
For  any  specified  instant  a  point  may  in  this  way  be  found, 
which,  being  both  so  far  outwards  and  so  far  downwards,  must 
be  the  position  of  the  ball  at  the  instant  in  question ;  and  by 
thus  finding  the  position  of  the  ball  at  several  separate  succeed- 
ing instants  of  time,  we  may  find  the  curved  path  which  a  ball 
fired  in  vacuo  would  traverse.  This  principle  of  the  inde- 
pendence of  simultaneously-acting  causes  was  an  experimental 
discovery  of  Galileo's :  before  his  time  it  was  held  as  self-evident 
truth  that  one  cause  must  cease  to  act  before  another  can  com- 
mence to  do  so ;  and  it  was  accordingly  believed  that  when  a 
projectile  was  shot  into  the  air,  the  force  of  projection  must  be 
expended  and  dissipated  before  any  tendency  to  fall  to  the  earth 
could  assert  itself. 


INTRODUCTORY.  5 

Experimentation.  —  When  we  learn  that  a  certain  pheno- 
menon is  due  to  a  congeries  of  causes,  we  may  arrange  matters 
so  as  to  prevent  one  of  the  ordinarily-acting  causes  from  pro- 
ducing its  effect,  and  then  we  may  observe  in  what  respect  the 
resultant  phenomenon  now  produced  differs  from  that  usually 
seen.  Thus  we  may  find  the  way  in  which  a  given  cause  acts. 
Again,  we  may  directly  arrange  matters  so  that  a  given  cause, 
and,  as  far  as  possible,  that  cause  alone,  shall  act,  and  we  may 
then  observe  what  happens.  The  principle  of  the  Constancy  of 
Nature  shows  us  that  like  causes  will  always  produce  like  results ; 
and  if  we  find  that  by  ingeniously  varied  interrogations  of  Nature 
we  have  obtained  as  reply  an  assurance  that  certain  causes  are 
allied  to  certain  effects,  we  feel  assured  that  the  same  causes  and 
the  same  effects  will  continue  to  be  so  allied.  This  assurance  is 
the  only  basis  of  the  art  of  Experimentation.  By  this  art  we 
become  acquainted  with  the  constant  modes  in  which  events 
follow  one  another  in  the  material  world,  these  modes  being  the 
Laws  of  Nature,  arbitrarily  appointed,  and  only  to  be  learned  by 
us  through  the  instrumentality  of  our  own  experimental  enquiry, 
or  else  through  attentive  consideration  of  the  varying  phenomena 
of  the  Universe,  "  experiments  made  at  Nature's  own  hand." 

Newton's  Laws  of  Motion  or  Axioms. — If  a  body  be  at  rest 
it  will  remain  at  rest :  if  in  motion  it  will  continue  to  move  until 
stopped  by  friction  or  some  external  force.  Here  we  find  the 
word  Force  meaning  not  only  that  which  causes  motion,  but  also 
that  which  arrests  motion.  Experiment  shows  us  that  this  is  true 
in  reference  to  bodies  which  are  at  rest,  for  they  remain  at  rest  if 
undisturbed ;  but  it  also  shows  that  among  bodies  which  are  in 
motion,  it  is  only  those  which  are  moving  in  a  Straight  Line 
that  retain  their  course  unaffected  when  allowed  to  move  unex- 
posed  to  the  action  of  any  disturbing  cause.  Bodies  Avhich  are 
moving  in  curved  paths,  such  as  sling-stones,  do  not  retain  their 
curved  paths  when  liberated,  but  continue  their  course  in  a  straight 
line  in  that  direction  in  which  they  happened  to  be  moving  at  the 
instant  of  release  from -constraint.  Hence  Newton,  in  his  First 
Law  of  Motion,  says,  "Every  body  tends  to  persevere  in 
its  state  of  Rest  or  of  Uniform  Motion  in  a  Straight 
Line  unless  in  so  far  as  it  is  acted  on  by  impressed 
Force,"  and  this  is  tersely  expressed  by  saying  that  "Matter 
has  Inertia/' 

If  a  single  force  act  upon  a  body  which  is  at  rest,  the  body 
will  begin  to  move  in  a  straight  line ;  and,  further,  the  greater 


6  PRINCIPLES   OF  PHYSICS. 

the  force,  the  more  rapid  will  be  the  motion  of  the  body  acted 
upon.  If  the  body  be  already  in  motion,  the  force  acting  upon 
it  will  cause  it  to  move  more  rapidly  or  more  slowly  in  the  same 
straight  line,  or  else  in  a  deflected  course.  Experiment  shows 
that  every  force  has  a  definite  direction  in  which  it  tends  to  cause 
a  body  to  move,  whether  that  body  be  already  under  the  action  of 
other  forces  or  not.  Thus  the  words  of  Newton,  in  his  Second 
Law  of  Motion,  are :  "Change  of  Motion  is  proportional 
to  the  impressed  Force,  and  takes  place  in  the  direc- 
tion of  the  Straight  Line  in  which  the  force  acts." 

The  word  Motion  in  this  law  is  now  rendered  Momentum  (p.  19). 

The  third  of  the  Laws  of  Motion  which  Newton  formulated 
as  axiomatic  is  the  following:  —  "To  every  Action  there 
is  always  an  equal  and  contrary  Reaction;  or  the 
Mutual  Actions  of  any  two  bodies  are  always  equal 
and  oppositely  directed."  The  truth  of  this  statement  is 
based  upon  experimental  evidence,  but  its  universal  applicability 
is,  after  consideration,  seen  to  be  reasonable  enough;  and  in 
this  sense  Newton  uses  the  word  Axiom. 

When  a  shot  is  fired  from  a  gun,  if  the  gun  be  free  to  move  there  is  con- 
siderable recoil,  the  shot  moving  forward  and  the  gun  backwards.  If  the 
gun  be  fixed  to  the  ground,  the  shot  is  apparently  the  only  thing  which 
moves.  If  the  shot  were  held  fast  and  the  gun  were  free  to  move,  the  gun 
would  move  backwards.  In  this  case  we  see,  then,  that  to  the  action  which 
impels  the  shot  forward  there  is  a  contrary  reaction  which  impels  the  gun 
backwards ;  and  in  the  sequel  we  shall  learn  what  evidence  there  is  for  the 
statement  that  that  reaction  is  equal  to  the  action. 

When  a  man  walks  on  firm  ground,  the  action  of  his  legs  in  locomotion 
tends  to  separate  his  body  from  the  ground  at  each  step.  The  action  which 
tends  to  raise  his  body  is  contrary  to  the  reaction  tending  to  depress  the 
earth,  and  at  every  step  the  earth  is  pushed  down  as  a  whole,. or  else  if  the 
soil  be  soft  it  yields  locally  and  the  foot  sinks.  Hence  the  difficulty  experi- 
enced in  getting  out  of  boggy  soil ;  the  soft  mud  yields  under  the  foot  at 
each  effort  made  by  the  individual,  so  that  every  step  causes  him  to  sink 
more  deeply. 

When  a  horse  is  loosely  harnessed  to  a  car,  it  may  sometimes  be  observed 
that  an  inexperienced  animal  starts  forward  quickly ;  but  suddenly  the  traces 
tighten,  the  car  is  jolted  forward,  and  the  horse  is  jolted  backwards. 

If  a  locomotive  with  a  heavy  train  be  suddenly  started,  it  will  be  seen 
that  its  wheels  may  uselessly  turn  round ;  it  has  given  a  sudden  pull  to  the 
carriages,  and  their  reaction  upon  it  is  equivalent  to  a  backward  pull  given 
to  a  moving  engine. 

The  earth  attracts  the  moon,  and  the  moon  equally  attracts  the  earth. 
The  former  attraction  mainly  keeps  the  moon  in  her  orbit,  and  the  latter  is 
one  of  the  causes  of  tidal  phenomena. 

When  a  stone  is  thrown  upwards  from  the  earth,  the  earth  is  thrown 
back  by  recoil,  and  moves  downwards  to  a  very  small  extent  so  long  as  the 


INTRODUCTORY.  7 

stone  continues  to  ascend  :  when  the  stone  is  at  its  highest  point  the  earth  is 
at  its  lowest,  and  as  the  stone  falls  the  earth  ascends  to  meet  it.  This  is,  of 
course,  not  the  result  of  direct  observation,  but  is  deduced  by  way  of  inference 
from  Newton's  third  law  of  motion,  which  is  confirmed  by  all  phenomena, 
terrestrial  and  astronomical,  by  which  it  can  be  put  to  the  test. 

The  next  statement  generally  applicable  is  that  of  the  Inde- 
structibility of  Matter.  This  is,  that  Matter,  as  we  at  present 
know  it,  cannot  be  destroyed  by  any  process  with  which  we  are 
acquainted.  The  limitations  of  this  statement  should  be  borne  in 
mind,  for  there  is  no  scientific  warranty  for  saying  that  Matter  is 
absolutely  indestructible,  and  more  than  one  consideration  indi- 
cates that  the  structure  of  Matter  may  be  such  as  to  denote  that 
in  its  present  form  it  has  had  a  beginning  and  may  have  an  end. 
Within  our  experimental  knowledge,  however,  Matter  cannot  be 
destroyed :  and  when  it  apparently  disappears,  as  when  a  candle 
is  burned  in  the  air,  Chemistry  charges  itself  with  the  explana- 
tion of  that  disappearance,  and  shows  what  new  forms  the  matter 
has  assumed. 

Another  principle  of  the  greatest  possible  use,  and  entirely 
the  result  of  experiment,  is  that  of  the  Indestructibility  or  the 
Conservation  of  Energy.  Energy  has  been  provisionally  defined 
as  the  Power  of  doing  Work ;  and  this  doctrine  states  that  this 
power  of  doing  work  may  alter  its  form  but  is  never  destroyed. 
A  coiled  watch-spring  possesses  power  of  doing  work  in  virtue  of 
its  distortion ;  when  it  uncoils,  it  seems  to  lose  this  power  of 
doing  work,  but  the  Energy  thus  lost  is  transferred  to  other 
bodies,  while  Heat,  Light,  or  Sound  produced,  Work  done, 
Electrical  Condition  set  up,  Friction  overcome,  etc.,  present  ths 
missing  Energy  in  several  apparently  dissimilar  forms,  which  may 
all  be  reduced,  however,  to  two  types :  Energy  due  to  Motion ; 
Energy  due  to  Displacement.  The  Energy  of  a  body  depends 
on  the  advantage  which  that  body  possesses  either  of  motion  or 
of  position :  the  loss  of  that  advantage  can  only  occur  through 
some  other  body  or  bodies  simultaneously  acquiring  either  mo- 
tion or  an  advantage  of  position.  If  Energy  disappear  in  one 
form,  it  will  reappear  in  one  or  several  others,  and  none  of  it  is 
ever  lost,  though  it  may  assume  such  a  form  that  it  is  no  longer 
a  power  of  doing  work  available  to  man,  namely,  the  form  of 
uniformly  diffused  Heat.  The  principle  of  the  Conservation  of 
Energy,  which  is  so  important  that  the  whole  of  NaturaJ.  Philo- 
sophy may  be  said  to  be  a  commentary  on  it,  will  be  better 
understood  when  the  laws  of  Energy  have  been  discussed,  as 
they  will  be  at  greater  length  in  Chapter  IV. 


IUITIVBR5I 

'.     • 


g  PRINCIPLES   OF   PHYSICS. 

A  corollary  to  this  principle  takes  the  form  of  a  statement 
of  the  belief  that  The  Perpetual  Motion  is  impossible :  if  the 
sum  of  the  Energy  in  the  Universe  be  constant,  no  machine  in 
which  this  energy  is  employed  in  doing  work,  in  which  friction 
is  overcome,  in  which  sound  is  produced,  and  so  on,  can  possibly 
go  on  for  ever,  for  the  reserve  of  energy  at  its  disposal  will  ulti- 
mately be  exhausted  and  become  useless  to  that  machine.  Even 
the  tides  will  ultimately  cease,  as  the  earth  loses  speed  —  we 
know  it  is  at  present  losing  an  aggregate  of  22  seconds  in  the 
course  of  a  century  —  in  its  rotation  round  its  own  axis. 

It  cannot  be  too  strongly  insisted  on  that  these  general 
principles,  the  Constancy  of  Nature,  the  Law  of  Causality, 
Galileo's  principle,  the  three  Laws  of  Motion,  the  Indestructibil- 
ity of  Matter  and  of  Energy,  are  of  no  value  for  us  except  in  so 
far  as  they  are  supported  by  experimental  evidence.  They  are 
grouped  together  here,  for  the  statement  of  them  is  necessary  for 
comprehension  of  the  results  which  have  been  obtained  through 
their  aid.  We  are  not  here  called  upon  to  go  through  the  steps 
by  which  they  have  been  arrived  at,  but  we  must  bear  in  mind 
that  no  a  priori  deduction  of  them  by  any  metaphysical  reasoning 
is  for  a  moment  admissible.  The  doctrine  of  the  Conservation  of 
Energy  is  very  simple  when  stated  as  the  result  of  experiment, 
and  its  simplicity  has  led  to  statements  that  the  contrary  is 
unthinkable,  and  that  a  belief  in  this  doctrine  is  deeply  grounded 
in  the  constitution  of  the  mind  of  man ;  but  all  conclusions 
derived  from  such  reasoning  must  be  regarded  with  suspicion, 
for  we  must  take  warning  by  the  example  of  the  ancients,  who 
believed  circular  motion  to  be  perfect  and  heavy  bodies  to  fall 
faster  than  light  ones,  until  experimental  evidence  was  adduced 
to  the  contrary.  The  truth  of  these  principles  must  be  proved 
by  their  perfect  accord  with  the  phenomena  which  we  may 
actually  observe,  and  by  their  enabling  us  to  predict  results  of 
hitherto  untried  experiments  which  agree  with  those  actually 
obtained.  Exact  science  depends  directly  for  its  facts  and  in- 
directly for  its  principles  upon  experimental  evidence,  and  the 
true  place  of  speculative  imagination  in  scientific  work  is  the 
conception  of  new  combinations  of  circumstances,  and  hence  of 
new  fields  of  experimental  Research,  as  also  the  construction  of 
Hypotheses,  which  explain  and  co-ordinate  observed  facts,  and 
which,  when  they  are  found  to  do  this  consistently  and  with  but 
a  few  reasonable  and  simple  assumptions,  are  raised  to  the  rank 
of  accepted  Theories. 


CHAPTER  I. 

TIME,   SPACE,    AND  MASS. 

So  far  as  man's  knowledge  of  phenomena  occurring  around 
him  has  become  accurate,  it  has  been  obtained  by  means  of 
precise  Measurement;  and  the  Fundamental  Units  in  terms 
of  which  every  measurement  must  be  executed  are  those  of 
Time,  Space,  and  Mass. 

The  unit  of  Time  is  usually  taken  as  one  Second,  and 
the  time  during  which  phenomena  appear  or  are  observed  is 
reckoned  in  seconds,  unless  motives  of  obvious  convenience 
cause  it  to  be  reckoned  in  minutes,  hours,  days,  years,  or  cen- 
turies. The  second  is  usually  a  second  of  mean  solar  time  — 
that  is  to  say,  the  -g-g-J^th  part  of  the  average  length  of  a  solar 
day. 

The  solar  day  is  the  period  which  elapses  between  the  sun's  crossing 
the  meridian,  or  being  situated  directly  south  (or  in  the  southern  hemisphere, 
directly  north)  of  a  place,  and  the  next  occasion  on  which  it  crosses  that 
line.  The  sidereal  day,  in  the  same  way,  is  the  interval  between  two 
successive  transits  of  any  fixed  star.  The  sidereal  days  are  shorter  than  the 
solar ;  they  are  nearly  constant  in  length,  for  a  sidereal  day  is  the  time  of  one 
complete  rotation  of  the  earth  round  its  axis  ;  but  the  solar  day  is  not  con- 
stant in  length.  A  clock  can  keep  time  with  the  stars,  and  keep  good 
"  sidereal  time ;  "  but  a  clock  of  ordinary  construction  does  not  always 
indicate  noon  when  the  sun  is  highest  in  the  heavens ;  it  is  sometimes 
apparently  14}  minutes  fast,  and  sometimes  appears  to  be  16^  minutes  slow. 
A  good  clock,  however,  is  one  which  measures  off  and  indicates  as  twenty- 
four  hours  a  period  of  time  equal  to  the  average  length  of  the  solar  day 
for  a  year  or  a  century  or  an  age,  and  such  a  clock  is  said  to  keep  "  mean 
solar  time ; "  while  the  Second  used  in  physical  measurements  is  the  second 
as  indicated  by  a  clock  such  as  this.  Astronomers  reckon  by  the  sidereal 
day,  which  is  equal  to  86164-092  mean  solar  seconds :  and  they  use  shorter 
pendulums  in  their  clocks,  so  that  these  keep  not  mean  solar,  but  sidereal 
time.  The  astronomer's  second  is  then  equal  to  (86164-092-86400)  =  0-99727 
mean  solar  second. 

Space.  —  When  a  single  point  moves  it  describes ^a  Line: 
if  it  travel  by  the  shortest  distance  between  two  points,  its  path 
is  a  straight  line ;  and  a  straight  line  is  an  example  of  space  of 


-J^Q  TIME,    SPACE,   AND   MASS.  [CHAP. 

one  dimension.  Movement  and  measurement  may  be  effected 
in  a  forward  or  a  backward  direction  along  it,  but  as  a  line  has 
neither  breadth  nor  thickness  there  can  be  no  other. 

Distance  along  a  straight  line  may  be  measured  in  one 
direction  arbitrarily  chosen ;  let  this  be,  for  instance,  the  direc- 
tion from  left  to  right ;  if,  then,  a  point  travel  towards  the  right 
its  motion  is  positive,  if  to  the  left,  negative.  If  it  move  a 
inches  to  the  right  and  then  b  inches  to  the  left,  its  distance 
from  the  starting-point  becomes  a  —  b;  while  if  it  first  go  b 
inches  to  the  left  and  then  a  to  the  right,  its  position  will 
become  —  b  +  a  from  that  point;  and  these  two  positions  are 
the  same,  for  a  -  b  =  -b  +  a.  Hence  we  learn  that  if  a  point 
move  backwards  and  forwards  by  varying  amounts  along  a  line, 
it  does  not  matter  in  what  order  it  performs  these  operations : 
the  spot  ultimately  arrived  at  will  be  the  same  in  all  cases. 

In  order  to  effect  measurements  along  lines,  we  require  a 
standard  of  length.  This  is  taken  as  the  Foot  or  the  M£tre. 
The  British  standard  yard,  which  is  equal  to  three  feet,  is 
denned  by  law  as  "  the  distance  between  the  centres  of  the 
transverse  lines  in  the  two  gold  plugs  in  the  bronze  bar  de- 
posited in  the  office  of  the  Exchequer"  at  the  temperature  of 
62°  F.  A  number  of  authorised  copies  of  this  have  been  made 
and  are  deposited  at  the  Royal  Mint,  the  Royal  Observatory  at 
Greenwich,  the  New  Palace  at  Westminster,  and  under  the 
care  of  the  Royal  Society  of  London. 

The  MStre  is  the  distance,  at  the  temperature  of  melting 
ice,  between  the  ends  of  a  platinum  rod  preserved  in  the 
Archives,  and  of  which  copies,  to  regulate  French  commerce, 
are  preserved  at  the  Minist^re  de  ITnte'rieure  in  Paris.  It  was 
originally  intended  to  represent  ithe  '(ten-millionth  part  of  the 
distance  from  the  Equator  to  tfhe  Pole :  the  measurements  of 
Delambre  and  Me*chain,  from  which  Borda  made  the  standard 
metre  according  to  a  law  of  the  French  Republic  passed  in 
1795,  have  been  found  not  to  be  quite  correct,  for  the  earth's 
quadrant  is  now  known  to  measure  10,000,880  metres. 

The  metric  system  of  measurement  of  length  is  decimal; 
each  metre  contains  10  decimetres,  100  centimetres,  or  1000 
millimetres :  1000  metres  make  a  kilometre,  which  is,  roughly 
speaking,  about  f  (f  f  foO  of  a  mile ;  one  metre  is  equal  to 
39-37043196  inches,  or  3-28087  feet ;  a  decimetre  very  nearly 
corresponds  to  4  inches  (really  3-937043196)  ;  a  millimetre  is 
very  nearly  equal  to  the  twenty-fifth  of  an  inch.  For  the  pur- 


I.] 


SPACE. 


11 


pose  of  physical  measurement  it  is  customary  and  convenient  to 
make  use  of  the  Centimetre*  (-3937043196  inch)  as  a  unit  of 
length.  One  English  foot  is  equal  to  3047972654  centimetres, 
and  an  inch  to  2-53993,  or  very  nearly  2-54  centimetres. 

A  plane  Surface  has  length  and  breadth  but  no  thickness, 
and  is  therefore  said  to  be  space  of  two  dimensions.  Two 
terms  are  always  necessary  for  the  precise  statement  of  the  posi- 
tion of  any  point  on  a  surface.  The  position  of  a  ship  at  sea  is 
determined  when  its  latitude  and  its  longitude  are  known. 

The  position  of  a  point  a  on  a  plane  surface  is  determined  by  choosing 
a  fixed  point  O  as  the  origin ;  then  two  axes,  Ox  and  Oy,  are  chosen,  gen- 
erally at  right  angles  to  one  another ;  aA  is  drawn  parallel  to  Ox,  and  aB 


Fig.i 


parallel  to  Oy,  and  the  point  a  is  said  to  be  situated  at  a  distance  OA  along 
the  axis  of  y,  and  OB  along  the  axis  of  x.  If  a  point  lie  at  the  same  time 
three  miles  to  the  north  and  four  miles  to  the  west  of  a  given  place,  its  true 
position  (at  the  distance  of  five  miles)  can  be  easily  indicated  on  a  chart. 
The  symbols  +  and  —  are  also  used  here  to  denote  that  the  measurement  is 
to  one  side  or  the  other  of  the  point  assumed  as  the  origin.  Points  to  the 
right  of  O  have  a  positive,  points  to  the  left  a  negative,  value  of  Ox ;  points 
above  O  have  a  positive,  points  below  a  negative,  value  of  Oy.  Thus  (Fig. 
1)  the  point  a1  has  abscissa  (or  line  cut  off  along  the  axis  of  x}  OB,  and 
ordinate  (cut  off  along  the  axis  of  y)  OA;  the  point  a"  has  abscissa  —OB 
and  ordinate  +OA;  the  point  a'"  has  abscissa —OB  and  ordinate  —  OA; 
that  at  a""  has  abscissa  -fOB  and  ordinate  —  OA. 

*  It  is  worth  remarking  that  a  French  ten-centime  piece  measures  3  centimetres 
across,  while  a  five-centime  piece  has  a  diameter  of  2?  centimetres.  Similarly,  an 
English  halfpenny  measures  an  inch,  while  a  penny  measures  an  inch  and  a  fifth. 


|2  TIME,    SPACE,    AND   MASS.  [CHAP. 

The  Area  of  a  Surface  may  be  measured  if  we  fix  upon  a 
standard  unit  of  area.  The  unit  of  length  may  be  made  use  of 
in  order  to  obtain  this.  If  a  square  be  constructed,  one  of  whose 
sides  is  one  foot  or  one  centimetre,  we  shall  have  a  unit-surface 
whose  area  is  known  as  one  square  foot  or  one  square  centi- 
metre;  and  the  areas  of  other  surfaces  may  be  measured  by 
comparison  with  these  standards. 

A  Solid  has  length,  breadth,  and  thickness,  and  is  said  to 
occupy  space  of  three  dimensions.  The  position  of  any 
point  in  tridimensional  space  requires  three  numerical  terms  for 
its  exact  statement.  The  position  of  a  balloon,  for  instance, 
will  be  definitely  known  if  the  latitude  and  longitude  of  the 
spot  over  which  it  stands  and  its  height  above  that  spot  be 
ascertained. 

Three  terms  are  also  required  to  define  the  position  of  a  star :  the 
telescope  has  to  move  so  much  "  in  azimuth  "  round  a  vertical  axis  ;  then  so 
much  in  "  altitude  "  round  a  horizontal  axis ;  and  thirdly,  the  distance  of 
the  star  in  a  straight  line  must  be  known. 

A  cube  whose  side  is  one  foot  or  one  centimetre  —  that  is, 
a  cubic  foot  or  a  cubic  centimetre  —  serves  as  the  unit  of 
volume.  For  convenience'  sake  other  units  of  volume  are  often 
chosen,  such  as  the  cubic  inch,  the  cubic  decimetre  (otherwise 
known  as  the  liquid  measure,  one  Litre),  the  cubic  metre,  and 
so  forth. 

The  remaining  fundamental  idea  involving  measurement  is 
that  of  Mass,  or  quantity  of  Matter.  The  notion  implied  in 
this  term  is  quite  distinct  from  that  of  Weight.  The  weight 
of  a  certain  quantity  of  matter  depends  upon  the  presence  and 
nearness  of  other  matter,  which  attracts  it  according  to  the  well- 
known  law  of  Gravitation.  This  may  and,  even  within  our 
terrestrial  observation,  does  vary ;  the  effect  of  gravity  on  a 
given  mass  —  that  is  to  say,  its  Weight  —  is  greater  as  we  near 
the  Poles  than  it  is  at  the  Equator;  and  the  weight  of  a 
substance  varies,  therefore,  according  to  local  causes,  while  the 
mass  or  quantity  of  matter  in  it  remains  the  same.  Cceteris 
paribus,  however,  equal  masses  will  everywhere  counterpoise 
one  another  in  a  balance,  and  we  may  define  the  unit  of  mass 
as  that  quantity  of  matter  which  will  counterpoise  in  a  balance 
a  certain  standard  mass  known  as  a  standard  Pound  or  Gramme. 

The  British  standard  Pound  is  a  piece  of  platinum  preserved 
in  the  same  place  as  the  standard  yard,  while  authorised  copies 
of  it  are  preserved  at  the  same  institutions.  The  French  stand- 


i.]  MASS.  13 

ard  is  the  Kilogramme  (=  1000  grammes),  made  of  platinum, 
and  preserved  at  the  Archives  in  Paris.  This  is  intended  to 
have  the  same  weight  as  a  cubic  decimetre  of  water  at  its 
temperature  of  maximum  density  —  that  is,  3*9°  C.  Since  a 
kilogramme  contains  a  thousand  grammes,  and  a  cubic  decimetre 
a  thousand  cubic  centimetres,  it  follows  that  the  gramme  is 
intended  to  be  equal  to  the  mass  of  one  Cubic  Centimetre 
of  water  at  3-9°  C.  Comparison  of  the  actual  standards  shows, 
however,  that  a  litre  of  water  weighs,  at  3-9°  C.,  1-000013  kilo- 
grammes, and  a  cubic  centimetre  of  water  at  3-9°  C.  weighs 
therefore  not  one  gramme,  but  1-000013  grm.  For  most  practical 
purposes  the  intended  value  may,  however,  be  taken  as  correct. 
The  British  pound  avoirdupois  weighs  7000  grains,  while  the 
standard  kilogramme  weighs,  according  to  Prof.  W.  H.  Miller, 
15432-34874  grains,  and  the  gramme  15-43234874  grains. 

It  may  be  noticed  that  the  British  fluid  ounce  of  water  at  62°  F.  weighs 
one  ounce  avoirdupois;  that  the  British  pint  of  water  (20  fluid  ounces) 
weighs  therefore  a  pound  and  a  quarter,  and  the  British  gallon  of  water 
ten  pounds.  A  French  franc-piece  weighs,  when  new,  five  grammes. 

In  British  measurements  the  Foot,  the  Pound  and  the 
Second  may  be  used  as  the  fundamental  units.  In  British 
Magnetic  Observatories  the  units  employed  till  lately  were  the 
Foot,  the  Grain  and  the  Second. 

The  C.G.S.  System.  —  For  the  international  convenience 
of  scientific  men  the  C.G.S.  or  Centimetre-Gramme-Second 
system  of  units  and  measurements  is  in  current  use. 

The  gramme  is  chosen  as  a  unit  rather  than  the  kilogramme,  the  cen- 
timetre rather  than  the  metre ;  firstly,  because  the  use  of  smaller  units 
diminishes  the  need  for  working  with  decimal  fractions ;  and,  secondly, 
because  on  the  C.  G.  S.  system  the  density  of  water  (p.  220)  is  equal  to  unity, 
which  is  a  distinct  advantage.  If  the  kilogramme  and  the  metre  had  been 
employed  as  units,  the  density  of  water  —  the  number  of  kilogrammes  in  a 
cubic  metre  —  would  have  been  1000. 

The  introduction  of  coherent  systems  of  units  for  the 
measurement  of  all  physical  quantities  has  been  an  enormous 
stride  in  advance.  When  we  have  a  problem  to  solve 
numerically,  if  we  take  care  to  put  in  all  the 
terms  in  C.G.S.  measurement,  the  answer  comes 
out  in  C.G.S.  units,  ready  for  use  without  further 
reduction. 


CHAPTER  II. 

NOTIONS   DERIVED   FROM  THE  PRECEDING. 

WHEN  a  physical  particle  changes  its  position,  it  effects 
Motion.  This  Motion  or  Change  of  Position  must  be  per- 
formed by  passing  along  a  definite  continuous  path  —  con- 
tinuous because  it  is  not  possible  for  any  physical  particle  to 
occupy  two  consecutive  positions  without  traversing  the  inter- 
mediate space. 

In  this  respect  the  path  of  a  physical  particle  differs  from  many  mathe- 
matical curves  which  abruptly  end  at  one  point  and  recommence  their  course 
at  another.  Obviously  the  path  described  by  the  moving  particle  may  have 
any  form,  straight  or  curved ;  and  the  shortest  possible  path  between  the 
initial  and  final  positions  is  a  straight  line. 

We  may  remind  the  reader  of  Newton's  use  of  the  word  Motion  in  the 
sense  of  Momentum  (pp.  6,  19). 

A  moving  body  may  travel  rapidly  or  slowly :  the  rate  at 
which  it  travels  along  its  path  is  called  its  Rate  of  Motion,  its  rate 
of  change  of  position,  its  Velocity.  The  Velocity  of  a  moving 
body  may  be  stated  in  units  of  length  per  unit  of  time,  e.g.  feet 
per  second;  and  a  body  is  moving  with  Unit  velocity  when 
it  moves  one  foot  per  second,  or  one  centimetre  per  second  (the 
latter,  the  C.G.S.  unit,  being  one  'Kine').  It  will  be  observed 
that  it  is  necessary  for  us  to  make  consistent  use  of  the  British 
or  of  the  C.G.S.  units  of  measurement,  and  not  to  use  them 
confusedly  within  the  limits  of  the  same  problem. 

A  body  which  moves  sixty  feet  in  five  seconds  has  a  mean 
velocity,  evidently,  of  twelve  feet  per  second.  The  velocity  is 
equal  to  sixty  divided  by  five  —  that  is,  to  the  whole  space 
traversed  divided  by  the  time  occupied  in  the  movement.  In 

• 

algebraical  language  this  may  be  expressed  thus:  v=-»  where 

c 

the  velocity,  space,  and  time  are  denoted  by  their  initial  letters. 
Multiplying  both  sides  of  this  equation  by  £,  we  get  vt=s;  the 

14 


[CHAP,  ii.]  VELOCITY.  15 

space  traversed  in  a  given  time  is  equal  to  the  velocity  per 
second  multiplied  by  the  number  of  seconds. 

If  we  consider  Motion  and  Velocity  in  any  one  particular 
direction,  we  may  emphasise  this  by  using  black-faced  type  for 
our  symbols  ;  our  equation  then  becomes  v  =  s/£  ;  the  Velocity 
in  any  given  direction  is  the  Space  traversed  in  that  direction 
divided  by  the  Time. 

A  Velocity  in  general,  without  reference  to  its  direction,  is 
sometimes  called  a  speed,  v;  while  the  term  velocity  is  then 
restricted  to  velocity,  v,  in  some  particular  Direction.  In  this 
volume  we  shall,  for  the  most  part,  distinguish  Speeds  in 
general  from  Velocities  in  particular  directions  by  the  use  of 
the  symbols  v  or  v,  as  required. 

Digression  as  to  mathematical  formulae  and  the  theory  of 
Dimensions.  —  Each  such  formula  is  a  kind  of  generalised  shorthand  blank 
form,  waiting  to  be  applied  to  particular  cases  by  being  consistently  filled  in 
with  appropriate  numbers.  In  words  at  full  length  we  may  affirm  that  the 
Number  expressing  a  speed  or  velocity  is  equal  to  the  Number  ex- 
pressing the  space  traversed  divided  by  the  Number  expressing  the  corre- 
sponding time  taken ;  all  these  being,  of  course,  systematically  measured  in 
consistent  units.  The  numbers  themselves  in  any  particular  case  we  may 
not  know  at  present,  and  in  the  meantime  we  may  not  even  care  to  know ; 
for  such  a  verbal  formula  is  of  a  higher  order  of  generality,  of  wider  value 
than  a  mere  statement  of  the  particular  numbers  in  any  particular  case.  By 
way  of  rough  jotting  we  may  shorten  the  phrase  "Number  expressing  a 
Velocity  "  down  to  the  simple  word  "  Velocity,"  and  so  on.  Then  we  have 
the  condensed  note  "  Velocity  =  Space  -t-  Time."  This  may  be  still  further 
shortened  by  using  initial  letters  only,  in  which  case  the  symbols  "  v  =  s  -4- 1," 

or  " v  =  -,"  or  " v  =  s/t"  suffice  to  express  the  law ;  or  we  may  agree  that 

these  unknown  numbers  shall  for  the  time  being  be  represented  by  letters 
arbitrarily  chosen.  Thus  if  we  agree  that  the  letter  a  shall  stand  for  "  num- 
ber expressing  velocity,"  or,  as  it  is  more  usually  phrased,  that  a  shall 
represent  velocity ;  and  similarly  that  b  shall  represent  space  traversed,  and 
c  the  corresponding  time,  the  condensed  expression  of  our  law  becomes 
a  =  b  +-  c.  To  apply  this  to  any  particular  case  we  must  know  what  the 
numerical  values  of  two  of  the  terms  actually  are ;  this  much  being  deter- 
mined, it  is  only  an  arithmetical  matter  to  find  the  numerical  value  of  the 
third  term.  For  example,  let  v  (the  number  expressing  a  velocity)  be  12 
(ft.  or  cm.  per  sec.),  and  'let  s  =  60  (ft.  or  cm.),  then  replacing  v  in  the 
equation  by  12  and  s  by  60  we  get  12  =  60  -*•  <,  and  t  cannot  have  any 
other  value  than  5  (sec.) ;  all  in  units  of  the  same  system. 

One  great  advantage  attending  the  use  of  mathematical  formulae  is 
their  susceptibility  to  algebraic  transformation.  The  above  equation  may  be 
written  s  =  vt  or  t  =  s/v,  either  of  which  modes  of  expression,  when  trans- 
lated into  words  at  full  length,  is  found  to  present  the  subject  from  so  fresh 
a  point  of  view  as  practically  to  amount  in  each  case  to  the  entmciation  of 
an  independent  truth. 

When  a  is  stated  by  a  formula  to  depend  upon  or  to  be  "  a  function  of  " 


16  DERIVED   NOTIONS.  [CHAP. 

6,  c,  d,  and  of  these  only,  it  seems,  when  put  into  words,  a  truism  to  affirm 
that  a  is  independent  of  variations  in  the  values  of  e,  f,  g,  etc. ;  yet  this 
often  leads  to  the  enunciation  of  valuable  principles,  e.g.  p.  21,  line  16. 

The  Theory  of  Dimensions.  —  The  number  expressing  a  Velocity  is 
the  number  expressing  a  Space  divided  by  the  number  expressing  a  Time ; 
v  =  s/t,  as  we  have  seen  before.  But  there  underlies  this  mode  of  expression 
a  tacit  understanding  that  we  adhere  consistently  to  some  known  system  of 
units.  The  numbers  must  vary  with  the  units  conventionally  employed, 
even  when  the  same  facts  have  to  be  expressed.  Consequently  we  may,  if 
we  have  in  our  minds  a  possible  change  of  units,  write  such  an  equation  as 
v[V]  =«  [S]  -f- 1  [T],  where  the  italic  initials  represent  numbers  and  the 
corresponding  bracketed  letters  the  respective  conventional  units.  If  v,  s, 
and  t  in  the  above  equation  become  all  =  1,  that  equation  becomes  [V] 
=  [S/T],  an  equation  which  refers  to  the  conventional  units  only.  Such  an 
equation  is  technically  known  as  an  equation  of  Dimensions.  Then  if  we 
change  our  conventional  units  from  [V],  [S],  and  [T]  to  others,  say,  [/V], 
[raS],  [nT],  the  last  written  equation  must  still  hold  good,  and  the  new 
unit  [/V]  is  equal  to  [mS/nT],  or  to  m/n  [S/T]  ;  that  is,  the  new  unit  of 
velocity  is  equal  to  m/n  times  the  old  unit.  The  numerical  value  of  any 
given  velocity  is,  inversely,  n/m  times  as  great  when  expressed  in  terms  of 
the  new  units  as  it  was  when  expressed  in  terms  of  the  old  units ;  that  is,  it 
varies  inversely  as  the  unit  employed ;  just  as  a  sum  of  £40,000  seems 
greater  (one  million)  when  expressed  in  the  smaller  French  unit,  the  franc. 
Let  us  now  set  ourselves  a  problem :  What  is  the  ratio  between  the  British 
and  the  C.G.S.  unit  of  velocity?  The  former  is  1  ft.  per  sec.,  the  latter  is 
1  cm.  per  sec.  Here  [V]  =  [S/T]  =  [Foot/Second]  =  [30478  cm. /second] 
=  30478 [cm. /second]  ;  the  British  unit  is  30478  times  the  C.G.S.  unit. 
Consequently  a  velocity  of  3047-8  cm./secs.  would  be  a  velocity  of  only  100 
if  measured  in  ft. /seconds. 

But  the  Equation  of  Dimensions  is  not  limited  to  this  interpretation 
and  use.  It  far  more  frequently  means,  in  actual  use,  to  adhere  for 
example's  sake  to  the  equation  [V]  =  [S/T],  that  the  Numerical 
Measure  of  any  Velocity  is  some  Number  of  Units  of  Space  (or  Length) 
divided  by  a  corresponding  Number  of  Units  of  Time  :  and  where  we  have, 
for  example,  the  Dimensions  of  a  Quantity  of  Electricity*  in  magnetic 
measure  given  as  [</]  =  [M^L*],  it  means  that  the  numerical  measure  of 
any  given  quantity  of  electricity  on  the  magnetic  system  of  measurement  is 
the  square  root  of  some  Number  of  Units  of  Mass  multiplied  by  the  square 
root  of  a  corresponding  Number  of  Units  of  Length,  the  Numbers  themselves 
necessarily  varying  inversely  as  the  units  employed. 

The  equations  of  Dimensions  thus  explained  were  an  invention  of 
Fourier's,  and  were  brought  into  prominence  by  Clerk  Maxwell.  Their  use 
is  twofold :  (1)  as  a  means  of  converting  physical  quantities  expressed  in 
one  set  of  units  into  the  same  quantities  expressed  in  other  units ;  and 
(2)  as  a  means  of  checking  our  equations,  for  the  dimensions  must  agree 
on  both  sides,  as  will  be  seen  in  very  simple  examples  on.  p.  60. 

Velocity  (resumed).  —  If  a  body  move  through  equal 
spaces  in  equal  times,  its  velocity  is  said  to  be  uniform. 

We  are  familiar  with  instances  in  which  a  body  such  as 
a  railway  train  is  said  to  be  running  at  a  certain  time  with  a 


ii.]  VELOCITY.  17 

velocity  of  (say)  thirty  miles  an  hour.  This  indicates  that  if 
the  train  ran  for  a  whole  hour  at  the  rate  at  which,  and  in  the 
same  direction  as,  it  was  travelling  at  the  instant  of  observation, 
it  would  at  the  end  of  an  hour  be  thirty  miles  away  from  the 
point  which  it  occupied  at  the  beginning  of  it.  But  the  train 
may  possibly  not  have  run  more  than  a  mile  on  the  whole. 
The  statement  means,  then,  that  during  (say)  a  minute  it  ran 
half  a  mile,  and  that  therefore  during  sixty  minutes  it  might, 
at  the  same  speed,  have  run  thirty  miles.  But  even  during  a 
minute  it  may  have  gained  or  lost  speed,  so  as  to  render  its 
motion  not  uniform  but  variable :  the  statement  would  be  still 
more  exact  if  we  knew  that  in  six  seconds  it  ran  the  twentieth 
of  a  mile,  or  in  one  second  the  hundred-and-twentieth ;  for 
when  the  interval  of  time  is  very  short,  there  is  less  possibility 
of  variation  during  that  interval,  and  the  speed  approximates 
more  nearly  to  uniformity.  Hence  the  variable  velocity  of  any 
moving  body  at  a  particular  instant  is  found  by  observing  the 
amount  of  motion  effected  in  a  certain  very  short  interval  of  time, 
and  finding  what  movement  would  be  effected  in  one  unit  of 
time  if  the  velocity  were  to  remain  uniform  during  that  period. 
If  a  body  move  over  a  certain  space,  s  (say  thirty  feet),  in 

time  t  (say  ten   seconds),  the   equation  v  =  -  =  —  =  3  feet  per 

6        _L  \J 

second  shows  what  the  mean  or  average  velocity  is  during  the 
motion.  The  mean  velocity  of  a  train  which  travels  fifteen 
miles  in  one  hour  is  a  quarter  of  a  mile  a  minute,  or  g^o  Par^  °^ 
a  mile  in  a  second,  although  during  some  seconds  or  minutes  it 
may  be  travelling  at  the  rate  of  sixty  miles  an  hour,  at  others 
may  be  standing  still,  and  at  others  may  be  actually  going 
backwards. 

All  velocities,  mean  and  constant,  uniform  and  variable,  may 
be  expressed  in  feet  or  in  centimetres  per  second,  and  can,  when 
so  expressed,  be  compared  with  one  another. 

All  measurable  velocities  are  Relative ;  we  know  nothing 
about  Absolute  velocities  in  space,  for  we  have  no  standard 
of  comparison. 

Problems. 

1.  If  a  body  move  144  feet  in  3  seconds,  what  will  be  its  mean  velocity? 

—  Ans.  48  feet  per  second. 

2.  In  the  previous  question :  What  will  be  its  mean  velocity  during  the 
second  second  if  it  travel  16  feet  in  the  first  second  and  80  feet  in  the  third  ? 

—  Ans.  48  feet  per  second. 


13  DERIVED  NOTIONS.  [CHAP. 

3.  A  body  moves  with  a  uniform  velocity  of  40  miles  1600  yards  per 
hour :  what  is  its  velocity  in  feet  per  second ;  and  how  many  feet  will  it 
traverse  in  10  seconds  ?  —  Ans.  60  feet  per  second;  600  feet. 

4.  A  railway  train  explodes  two  detonating  signals  placed  on  the  rails  at 
a  distance  from  one  another  of  176  feet ;  an  interval  of  exactly  2  seconds 
elapses  between  the  explosions.     Compare  the  velocity  during  that  interval 
with  the  mean  velocity,  which  is  indicated  by  the  statement  that  the  train 
takes  an  hour  and  a  half  to  perform  the  journey  between  two  stations  45 
miles  distant  from  one  another.  —  Ans.  It  is  twice  the  mean  velocity. 

5.  Which  is  the  greater  velocity,  40  miles  an  hour  or  12  metres  per 
second  ? 

6.  A  train  travels  10  miles  at  a  velocity  of  20  miles  per  hour ;  then  4 
miles  at  an  average  rate  of  30  miles  per  hour ;  then  6  at  a  uniform  rate  of 
40  miles  per  hour ;  it  takes  1  mile  to  come  to  rest,  running  at  an  average 
speed  of  20  miles  an  hour ;  it  stands  for  7  minutes ;  it  starts  and  runs  for  20 
minutes  at  the  average  speed  of  21  miles  an  hour.     What  has  been  its  mean 
velocity?  —  Ans.  32  feet  per  second. 

Acceleration.  — When  the  rectilinear  velocity  v  of  a  moving 
body  varies,  the  Rate  of  Change  of  Velocity  is  called  its 
Acceleration.  In  popular  language  this  word  indicates  increase 
of  speed,  but  it  is  in  this  connection  used  to  signify  the  rate  of 
change  of  the  velocity,  whether  that  change  be  an  increase  or  a 
diminution.  If  a  body  be  moving  at  the  rate  of  ten  feet  a  second 
at  the  beginning  of  a  certain  second  of  time,  and  at  the  end  of 
that  second  be  found  to  be  moving  in  the  same  line  at  the  rate 
of  eleven  or  of  nine  feet,  it  is  said  that  its  motion  has  been 
accelerated  during  that  second,  positively  in  the  former  case, 
negatively  in  the  latter,  by  one  foot  per  second.  Acceleration 
is  usually  indicated  by  the  symbol  «,  and  the  Unit  of  Accel- 
eration is  the  acceleration  observed  when  a  body  alters  its 
velocity  in  a  given  direction  by  one  unit  of  velocity  every  second. 
A  body,  then,  which  has  its  velocity  in  a  given  direction  in- 
creased or  diminished  by  one  foot  in  one  second,  two  feet  in  two 
seconds,  and  so  forth,  is  undergoing,  in  that  direction,  a  unit 
acceleration,  in  British  units ;  if  by  one  cm.  per  sec.,  n  cm.  in  n 
seconds,  it  is  undergoing  a  unit  acceleration  (one  4  Spoud ')  in 
C.G.S.  units. 

The  initial  velocity  may  be  zero,  the  body  being  originally 
at  rest ;  in  such  a  case  the  body  will  undergo  unit  acceleration 
in  a  given  direction,  if  in  that  direction  it  acquire  unit  velocity 
in  one  second,  or  a  velocity  of  n  units  in  n  seconds. 

There  are  some  cases  in  which  the  apparent  effect  of  acceleration  is  to 
change  the  direction  of  motion ;  but  there  is  no  essential  difference  between 
such  cases  and  those  upon  which  the  definition  here  given  is  based ;  and  such 
a  result  will  be  readily  understood  after  we  have  discussed  the  Composition 
of  Velocities  and  of  Accelerations. 


ii.]  ACCELERATION.  19 

If  a  body  move  with  velocity  v0,  and  at  the  end  of  t  seconds 
with  velocity  vt,  the  total  change  of  velocity  during  t  seconds 
is  v,  —  V0,  the  time  during  which  this  change  is  effected  is  t, 

TT     "tr 

and  the  acceleration  per  second  is  — Q.     This  is  the  mean 

acceleration,  in  the  direction  of  motion,  during  the  time  t: 
and  the  acceleration  may  during  that  period  t  be  uniform  or 
variable,  but  an  approximation  to  its  value  at  any  instant  may 
be  found  by  making  the  interval  t  as  short  as  possible. 

Accelerations,  being  measured  by  the  velocities  imparted 
per  unit  of  time,  are  stated  in  terms  of  units  of  length  per 
second,  per  second. 

Problems.  —  1.  A  body  starts  from  rest  under  the  influence  of  a  force 
which  produces  acceleration  a  =  2  ft.-per-sec.  per  second :  when  will  it  have 
a  velocity  of  1000  feet  per  second?  —  Ans.  At  the  end  of  the  500th  second. 

2.  A  body  travels  at  12  feet  per  second ;  in  10  seconds  it  is  moving  7  feet 
per  second  :  what  is  the  mean  acceleration  ?  —  Ans.  —  |  ft.-per-sec.  per  second. 

3.  If  in  the  last  question  the  acceleration  had  been  +  £  ft.-per-sec.  per 
second,  what  would  have  been  the  rate  of  movement  at  the  end  of  10 
seconds?  —  -4ns.  17  feet  per  second. 

4.  A  body  moves,  in  the  first  second  during  which  it  is  under  observation, 
through  a  space  of  16  feet ;  in  the  fourth  second  through  112  feet :  what 
is  the  acceleration  per  second?  —  Ans.  32  ft.-per-sec.,  so  that  during  con- 
secutive seconds  it  moves  16,  48,  80, 112  feet.     At  the  end  of  each  successive 
second  it  moves  with  a  velocity  of  32,  64,  96, 128  feet  per  second  respectively. 

5.  A  body  as  it  moves  is  made  to  record  its  own  velocity  :  it  is  found 
that  at  a  certain  instant  it  is  moving  at  the  rate  of  112  feet  a  second;  after 
an  interval  of  ^V  second  its  velocity  is  114  feet  per  second  :  what  is  its  accel- 
eration ?  —  A  ns.  Vt  ~  v°  =  114~112  =  40  ft.-per-sec.  per  second. 

t  A 

6.  A  particle  moves,  during  the  first  second,  with  diminishing  velocity, 
at  the  mean  rate  of  10  centimetres  per  second ;  the  next  second  it  moves  at 
the  mean  rate  of  8  centimetres  per  second ;  the  acceleration  is  constant :  how 
far  will  it  travel,  and  what  will  it  do  when  it  has  come  to  rest?  —  Ans.  It 
will  go  on  for  5£  seconds,  will  traverse  30-25  centimetres,  and  will  return, 
arriving  at  every  point  on  its  previous  path  with  the  same  speed  as  that  with 
which  it  left  it,  and  will  retrace  the  30-25  centimetres  in  another  5£  seconds, 
passing  the  starting  point  with  a  reversed  velocity  of  11  centimetres  per  second. 

Momentum. —  When  a  body  whose  mass  is  m  moves  with  a 
speed  or  velocity  v,  it  is  said  that  the  total  Momentum  or  Quan- 
tity of  Motion  is  the  product  of  these  two  terms,  namely,  mv 
units  of  Momentum.  The  greater  the  velocity  or  the  greater 
the  mass  moved,  the  greater  the  quantity  of  motion.  The 
C.G.S.  unit  of  momentum  is  called  a  Bole. 

Force.  —  When  a  body  which  is  at  rest  is  set  in  motion,  or 
one  which  is  in  motion  is  accelerated  (positively  or  negatively) 


20  DERIVED   NOTIONS.  [CHAP. 

or  deflected  from  its  straight  course,  we  commonly  attribute 
these  effects  to  impressed  force,  or  simply  to  Force.  This  is 
sometimes  defined  as  any  Cause  which  tends  to  alter  a  body's 
state  of  rest,  or  of  uniform  motion  in  a  straight  line.  It  is  bet- 
ter defined  as  it  is  by  Newton,  not  as  a  cause,  an  existing  reality 
of  any  kind,  but  simply  as  an  observed  Phenomenon,  a  measurable 
Action  upon  a  body,  under  which  the  state  of  rest  of  that 
body,  or  its  state  of  uniform  motion  in  a  straight  line,  suffers 
change. 

The  presence  and  the  mutual  influence  of  at  least  two  bodies  are  always 
essential  to  the  production,  in  any  one  of  them,  of  those  effects  of  displace- 
ment which  we  commonly  attribute  to  Force ;  and  such  displacement  is 
always  associated  with  a  transformation  or  a  redistribution  of  Energy. 

Forces  considered  as  measurable  Actions  are  measured  by  the 
Masses  set  in  motion,  and  by  the  Velocities  imparted  to  them  in 
unit  of  time — that  is  to  say,  by  their  Accelerations;  and  the 
equation  F  =  ma  enables  us  to  measure  any  Force  F, 
as  the  product  of  a  Mass  m  into  the  Acceleration  a 
imparted  to  it,  as  found  by  observation. 

Any  observed  acceleration  must  necessarily  be  acceleration 
in  some  particular  direction,  at  any  rate  at  the  instant  of  obser- 
vation. We  shall  have  occasion  from  time  to  time  to  emphasise 
this  by  writing  a  as  a,  where  the  black-faced  type  draws  atten- 
tion to  the  fact  that  the  quantity  symbolised  is  a  directed 
quantity ;  and  when  we  do  so,  our  equation  becomes  F  =  ma ; 
i.e.  the  Force  in  a  particular  Direction  is  equal  to  the  Mass  m  x 
the  Acceleration  acquired  in  that  direction. 

Force  may  also  be  measured  as  an  Observed  Rate  of  Change  of  Momen- 
tum. Acceleration  a  —  rate  of  change  of  velocity  v;P  =  mxo  =  mx  rate 
of  change  of  v  =  rate  of  change  of  rav  =  rate  of  change  of  momentum.  A 
uniform  Force  is  therefore  numerically  equal  to  the  Amount  of  Momentum 
gained  or  lost  by  a  body  during  each  Second.  (See  further  p.  41  and  p.  47.) 

The  product  of  a  force  acting  into  the  time  during  which  it  acts  measures 
the  momentum  imparted  during  that  time ;  and  this  product  is  known  as 
Impulse  —  a  term  of  frequent  use  in  the  study  of  the  working  of  machinery. 

By  a  convenient  form  of  speech  a  given  Force  is  said  to  act 
u  pon  a  given  body  and  to  impart  to  it  a  given  acceleration.  It 
must  be  constantly  borne  in  mind,  however,  that  a  Force  is  not  a 
physical  entity,  and  the  word  Force  is  not  in  itself  an  explana- 
tion of  anything.  Force  can  never  be  measured  until  we  already 
know,  absolutely,  or  by  comparison,  the  mass  acted  upon  and  the 
acceleration  actually  imparted  to  it;  and  force  may  be  increased 
or  diminished  by  varying  the  arrangement  of  the  bodies  to  whose 


ii.]  FORCE.  21 

mutual  actions  it  corresponds ;  as  in  the  case  of  the  Hydraulic 
Press,  where  the  ordinary  action  presents  an  apparent  increase 
of  Force,  while  if  the  action  be  reversed,  Force  seems  to  be 
destroyed. 

If  a  body  weighing  three  pounds  be  set  in  motion  so  as  at  the  end  of  one 
second  to  have  had  a  velocity  of  four  feet  per  second  imparted  to  it,  then 
F  =  ma  =  3  x4  =  12Poundalsor  British  units  of  force.  The  same  force 
would  have  imparted  to  a  two-pound  mass  an  acceleration  of  six  feet,  to  a 
one-pound  mass  an  acceleration  of  twelve  feet,  and  to  a  twelve-pound  mass 
it  would  have  imparted  in  one  second  a  speed  of  one  foot  per  second. 

If  the  mass  moved  be  a  unit,  and  the  acceleration  acquired  by 
the  mass  be  unity,  the  product  ma  =  F  is  also  unity ;  and  hence 
the  Unit  of  Force  is  to  be  defined  as  that  observed  when  a  unit 
of  mass  is  found  to  acquire  unit  velocity  in  the  course  of 
one  second. 

It  will  be  observed  that  this  definition  of  the  Unit  of  Force  is  absolute, 
is  not  affected  by  local  variations  in  the  intensity  of  gravity,  and  is  hence 
everywhere  the  same. 

If  the  unit  of  length  chosen  be  the  centimetre,  and  the  unit 
of  mass  the  gramme,  the  Unit  of  Force  will  in  one  second 
cause  a  gramme-mass  to  acquire  a  velocity  of  one  centimetre  per 
second  ;  and  the  unit  of  Force  so  defined  is  called  a  Dyne.  Any 
force  may  be  stated  to  be  equal  to  so  many  dynes. 

One  million  dynes  make  one  Megadyne. 

Problems. 

1.  A  certain  force  acts  upon  m  units  of  mass  of  matter,  and  at  the  end 
of  a  second  that  mass  is  found  to  be  moving  with  a  velocity  of  32  feet  per 
second :  what  velocity  will  be  produced  if  the  same  force   act  upon  32  m 
units  of  mass  for  the  same  period?  —  Ans.  One  foot  per  second. 

2.  How  many  dynes  of  force  are  required  to  set  a  mass  weighing  50 
kilogrammes  in  motion  with  a  velocity  of  12  metres  per  second,  the  force 
being  supposed  to  act  for  precisely  one  second?  —  Ans.  60,000,000. 

3.  How  many  dynes  are  required  to  make  a  gramme-mass  move  with  a 
velocity  of  9-81  metres  per  second,  the  force  measured  in  dynes  being  sup- 
posed to  act  for  precisely  one  second  ?  what  if  it  act  for  two  seconds  ?  — 
Ans.  981  dynes;  490-5  dynes. 

4.  Compare  the  velocities  produced  by  the  action  on  masses  of  2  kilo- 
grammes, 750  grammes,  and  one  gramme  respectively,  of  forces  measuring 
respectively  300,000,  112,500,  and  150  dynes.  —  Ans.   All  equal;  150  centi- 
metres per  second  if  the  action  endure  for  one  second. 

5.  Equal  forces  act  upon  the  masses  specified  in  the  last  question :  what 
will  be  the  relative  accelerations  produced  ?  —  Ans.  3:8:  6000. 

Weight.  —  Experiments  made  near  the  earth's  surface  show 
that  every  mass  of  matter  acquires,  if  gravity  act  freely  upon 


22  DERIVED   NOTIONS.  [CHAP. 

it  for  one  second,  a  downward  velocity  of  nearly  981  cm.  (32-2 
feet)  per  second.  This  downward  acceleration  is  experimentally 
found  to  be  independent  of  the  nature  and  of  the  size  of  the 
falling  body;  but  it  is  not  the  same  at  all  parts  of  the  earth's 
surface,  for  between  the  Poles  and  the  Equator  it  presents  a 
difference  of  about  |  per  cent.  Whatever  be  the  local  value 
of  this  gravitational  acceleration  #,  we  find  that  the  equation 
F  =  ma  takes  the  form  G  =  mg,  where  G  is  the  local  Force  or 
measurable  Action  or  downward  Pull  of  Gravity  upon  a  body 
of  mass  m;  and  this  is  known  as  the  Weight  of  that  body. 
Necessarily,  if  the  downward  acceleration  g  differs  from  place 
to  place,  the  local  weight  of  a  given  mass  will  vary  in  like 
proportion. 

But  if  we  take  the  acceleration  due  to  terrestrial  gravita- 
tion as  g  =  981,  which  is  somewhat  nearer  its  Paris  than  its 
Greenwich  value  (see  p.  205),  the  Gravitational  Force  acting  on 
a  gramme-mass,  being  its  Mass  x  its  Acceleration,  is  (1  gramme 
x  981  units  of  acceleration),  or  981  Dynes.  The  Weight  of  a 
gramme-mass,  that  is  to  say,  is  equal  to  981  dynes ;  and  con- 
versely, a  C.G.S.  unit  of  Force,  one  Dyne,  is  equal  to  the 
Weight  of  1/981  gramme. 

Similarly,  if  we  use  British  units,  we  find,  writing  pound  for  gramme, 
and  32-2  feet  for  981  cm.,  that  the  Weight  of  a  pound-mass  is  equal  to  32-2 
British  units  of  Force,  or  Poundals ;  the  Poundal  is  therefore  nearly  equal 
to  the  Weight  of  half-an-ounce. 

Fig.2. 


B 

If  a  unit-mass  be  divided  into  two  parts,  one  of  which,  A,  weighs 
(g-l)/g  units,  and  the  other,  B,  l/g  unit,  and  if  the  weight  of  the  smaller 
be  employed  to  set  both  in  motion,  then  the  whole  mass  set  in  motion  is 
m  =  l,  and  the  Force  acting  is  the  Weight  of  l/g  unit-mass,  that  is,  l/g  x  g 
=  1.  Hence,  m  =  1,  F  =  1 ;  and  since  F  =  ma,  the  Acceleration  must  also 
be  unity,  and  at  the  end  of  a  second  the  whole  mass  would,  apart  from 
friction,  be  found  to  be  moving  with  unit  velocity. 

The  local  Weight  of  a  gramme-mass  is  called  the  local  Intensity 
of  Gravity;  and  this  is  equal  to  g  dynes. 

We  can  state,  then,  that  any  given  Force  is  equal  to  the 
Weight  of  so  many  units  of  mass  at  a  certain  definite  place. 


,,] 


WEIGHT.  23 


The  engineer's  unit  of  Force  is  the  Weight  of  1  Ib.  or  that  of  1  kilo- 
gramme. He  accordingly  speaks  of,  say,  a  "Force  of  8  Ibs.,"  where  the 
physicist  would  say  a  "  Force  equal  to  the  local  Weight,  at  some  particular 
place,  of  a  Mass  of  8  Ibs."  His  unit  of  force  is  therefore  a  variable  unit  : 
whereas  the  physicist's  unit  of  force  does  not  in  any  way  depend  on  local 
variations  in  the  force  of  gravity.  Again,  if  we  seek  the  analogue  of  the 
equation  F  =  mo,  we  find  it  to  take  the  form  P=ma/g:  and  this  is  awkward. 
Still,  if  properly  understood,  such  expressions  as  •'  a  Force  of  8  Ibs."  are 
compendious  and  not  wanting  in  convenience,  even  though  they  do  lead 
to  encumbering  some  of  the  engineer's  formulae  with  an  unnecessary  divisor 
g;  they  facilitate  the  immediate  expression  of  certain  results  in  foot-pounds 
or  in  kilogramme-metres  (p.  41)  ;  and  the  error  which  they  may  introduce, 
through  the  variability  of  the  engineer's  unit  of  force  from  place  to  place, 
is  practically  well  within  one-half  per  cent. 

Stress.  —  The  word  Force  is  limited  to  the  case  in  which 
some  movement  of  masses  or  of  particles  is  produced,  varied, 
or  checked:  what  is  popularly  known  as  the  force  tending  to 
bring  a  spring  back  to  its  original  form,  but  not  actually  doing 
so,  is  a  Stress;  and  the  condition  of  the  spring  under  such 
circumstances  is  a  Condition  of  Stress.  A  spring,  when  its 
form  is  altered,  tends  to  resume  its  original  form,  and  it  exerts 
a  pressure  or  a  pull  upon  any  object  so  placed  as  to  prevent  its 
doing  so  ;  but  this  object  also  exerts  continuously  an  equal  but 
opposed  pressure  or  pull  upon  the  spring.  This  mutual  pressure 
or  pull  will  cause  motion  if  the  bodies  pressed  upon  or  pulled 
become  free  to  move  ;  if  not,  the  pressure  or  pull  is  continuously 
applied  without  producing  movement,  and  such  an  inactive 
Mutual  Pressure  or  Pull  is  called  a  stress. 

In  popular  language  a  Stress  is  called  a  Strain,  as  where  it  is  said  that 
a  bridge  or  wire  being  exposed  to  too  great  a  strain  gives  way  and  breaks  or 
snaps.  Properly  the  word  Strain  means  Deformation  of  a  body. 

Every  such  stress  implies  at  least  two  fixed  points  ;  these  are  either 
pressed  together,  or  else  the  material  stretched  between  them  is  in  a  condi- 
tion of  tension.  In  the  former  case,  when  the  condition  of  stress  ceases,  the 
body  previously  compressed  expands  ;  in  the  latter  case,  when  set  free  it  con- 
tracts. If  both  ends  of  a  stretched  body  be  simultaneously  liberated,  the 
resultant  movement  is  towards  the  centre  ;  if  one  end  only  be  set  free,  the 
movement  is  towards  the  end  which  remains  fixed  ;  and  conversely  for  a 
body  exposed  to  compression; 

Stress  therefore  always  implies  mutual  Action  and  Reaction; 
and  we  might,  with  Tait,  paraphrase  Newton's  third  Law  thus  : 
"  Every  action  between  two  bodies  is  a  Stress."  A  stress  is 
always  numerically  equal  to  either  the  Action  or  the  Reaction,  as 
also  to  the  Force  which  is  necessary  to  produce  it,  or  to  that 
which  is  developed  when  the  condition  of  stress  comes  to  an  end. 
Stress  cannot  be  said  to  be  either  positive  or  negative  in  the  line 


24  DERIVED  NOTIONS.  [CHAP. 

of  its  application,  for  it  depends  on  extraneous  circumstances 
which  point  or  part  of  the  stressed  body  shall  be  set  free,  and 
therefore  what  shall  be  the  direction  of  the  resultant  movement; 
but  it  has  a  numerical  magnitude,  for  it  can  be  measured  in 
dynes ;  and  it  may  be  numerically  specified  either  (1)  as  Total 
Stress  F  or  (2)  as  Stress  /  per  Unit  of  Area  of  the  common 
bounding  surface  between  two  bodies  under  mutual  Action  and 
Reaction. 

In  the  first  sense,  Stresses  or  total  stresses  may  be  measured 
as  equal  to  the  Forces  which  produce  them.  A  spring  is  pressed 
upon  by  a  certain  known  weight ;  it  yields  to  a  certain  extent ; 
it  is  then  caught  by  a  ratchet,  and  the  weight  is  removed.  The 
Force  necessary  to  cause  the  given  yielding  of  the  spring  is 
known,  for  it  is  the  numerical  value,  in  dynes,  of  the  Weight  of 
the  mass  employed;  the  Total  Stress  established  in  the  spring 
is  numerically  equal  in  dynes  to  the  Force  used.  The  whole 
upward  pressure  of  the  spring  on  the  ratchet  must  be  numeri- 
cally equal  to  the  weight  of  the  mass  removed ;  so  must  the 
downward  pressure  of  the  ratchet  on  the  spring.  The  opposite 
extremity  of  the  spring  imposes  an  equal  and  downward  pres- 
sure on  its  support,  opposed  to  which  is  an  equal  upward 
pressure  of  the  support  upon  the  spring. 

In  the  second  sense,  the  numerical  value  of  the  stress  (i.e. 
Stress  per  Unit  of  Area)  is  obtained  by  dividing  the  total  stress 
(in  dynes)  by  the  area  (in  sq.  cm.)  over  which  it  is  distributed; 
and  this  is  otherwise  known  as  the  Intensity  of  the  Stress. 

In  the  sequel  we  shall,  except  where  the  context  makes  it 
plain,  avoid  the  use  of  the  unqualified  Avord  Stress,  and  shall 
endeavour  to  make  it  clear  whether  in  any  particular  instance 
we  refer  (1)  to  a  Condition  of  Stress,  (2)  to  a  Total  Stress  of 
so  many  dynes,  or  (3)  to  a  Stress  of  so  many  dynes  per  square 
centimetre. 

Pressure.  —  Suppose  a  heavy  slab  of  iron  weighing  100  kilo- 
grammes to  be  laid  upon  a  flat  slab  of  indiarubber  of  sufficient 
size ;  and  let  its  under  surface  be  flat  and  have  an  area  of  1000 
sq.  cm.  Its  total  weight  is  G  =  100,000  grms.  x  981  =  98,100,000 
dynes  ;  and  this  is  distributed  over  the  underlying  surface  of  the 
indiarubber  as  a  Total  downward  Pressure  P  of  98,100,000  dynes. 
When  the  arrangement  is  that  specified,  the  indiarubber  suffers 
(over  1000  sq.  cm.)  a  downward  pressure  p  =  98,100  dynes  on 
each  sq.  cm.  acted  upon ;  but  it  exerts  on  the  iron  an  upward 
pressure  of  equal  amount,  for  the  pressure  is  mutual. 


ii.]  PRESSURE.  25 

Now  let  the  metal  slab  be  mounted  on  four  legs,  whose  joint 
cross-area  is,  say,  20  sq.  cm. ;  and  let  the  whole  again  stand  upon 
the  indiarubber.  The  total  pressure  P  is  the  same  as  at  first ; 
but  it  is  now  distributed  over  an  area  of  only  20  sq.  cm.,  for 
which  reason  the  indiarubber  and  the  metal  are  now  subject 
to  a  mutual  pressure  p'  =  4,905,000  dynes  per  sq.  cm.  across 
the  area  of  contact. 

Here,  therefore,  we  have  again  a  number  of  different  mean- 
ings. The  word  Pressure  may  mean :  — 

(1.)  Between  two  objects  having  a  common  bounding  sur- 
face, a  Total  Mutual  Pressure  P  of  so  many  dynes  over 
that  whole  surface,  and  at  right  angles  to  that  surface. 

(2.)  From  the  point  of  view  of  one  of  the  objects,  the  T o  tal 
Pressure  P  suffered  by  it  or  exerted  by  it  across  the  whole 
area  of  mutual  contact  and  at  right  angles  to  that  surface ;  the 
same  number  of  dynes  over  the  whole  area  as  in  the  preceding 
case. 

(3.)  The  Mutual  Pressure  per  Unit  of  Area  of  the 
common  surface  and  at  right  angles  to  that  surface.  (The 
"  Intensity  of  Pressure,"  p  dynes  per  sq.  cm.) 

(4.)  The  Pressure  of  the  one  body  on  the  other  across  the 
common  bounding  surface,  measured  in  dynes  per  sq.  cm.,  or 
"Barads,"and  at  right  angles  to  that  surface  (Pressure  of  A 
on  B,  or  of  B  on  A  per  Unit  of  Area). 

(5.)  In  the  interior  of  a  mass  subjected  to  a  uniform  stress 
of  so  many  dynes  per  unit  area  of  its  bounding  surface,  acting 
inwardly,  there  is  a  mutual  pressure  of  all  the  parts  of  the  mass 
upon  one  another,  in  all  directions:  this  is  an  undirected  or 
Hydrostatic  Pressure;  and  if  the  applied  stress  or  pressure 
be  p  dynes  per  sq.  cm.,  at  right  angles  to  the  bounding  surface, 
this  hydrostatic  pressure  is  equal  to  p  =  p  dynes  per  sq.  cm. 
across  any  plane  arbitrarily  chosen  in  the  mass  considered. 

The  distinction  between  P,  the  Total  Pressure,  and  p,  the  Intensity  of 
Pressure,  is  a  matter  of  importance.  Compare  a  railway  tie  or  sleeper,  on 
which  the  rail  rests  by  a  wide  chair-bearing,  with  one  on  which  the  rail  rests 
by  a  narrow  bearing ;  when  the  train  crosses  the  latter,  the  whole  pressure  is 
borne  by  a  small  area,  and  the  sleeper  may  give  way  locally.  A  knife  or  a 
chisel  is  an  instrument  for  producing  great  intensity  of  pressure ;  when  we 
"  sharpen  "  it,  we  diminish  the  area  of  its  edge. 

We  shall  endeavour  to  make  it  consistently  clear  in  which 
sense  the  word  Pressure  is  used  in  each  particular  case*' 

Tension.  —  If  a  mass  of,  say,  100  kilogrammes  be  hung  by 
a  metallic  rod  of  1  sq.  cm.  cross-section,  the  metallic  rod  is  under 


26  DERIVED   NOTIONS.  [CHAP,  n.] 

longitudinal  tension  amounting  to  98,100000  dynes  across  that 
one  sq.  cm.  of  cross-section.  If  the  same  mass  had  been  sus- 
pended by  a  metallic  rod  of,  say,  10  sq.  cm.  cross-section,  the 
tension  would  have  amounted  to  9,810000  dynes  per  sq.  cm.  of 
cross-sectional  area.  In  both  these  cases  the  Total  Tension  is 
equal  to  the  Weight  of  the  mass  suspended,  viz.  98,100000 
dynes  ;  and  it  is  irrespective  of  the  transverse-sectional  area  of 
the  rod  which  is  being  acted  upon.  Here  again  we  have  thus  to 
distinguish  between  a  Total  Tension  (of  T  dynes)  and  a  Tension 
per  Unit  of  cross-sectional  Area  (T/Area  =  t  dynes  per  sq.  cm.) ; 
and  here  again  we  shall  have,  in  the  sequel,  to  make  it  plain  to 
which  of  these  reference  is  being  made  in  any  particular  case. 

As  a  rule  the  phrase  The  Tension  of  a  Cord  is  supposed  to  mean  the 
Total  Tension  T  acting  across  any  transverse-section  of  a  cord  ;  this  is  the 
same  at  all  parts  of  a  cord  stretched  between  two  points,  whatever  may  be 
the  local  variations  of  the  thickness  of  that  cord.  Pretty  obviously  the 
thinnest  part  of  a  cord  thus  stretched  between  two  supports  is  exposed  to 
the  greatest  tension  per  unit  of  cross-sectional  area;  for,  the  Total  Tension 
being  uniform,  it  necessarily  follows  that  where  the  cross-area  is  least,  there 
the  Tension  per  Unit  of  Area  (i.e.  the  quotient  (Total  Tension  -f-  Cross- Area), 
otherwise  known  as  the  Intensity  of  Tension  or  the  Traction  t)  is  the 
greatest ;  and,  accordingly,  a  stretched  string  is  most  liable  to  snap  where  it 
is  thinnest. 

It  is  scarcely  necessary  here  to  point  out  for  the  sake  of  clearness  that 
there  are  three  other  distinct  meanings  of  the  same  word  Tension,  which 
will  duly  come  up  in  their  respective  places.  These  are :  (1)  the  Surface- 
tension  of  a  Liquid,  p.  272 ;  (2)  Electric  Tension,  a  name  given  to  the  self- 
repulsion  of  electrified  surfaces,  p.  582 ;  (3)  "  in  Tension,"  an  old-fashioned 
and  obsolete  phrase  denoting  a  certain  arrangement  of  cells  in  a  galvanic 
battery,  p.  640. 


CHAPTER  III. 

MEASUREMENTS. 

IN  the  foregoing  chapters  we  have  become  acquainted  with  the 
units  of  Space,  Time,  and  Mass,  and  with  those,  derived  from  the 
preceding,  of  Velocity,  Acceleration,  Momentum,  and  Force; 
and  it  is  for  us  now  to  ascertain  what  principles  are  made  use  of 
in  the  various  measurements  effected  in  terms  of  these  units. 

In  the  Measurement  of  Lengths  two  main  methods  are 
resorted  to  —  Line  measurement  (mesure  a  traits)  and  End 
measurement  (mesure  a  bouts').  The  former  is  the  method  in 
habitual  use  among  carpenters,  who  lay  off  so  many  feet  and 
inches  by  the  aid  of  their  pocket-rule ;  the  latter  is  the  method 
which  they  use  when  they  measure  the  width  of  a  cavity  by 
means  of  a  pair  of  callipers,  which  they  open  until  it  exactly 
fits  the  space. 

Line  Measurement.  —  The  length  of  any  line  may  be  measured  by  a 
graduated  scale,  which  may  be,  like  the  carpenter's  pocket-rule,  somewhat 
roughly  graduated ;  or  instruments  on  the  same  principle  may  be  made  use 
of,  which  are  finely  and  very  accurately  divided.  The  measurement  is  effected 
by  observing  the  nearest  coincidence  of  the  marks  on  the  scale  with  the 
length  of  the  object,  and  by  then  reading  off  the  value  on  the  scale.  This 
is  a  familiar  operation,  but  it  will  be  observed  that  it  depends  on  the 
accuracy  of  the  sight.  The  eye  requires  to  be  held  directly  first  over  one, 
then  over  the  other  end  of  the  object  to  be  measured,  together  with  the  cor- 
responding part  of  the  scale ;  for  the  true  coincidences  would  be  disturbed  if 
the  scale  and  object  were  looked  at  obliquely.  It  is  found  that  in  estimating 
measurements  which  are  very  small  the  eye  gets  confused ;  and  besides  this, 
the  difficulty  of  making  accurately  and  very  finely  graduated  scales  increases 
with  the  accuracy  required.  Thus  it  happens  that,  while  the  ordinary  24- 
inch  rule  divided  into  various  fractions  of  an  inch,  or  Whitworth's  very 
convenient  20-inch  rule,  decimally  divided,  or  a  measure  divided  to  half  or 
quarter  millimetres,  may  be  used  for  measurements  involving  differences  of 
the  hundredth  of  an  inch,  it  is  only  with  difficulty  that  they  can  so  be 
applied,  and  it  is  much  more  convenient  in  cases  involving  minute  measure- 
ments, such  as  observations  of  the  height  of  the  barometer  or  thermometer, 
to  use  a  contrivance  called  a  Vernier.  This  is  a  subsidiary  scale  which 
slides  up  and  down  past  the  main  scale,  and  is  differently  divided  from  it. 

27 


28 


MEASUREMENTS. 


[CHAP. 


Fig.S. 


Barometer 


Sextant 


30 


A  — 


—     6 


There  are  two  kinds  of  Vernier  in  use,  those  known  as  the  Barometer- 
Vernier  and  the  Sextant-Vernier,  which  require  separate  descriptions. 
The  Barometer- Vernier  is  thus  graduated  :  a  line  is  set  off  on  the  vernier 
equal  to  eleven  divisions  on  the  main  scale.  This  line  is  divided  into  ten 
equal  parts.  Each  of  these  parts  is  therefore  equal  to  ITL  division  on  the 
main  scale.  If  the  main  scale  be  divided  to  tenths  of  an  inch,  the  difference 
between  a  division  on  the  main  scale  and  one  on  the  vernier  is  T^7  inch. 
Suppose  the  object  measured  to  be  more  than  29-5  inches  and  less  than  29-6 
inches  on  the  scale.  The  zero  of  thevernieris  laid,  as  exactly  as  possi- 
ble, opposite  to  the  point  whose  position  is  to  be  found,  the  extremity 

of  the  object  to  be  measured,  or  the  height 
of  the  mercurial  column  in  the  barome- 
ter ;  then  on  looking  down  the  vernier  it 
will  be  found  that  at  some  point  there  is 
a  coincidence  between  a  graduation  mark 
on  the  vernier  and  one  on  the  main  scale. 
The  number  of  that  mark  on  the  vernier 
is  noted,  and  that  is  the  figure  required 
in  the  second  place  of  decimals.  For 
example,  let  the  point  A  be  above  29-5, 
below  29-6  inches ;  the  zero  point  0  of  the 
vernier  is  brought  opposite  to  it:  the 
point  6  of  the  vernier  coincides  with  a 
division  of  the  main  scale ;  the  length  is 
29-56. 

In  the  Sextant- Vernier,  which  is  the 
form  more  usually  found  in  instruments 
of  Continental  make,  the  divisions  on  the 
vernier  run  in  the  same  direction  as  those 
on  the  main  scale.  A  line  is  set  off  on 
the  vernier  equal  to  nine  divisions  of  the 
main  scale  ;  this  is  divided  into  ten  parts, 
each  of  which  is  equal  to  nine-tenths  of 
a  division  of  the  main  scale.  In  the  same 
way  the  vernier  is  moved  until  its  zero  point  is  brought  opposite  the  end  of 
the  object  to  be  measured,  and  the  mark  up  the  vernier  which  first  coincides 
with  a  division-mark  on  the  main  scale  gives  the  figure  as  before. 

Frequently,  as  in  the  sextant,  a  little  magnifying  glass  is  so  placed  that 
the  zero  point  of  the  vernier  may  be  by  its  aid  brought  more  accurately 
opposite  the  object  to  be  measured.  When  still  greater  accuracy  is  required, 
a  microscope  is  so  placed  as  to  ensure  the  greatest  possible  completeness  of 
coincidence  between  the  zero  point  of  the  vernier  and  the  end  of  the  object 
to  be  measured,  both  of  which  are  simultaneously  brought  into  the  centre 
of  the  field. 

The  Cathetometer  is  an  instrument  whereby  vertical  heights  are 
measured.  It  consists  of  a  vertical  rod  on  which  a  -finely-graduated  scale 
is  engraved.  This  carries  a  sliding  piece  to  which  is  attached  a  telescope. 
In  this  telescope  is  fixed  a  pair  of  spider  threads  or  fine  platinum  wires 
arranged  at  right  angles  to  one  another,  and  so  placed  (in  the  focus  of  the 
eyepiece)  as  to  be  visible  simultaneously  with  the  object  looked  at  through 
the  lenses.  The  telescope-carrier  is  placed  in  such  a  position  on  the  vertical 
rod  that  the  lower  end  of  the  object  to  be  measured  is  seen,  when  looked  at 


29 


10 


in.]  LENGTH.  29 

through  the  telescope,  to  coincide  exactly  with  the  point  of  crossing  of  the 
spider-threads  in  the  field  of  view ;  then  it  is  slid  up  until  the  upper  end 
of  the  object  to  be  measured  appears  to  coincide  with  the  same  point ;  the 
distance  along  which  the  carrier  has  been  slid  along  the  vertical  rod  indicates 
the  height  of  the  object  to  be  measured.  Provision  must  be  made  in  the 
construction  of  the  apparatus  for  ensuring  that  the  vertical  rod  is  quite  per- 
pendicular to  the  horizon ;  this  is  effected  by  making  it  stand  upon  three 
screws  whose  heights  can  be  adjusted  until  a  spirit-level  shows  the  base  of 
the  apparatus  to  be  quite  horizontal. 

It  may  be  necessary  to  compare  a  standard  measure  with  the  length  of  a 
body  which  is  very  nearly  of  the  same  length  as  the  standard.  In  this  case, 
a  microscope  may  be  placed  at  each  end.  Coincidence  as  perfect  as  possible 
is  established  between  the  images  of  the  object  and  the  standard  measure  in 
the  field  of  the  first  microscope.  If,  then,  the  coincidence  be  perfect  in  the 
field  of  the  second  microscope,  the  object  is  of  precisely  the  same  length  as 
the  standard.  This  but  rarely  occurs,  and  the  object  in  view  frequently  is 
to  ascertain  what  the  error  amounts  to.  The  second  microscope  is  provided 
with  spider  threads  in  the  focus  of  the  eyepiece,  and  the  end  of  the  object 
is  brought  exactly  under  the  apparent  crossing-point  of  these  threads ;  then 
the  microscope  is  moved  along  until  the  end  of  the  standard  appears  to  be  in 
the  same  position  ;  the  extent  to  which  the  microscope  has  been  moved  indi- 
cates the  difference  between  the  two  lengths  compared.  Since,  however,  the 
amount  to  which  the  microscope  has  been  moved  may  be  exceedingly  small 
and  difficult  to  measure,  the  methods  hitherto  described  may  be  insufficient 
in  accuracy,  and  we  have  to  resort  to  those  more  delicate  devices  which 
depend  on  the  properties  of  the  Screw. 

The  Screw,  as  will  be  seen  on  examination  of  any  specimen,  presents 
a  spiral  coiled  round  a  cylinder.  If  a  screw,  having  twenty  threads  to  the 
inch,  be  inserted  in  a  fixed  body,  and  turned  round  exactly  once,  its  point 
will  have  advanced  the  twentieth  part  of  an  inch.  If  the  head  of  the  screw 
be  connected  with  a  pointer  fixed  on  it  at  right  angles,  which  can  indicate 
on  a  graduated  circle  the  amount  of  rotation  of  the  screw,  there  will  be  no 
difficulty,  even  with  roughly  made  apparatus,  in  causing  the  screw  to  execute 
a  rotation  of  half  a  circle,  a  quadrant,  45°,  5°,  or  even  1°.  If  a  screw,  then, 
which  has  twenty  turns  to  the  inch  be  turned  through  one  degree  (3^ 
of  a  complete  turn),  its  point  will  have  advanced  or  been  retracted  by 
sio  x  sV  =  7200  inch.  But  this  is  rough  measurement.  By  making  the 
head  of  the  screw  part  of  a  large  wheel  with  graduated  circumference  and 
using  a  fixed  vernier,  rotation  of  the  screw  to  the  extent  of  half-a-minute  of 
arc  can  be  easily  observed,  and  this  would  correspond  to  onward  motion  on 
the  part  of  the  point  of  the  screw  of  y^V^o-  inch.  The  principle  of  the 
screw  thus  enables  us  to  detect  and  to  measure  very  small  quantities  of 
motion.  If  the  second  microscope  in  the  last  paragraph  be  connected  with 
a  graduated  screw  of  this  kind,  the  amount  of  its  motion,  indicating  the 
difference  of  length  of  the  two  objects  measured,  can  be  very  exactly 
determined. 

In  Sir  Joseph  Whitworth's  measuring  machines  advantage  is  taken  of 
another  principle  for  producing  and  measuring  very  slight  motion.  The 
screw  (twenty  threads  to  the  inch)  is  driven  by  a  "  worm-wheel,"  a  wheel 
bearing  200  teeth  on  its  circumference  :  this,  is  propelled  by  a  tangent-screw, 
a  screw  whose  threads  fit  between  the  teeth  of  the  worm-wheel :  each  turn 
of  the  tangent-screw  sends  each  tooth  of  the  worm-wheel  forward  into  the 


30  MEASUREMENTS.  [CHAP. 

position  previously  occupied  by  the  tooth  immediately  before  it  —  that  is  to 
say,  causes  the  worm-wheel  itself  to  revolve  through  the  two-hundredth  part 
of  360°,  and  to  press  the  point  of  the  screw  forward  by  the  four-thousandth 
of  an  inch.  But  the  tangent-screw  is  itself  driven  by  a  wheel  divided  into 
250  parts,  so  that  if  this  wheel  be  turned  round  only  one  division,  the 
tangent  screw  is  rotated  ^  of  a  turn,  and  the  point  of  the  "  wormrwheel " 
screw  is  thus  pressed  forward  the  250th  part  of  ?oW>  *•«•  the  millionth  part 
of  an  inch. 

If  a  screw  be  fixed  at  each  end  so  that  it  can  rotate  but  not  progress, 
the  "  thread  "  of  the  screw  will  appear  to  travel  when  the  screw  itself  is 
turned.  If  any  object  (the  slide-rest  of  a  lathe,  or  the  like)  have  a  female 
screw*  cut  in  it,  and  be  by  means  of  that  screw  fitted  upon  a  rotary  but 
otherwise  fixed  male  screw ;  and  if  it  be  then  placed  between  guides  so  as 
to  be  free  to  move  backwards  and  forwards  along  the  fixed  screw  but  in  no 
other  direction :  if  then  the  fixed  screw  be  rotated,  the  object  borne  by  it 
will  travel  along  it  in  one  direction  or  the  other,  according  to  the  sense  of 
the  rotation.  This  mechanism  will  be  thoroughly  understood  on  looking 
at  the  traversing-screw  and  slide-rest  of  a  lathe.  If  the  travelling  carrier 
bear  a  pencil  or  a  diamond,  and  mark  paper  or  glass  at  equal  intervals,  as 
indicated  by  equal  rotations  of  the  driving  wheel,  we  shall  have  a  con- 
trivance illustrating  the  main  principle  of  the  Dividing  Engine  which 
is  used  for  graduating  thermometer-tubes,  etc. 

End  Measurement.  —  If  a  couple  of  rods,  exactly  ten  feet  in  length, 
be  placed  on  the  ground  end  to  end ;  if  then  the  first  rod  be  taken  up  and 
carefully  laid  down  endways  at  the  other  end  of  the  second  ;  and  if  the  second 
be  taken  up  and  placed  in  the  same  way  beyond  the  first  and  just  in  contact 
with  it,  and  so  on :  then  a  very  accurate  setting  off  of  any  multiple  of  ten 
feet  can  be  easily  effected,  provided  that  the  rods  themselves  be  exactly 
ten  feet  long.  Measurement  of  a  given  length  can  also  be  thus  effected : 
if  there  be  an  odd  number  of  feet  and  inches,  they  can  be  measured  by  a 
set  of  smaller  rods,  or  by  an  ordinary  tape  measure. 

In  measuring  or  setting-off  in  this  way,  it  is  plain  that  we  depend  upon 
the  sense  of  touch  for  the  perception  of  the  contacts  set  up  between  the 
ends  of  the  rods.  The  sense  of  touch  is  found  to  give  more  satisfactory 
results  in  many  ways  than  the  sense  of  sight;  for  if  one  object  be  intended 
to  fit  into  another,  and  have  a  diameter  ^^  inch  less  than  what  is  exactly 
necessary,  its  fit  will  be  perfectly  loose.  The  eye  could  not  perceive  this 
directly  without  the  intervention  of  lenses. 

The  Callipers  used  by  carpenters  can  be  opened  out  so  as  exactly  to 
fit  into  a  cavity,  or  exactly  to  grasp  an  object.  They  are  usually  made  so 
that  the  one  end  serves  for  inside,  the  other  for  outside  measurement.  They 
are  useful  in  comparing  the  dimensions  of  objects  which  should  be  of  the 
same  size ;  but  it  is  difficult  to  take  very  accurate  measurements  off  a  scale 
with  them. 

Gauges  are  made  of  known  sizes,  and  the  size  of  the  object  to  be 
measured  is  compared  with  that  of  the  gauge  by  trying  the  fit.  If  the 
gauge  be  made  conical,  then  from  the  extent  to  which  it  penetrates  a  given 
aperture  can  the  width  of  that  aperture  be  determined. 

*  A  screw  cut  out  of  a  solid  mass,  through  which  another  screw,  the  "male," 
passes.  In  the  ordinary  nut  and  bolt,  the  bolt  bears  the  male  screw,  the  nut  the 
female. 


III.] 


LENGTH. 


31 


The  Spherometer  consists  of  a  disc  of  metal  with  graduated  circum- 
ference. This  is  supported  on  three  equal  legs,  which  are  furnished  with 
hard  steel  points,  equidistant,  and  rounded  off  so  as  not  to  pierce  any  object 
on  which  the  instrument  is  set.  In  the  axis  of  it  is  a  screw,  the  steel  point 
of  which  is  also  rounded,  and  forms  a  fourth  foot.  Any  instrument  which 
stands  on  three  feet  is  certain  to  be  steady,  because  three  points  are  always 
in  some  one  plane  :  one  which  stands  on  four  feet  will  only  be  steady  if  the 
point  of  the  fourth  foot  be  exactly  in  the  same  plane  as  the  other  three. 
If  it  be  above  this  plane  the  instrument  does  not  rest  on  it  at  all ;  if  it  be 
below  this  plane  the  instrument  can  never  stand  on  more  than  three  feet  at 
a  time,  and  may  be  rocked  from  one  set  of  three  to  another.  If  the  sphe- 
rometer  be  set  upon  a  piece  of  glass,  it  will  stand  steadily;  if  the  central 
screw  be  turned  so  as  to  bring  down  the  fourth  foot,  the  instrument  will  be 
easily  rocked  if  it  be  brought  down  too  far.  The  hand  in  perceiving  and 
the  ear  in  hearing  this  rocking,  just  at  its  commencement,  concur  in  detect- 
ing very  small  motions  of  the  screw  just  at  that  part  of  its  movement. 
The  instrument  also  becomes  easy  to  spin  on  its  centre-screw.  By  means 
of  a  pointer  attached  to  the  head  of  the  screw  the  exact  position  of  the 
screw  which  corresponds  to  the  commencement  of  rocking  can  be  observed 
on  the  graduated  scale.  Suppose  the  thickness  of  a  piece  of  microscopic 
cover  glass  is  to  be  determined.  It  is  placed  under  the  fourth  foot.  This 
central  foot  of  the  spherometer  is  brought  down  upon  it  until  the  whole 
rocks;  the  central  screw  is  then  raised  until  the  rocking  ceases;  it  is  turned 
back  again  till  it  just  commences,  and,  as  before,  the  position  of  the  screw 
corresponding  to  the  commencement  of  rocking  can  be  observed  by  means 
of  the  pointer  and  the  graduated  scale.  If  the  pointer  had  stood  at  75° 
when  the  instrument  stood  on  the  plain  glass,  and  at  3°  when  the  central 
point  was  on  the  piece  of  thin  glass,  the  difference  of  position  of  the  pointer 
corresponds  to  72°,  or  /^  of  the  circumference ;  and  if  the  screw  itself  have 
twenty  turns  to  the  inch,  the  thickness  of  the  glass  is  ^  x  ^o  =  -reo  inch. 

The  curvature  of  a  lens  may  be  determined  by  this  instrument,  for  if 
the  lens  ABD  be  placed  under  a  spherometer,  Fig.  4  shows  that  the  amount 
of  curvature  determines  the  F1  4 

length  of  the  line  DE ;  and 
the  radius  r  of  the  sphere 
of  which  the  lens  may  be  — — 
considered  a  part  is  related 
to  the  line  DE  (represented 
by  /)  and  the  distance  a 
between  the  equidistant 
tripod  feet,  by  the  formula 
2r  ^  ^/3T.  +  I 

In    W  h  i  t  w  o  r  t  h  '  s 
Measuring    Engine    a    A  s  ^> B 

bar  representing    the  unit 

of  length  is  placed  between  two  jaws,  which  are  made  to  move  towards  one 
another  so  as,  without  pressure,  just  to  grasp  it :  they  are  then  separated 
from  one  another,  and  the  standard  unit  removed :  the  bar  to  be  measured 
is  placed  instead  of  it,  and  the  jaws  are  again  brought  together  so  as  to 
grasp  it  in  the  same  way.  The  jaws  are  brought  together  by  fine  screw 
adjustments,  such  as  those  previously  described,  so  that  the  difference  of 
the  millionth  part  of  an  inch  in  two  bars  of  metal  can  be  detected.  The 


32 


MEASUREMENTS. 


[CHAP. 


precise  position  at  which  the  jaws  grasp  objects  without  pressure  is  deter- 
mined by  a  plane  piece  of  metal,  which  is  included  along  with  them  between 
the  jaws,  with  its  edges  in  a  vertical  plane.  If  the  grasp  be  too  loose,  this 
piece  of  metal  can  be  moved  freely,  and  will  fall  back  when  lifted  and  let 

go;    if    the    grasp   be    too 
Fig.5. 

A 


tight,  this  metal  plane  can- 
not be  moved;  if  it  be 
exact,  the  metal  plane  can 
be  raised,  and  will  remain 
in  any  position  in  which  it 
may  be  placed. 

Another  plan  by  which 
an  alteration  in  the  length 
of  a  bar  may  be  determined 
is  the  Optical.  The  end 
A  of  a  bar  AB  rests  against 
a  strong  framework  at  B, 
so  that  any  alteration  in  its 
length  may  only  affect  the 
position  of  the  point  A. 
At  A  the  bar  is  in  contact 
with  a  lever  CD,  jointed  at  E,  and  bearing  a  mirror  at  D.  A  lamp  at  X 
casts  a  ray  of  light  on  the  mirror ;  this  is  reflected  to  a  screen  SS'.  If  B  A 
alter  in  length,  or  if  another  bar  of  slightly  different  length  be  substituted 
for  it,  the  bar  CD  assumes  another  position,  and  the  spot  of  light  on  the 
screen  SS'  is  deflected.  From  the  amount  of  deflection  may  be  calculated 
the  alteration  in  length  of  the  bar  BA. 

Good  linear  measurement,  in  whatever  way  effected,  ought 
to  present  an  error  less  than  YOTOQ-  ^'  °r  one"m^^on^-Q  °^  the 
whole. 

Measurement  of  Surface.  —  If  a  surface  be  bounded  by 
straight  lines  at  right  angles  to  one  another,  the  parallelogram 

mav  be  measured  by 
the  product  of  two 
adjacent  sides :  if  it 
be  of  any  other  form 
bounded  by  straight 
lines,  it  can  be  broken 
up  into  triangles,  and 
its  area  be  found  by  the 
rules  of  trigonometry : 
if  its  boundary  be  a 
regular  curve,  its  area 
can  generally  be  found :  but  if  the  surface  be  bounded  by  an 
irregular  curve,  the  determination  of  the  area  involves  the 
following  principle. 

Let  the  figure  YXO  be  bounded  by  the  two  rectangular 


Fig.6. 


a  b  c  de 


III.]  1 


SURFACE. 


33 


straight  lines  OY  and  OX,  and  the  curve  YABCDEX.  Find  its 
area.  Draw  a  series  of  lines  parallel  to  OY ;  these  will  cut  the 
curve  in  the  points  A,  B,  C,  D,  E,  and  so  forth.  Then  the  area 
YXO  is  divided  into  a  number  of  narrow  parallelograms,  O  YAa, 
AabB,  BfoC,  etc.  Each  of  these  is  equal  to  the  product  OY  x  Oa, 
aA  x  ab,  etc. :  these  being  all  found  and  added  together  give  the 
area  of  the  surface. 

If  now  the  surface  be  completely  bounded  by  an  irregular 
curve,  as  in  Fig.  7,  the  area  ABCDEA  is  first  found  by  the 
above  method,  then  the  area  A5CDEA.  The  difference  be- 
tween these  represents  the  area  of  the  curved  surface  ABCb. 
This  method  is  very  dif- 
ficult in  actual  practice, 
but  all  the  mathematical 
methods  of  integration 
are  based  upon  this  prin- 
ciple. For  actual  work 
a  convenient  means  of 
measurement  of  surface, 
which  gives  very  fair  re- 
sults, and  which  is  spe- 
cially useful  in  those 
cases  in  which  mechan- 
ical contrivances  have 
registered  their  own  per- 
formances on  paper,  is 
the  following :  — 

The  paper  on  which  the  curve  is  drawn  is  laid  on  a  flat 
board,  arid  the  outline  of  the  surface  very  carefully  traced  by  a 
sharp-pointed  penknife,  so  as  to  cut  out  the  part  of  the  paper 
bounded  by  that  outline :  this  is  then  weighed  and  its  weight 
compared  with  that  of  a  standard  area,  say  a  square  inch  of  the 
same  paper.  This  method  is  not  unexceptionable,  but  it  often 
gives  a  very  useful  approximation  to  the  value  required. 

An  instrument  called  a  plani meter  is  also  used  for  this 
purpose. 

Measurement  of  Volume.  —  The  volume  of  a  substance  may 
often  be  found  by  calculation  from  its  form  if  that  be  a  known 
geometrical  figure  ;  but  the  volume  of  a  mass  of  irregular  figure 
is  best  ascertained  by  the  rough  method  of  immersing^it  in  water 
or  any  liquid  which  will  not  affect  it,  and  by  observing  how  much 
more  bulk  the  whole  now  occupies  than  the  water  alone  had  done. 

D 


34  MEASUREMENTS.  [CHAP. 

If,  for  instance,  a  piece  of  metal  be  placed  with  three  fluid  ounces  of 
water  in  a  measure,  and  if  the  whole  measure  exactly  four  fluid  ounces,  the 
piece  of  metal  must  occupy  exactly  the  same  bulk  as  one  fluid  ounce  or  ^¥ 
British  gallon  of  water;  that  is,  since  a  gallon  of  water  occupies  277-274 
cubic  inches,  (277-274  H-  80)  or  3-466  cubic  inches ;  and  so  for  fractional 
parts  of  the  units  of  liquid  measure.  The  volume  of  a  flask  may  be  ascer- 
tained, in  cub.  cm.,  by  weighing  the  water  it  can  contain ;  1  gramme,  at 
3*9°  C.,  occupies  1  cub.  cm. 

Measurement  of  Time.  —  It  is  not  possible  or  necessary  to 
do  more  in  treating  of  this  than  to  suggest  one  or  two  leading 
principles.  A  simple  water-dropper,  consisting  of  a  vessel  of 
water  in  the  bottom  of  which  there  is  a  minute  hole,  through 
which  the  water  falls,  drop  after  drop,  into  a  dish,  was  used 
anciently  under  the  name  of  the  Clepsydra.  The  water  which 
fell  through  was  kept  in  the  lower  vessel:  the  amount  there 
accumulated,  or  equally  the  loss  of  level  in  the  upper  vessel, 
indicated  approximately  the  lapse  of  time.  It  was  found,  how- 
ever, that  the  flow  of  water  from  a  vessel  of  this  description  was 
far  from  uniform.  The  use  of  Wheelwork  set  in  motion  by 
some  constantly  acting  force  was  a  fruitful  suggestion  :  setting 
the  wheels  to  indicate  the  amount  of  their  own  rotation  by  means 
of  pointers  connected  with  their  axles  was  a  plan  early  adopted ; 
the  train  of  wheelwork  was  set  in  motion  by  a  falling  weight ; 
but  there  wanted  yet  some  regulating  contrivance  by  which  the 
motion  might  be  rendered  uniform.  A  heavy  flywheel  was 
adapted  to  the  mechanism,  but  without  the  desired  result  being 
.  fully  attained;  and  it  was  only  after  Galileo's  observation  of  the 
fact  that  the  Pendulum  oscillates  from  side  to  side  in  almost 
exactly  equal  periods  of  time,  whether  its  arc  of  oscillation  be 
great  or  small,  that  it  was  suggested  that  this  property  of  the 
pendulum  might  be  rendered  available  for  regulating  clockwork. 
This  was  effected  by  Huyghens ;  and  the  action  of  all  pendulum 
clocks,  however  various  the  trains  of  wheelwork,  depends  on  their 
regulation  by  an  isochronously  —  i.e.  in  equal  times  —  oscil- 
lating pendulum.  The  simplest  mode  in  which  this  regulation 
may  be  effected  is  the  following :  —  One  of  the  wheels  of  the  train 
of  mechanism  bears  on  its  circumference  an  appropriate  number 
of  teeth.  The  descent  of  the  weight  would,  if  there  were  no 
pendulum  attached,  cause  the  mechanism  to  run  on  continuously 
until  the  weight  had  run  down  to  its  lowest  possible  point ;  but 
at  every  stroke  of  the  pendulum  one  of  the  teeth  of  the  wheel  is 
caught  and  the  progress  of  the  wheelwork  arrested. 

The  isochronism  of  the  oscillations  of  the  pendulum  is  not 


in.]  TIME.  35 

sustained ;  variations  in  the  external  temperature  cause  changes 
in  the  length  of  the  pendulum,  and  hence  in  its  rate  of  motion. 
The  contrivances  by  which  compensation  is  made  for  this  cause 
of  error,  so  that  the  rate  of  oscillation  is  maintained  practically 
uniform,  will  be  explained  under  Heat. 

The  measurement  of  small  intervals  of  time  is  of  great  im- 
portance. A  tuning-fork,  if  a  writing-point  be  attached  to  it, 
will,  when  vibrating,  describe  wavy  lines  on  a  piece  of  smoked 
glass  or  paper  drawn  under  the  writing-point.  If  the  tuning- 
fork  vibrate  400  times  per  second,  the  time  taken  to  draw 
each  wave  on  the  paper  must  be  the  four-hundredth  part  of 
a  second;  and  if  any  other  phenomenon  be  so  produced  and 
arranged  as  to  record  its  own  performance  by  a  line  on  the  paper 
or  the  glass,  parallel  to  the  wavy  line  ot  the  tuning-fork,  its 
duration  may  be  estimated  by  counting  the  number  of  recorded 
vibrations  of  the  tuning-fork  to  which  that  duration  corresponds. 

Measurement  of  Mass.  —  Masses  are  compared  with  one 
another  by  means  of  the  Balance.  The  accurate  and  expeditious 
use  of  a  delicate  balance  involves  attention  to  certain  practical 
rules,  which  will  be  found  set  forth  in  Walker's  Balance. 

Measurement  of  Force.  —  There  are  four  main  methods  of 
measuring  any  force.  These  may  be  stated  as  — 

1.  Direct  Observation  of  Mass  and  Acceleration. 

2.  Direct  Counterpoising. 

3.  Indirect  Counterpoising. 

4.  The  Method  of  Oscillations. 

The  first,  the  method  of  direct  observation  of  the  mass  moved 
and  the  acceleration  imparted  to  it  by  the  force  to  be  measured, 
is  based  on  the  equation  F  =  wa;  and  if  m  the  mass  and  a  the 
acceleration  be  known,  F,  the  Force  acting,  can  easily  be  found. 
This  method  presents,  however,  serious  practical  difficulties  in 
the  observation  of  the  acceleration  produced. 

One  important  problem  to  be  solved  by  this  method  is  the  determination 
of  the  force  with  which  Gravity  acts  upon  a  unit  mass  of  matter  at  any  place. 
The  equation  F  =  ma  shows  that  if  we  use  a  unit  mass,  F  =  a ;  thus  we  need 
only  find  the  acceleration  produced.  This  is  effected  roughly  byAttwood's 
Machine.  In  this  the  weight  of  one  gramme  is  used  as  the  force  which 
sets  a  larger  mass  in  motion.  If  it  set  only  its  own  mass  in  motion,  a  velocity 
is  acquired  so  great  as  not  to  be  easily  observed :  if  this  limited  force  set  a 
larger  mass  in  motion,  the  speed  acquired  is  less,  varying  inversely  as  the 
aggregate  mass,  for  a  =  F/m.  If  a  gramme  in  falling  set  a  mass  of  100 
grammes  (including  its  own  substance)  in  motion,  it  can  only  acquire  a 
velocity  one-hundredth  that  which  it  would  have  acquired  if  it  had  fallen 


36 


MEASUREMENTS. 


[CHAP. 


Fig.8. 


alone.  The  essential  part  of  Attwood's  machine  consists  of  a  wheel  over 
which  two  masses  are  suspended.  Let  these  masses  be  49^  and  50£  grammes. 
The  total  mass  set  in  motion  is  100  grammes,  and  the  force  acting  is  the  ex- 
cess in  weight  of  the  heavier  mass  over  the  lighter  —  that  is,  50^  —  49^  =  the 
weight  of  one  gramme.  Let  this  gramme  not  be  a  fixed  part  of  the  heavier 
mass,  but  merely  a  piece  of  wire  which  can  be  removed  by  making  the 
weighted  mass  fall  through  a  metal  ring.  A  pendulum  which  beats  seconds 
regulates  a  timepiece ;  attached  to  the  wheelwork  of  the  timepiece  is  an 
"  excentric,"  which  works  a  lever ;  this  lever,  at  a  pre-arranged  instant, 
pushes  or  pulls  away  a  little  plate  which  supports  the  heavier  mass ;  this 

mass  suddenly  finds  itself  freely 
exposed  to  the  action  of  gravity ; 
the  excess-weight  of  the  little 
gramme -load  imparts  to  the 
whole  mass  a  certain  velocity; 
the  ring  is  placed  at  such  a  posi- 
tion as  to  catch  the  wire  exactly 
at  the  end  of  one  second,  this  be- 
ing indicated  by  the  sound  of  the 
timepiece  and  pendulum  coincid- 
ing with  the  click  of  the  wire 
upon  the  ring  which  catches  it  as 
it  falls.  Thereafter  there  is  no 
unbalanced  force  acting,  and  the 
mass  of  99  grammes  continues  to 
move  uniformly  according  to  the 
first  law  of  motion.  Its  speed 
can  be  observed  by  comparing 
the  distance  it  travels  with  the 
ticking  of  the  timepiece.  This 
is  done  by  placing  a  little  plate 
to  receive  the  falling  body.  A 
slight  sound  will  be  made  by  the 
falling  body  touching  this  plate. 
If  this  sound  and  that  of  the  pendulum  coincide,  the  plate  is  in  the  right 
position ;  if  not,  that  position  must  be  found  by  a  process  of  trial  and  error. 
It  is  found  that  if  a  pair  of  masses  of  49 1  grammes  each  be  suspended  over 
the  pulley,  and  one  of  them  be  loaded  with  one  gramme  so  that  the  whole 
mass  to  be  moved  weighs  100  grammes ;  if  the  overweight  be  taken  off  at 
the  end  of  one  second  by  a  ring ;  if  the  balanced  masses  be  allowed  to  move 
onward  with  their  then  acquired  velocity  for  one  second ;  if  a  plate  be  so 
adjusted  under  the  ring  as  to  check  this  motion  precisely  at  the  end  of  a 
second  —  it  is  found  that  that  plate  must  be  9-81  centimetres  below  the  ring. 
This  shows  that  the  force  acting  (the  weight  of  one  gramme),  acting  for  one 
second,  is  able  to  impart  to  a  mass  of  100  grammes  a  velocity  of  9-81  cm.-per- 
sec.  Hence  by  the  equation  F  =  ma,  F,  which  is  equal  to  the  Weight  of  one 
gramme,  is  equal  to  100  grammes  x  9-81  cm.-per-sec.  per  second  —  981  dynes. 
This  method  can  give  no  more  than  an  approximation  to  the  value  required. 
Much  greater  accuracy  is  attained  by  the  use  of  the  Pendulum.  The 
time  of  oscillation  of  a  pendulum,  as  we  shall  afterwards  learn,  varies 
inversely  as  the  square  root  of  the  force  of  gravity  at  the  place  where  the 
observation  is  made.  The  time  of  the  oscillation  of  any  pendulum  can  be 


491 


in.]  FORCE.  37 

very  accurately  learned  by  observing  the  time  taken  to  perform  a  certain 
sufficiently  large  number  of  oscillations,  and  dividing  that  time  by  the  whole 
number  of  oscillations.  From  this  observation  can  be  deduced  the  local 
acceleration  of  gravity. 

Measurement  of  Force  by  Direct  Counterpoising.  —  In  an 

ordinary  balance  whose  arms  are  perfectly  equal,  the  force  with 
which  gravity  acts  on  the  mass  in  one  pan  is  equal  to  that  with 
which  it  acts  on  the  mass  in  the  other.  For  one  of  these  we 
may  substitute  another  force  of  any  kind  but  of  equal  amount. 
If,  for  instance,  we  use  a  balance  with  glass  pans,  we  may  lay  one 
of  the  glass  pans  on  the  surface  of  mercury  and  determine  what 
mass  must  be  put  in  the  other  pan,  to  pull  the  first  from  the 
mercury.  Let  this  be  47  grammes,  and  the  area  of  the  glass  pan 
25  square  centimetres.  Then  a  force  equal  to  the  weight  of  47 
grammes  is  necessary  to  pull  25  square  centimetres  of  the  surface 
of  glass  away  from  mercury — that  is,  1-88  gramme  per  square 
centimetre ;  and  the  force  of  adhesion  between  mercury  and 
glass  is,  for  every  square  centimetre,  equal  to  the  weight  of  1-88 
grammes  —  that  is,  a  force  of  1-88  x  981  =  1844-28  dynes. 

A  soap  film  tends  to  contract.  If  we  find  how  much  mass 
must  be  suspended  on  a  soap  film  of  a  certain  size  in  order  to 
prevent  it  from  contracting,  the  force  of  contraction  will  be  equal 
to  the  weight  of  the  mass  which  the  film  supports,  and  that  force 
can  hence  be  measured  in  absolute  units. 

This  method,  as  well  as  the  next,  lends  itself  so  readily  that  no  special 
explanation  is  necessary,  to  the  measurement  of  stresses,  pressures,  tensions. 

Measurement  of  Force  by  Indirect  Counterpoising. — Let 

us  suppose  that  we  have  access  to  a  standard  unit  of  mass.  This 
is  hung  upon  a  spiral  or  spring  of  steel  wire.  It  is  observed  to 
lengthen  the  spring  by  a  certain  measured  amount.  If  another 
mass  be  hung  upon  the  same  spiral,  and  if  the  lengthening  pro- 
duced be  the  same,  the  inference  is  that  the  action  of  gravity 
upon  the  second  mass  is  equal  to  that  on  the  first,  and  hence,  if 
the  two  observations  be  made  at  the  same  place,  that  the  second 
mass  is  itself  equal  in  'quantity  to  the  first.  This  is  the  principle 
of  the  Spring  Balance.  Different  known  masses  may  be  sus- 
pended on  such  a  spiral,  and  the  elongations  produced  may  be 
recorded  on  a  scale  attached  to  the  instrument.  If  a  mass  of 
unknown  amount  be  attached  to  the  spiral,  its  weight  may  be 
found  by  reading  on  the  scale  the  number  of  standard  pounds 
and  ounces,  etc.,  requisite  to  produce  the  same  distortion  as  the 
unknown  mass  causes  when  hung  upon  the  spring. 


38  MEASUREMENTS.  [CHAP. 

The  instrument  known  in  one  form  as  a  spring  balance  is 
known  in  another  as  a  Dynamometer.  The  form  of  the  steel 
spring  used  is  quite  independent  of  the  principle  involved,  which 
is  that  if  two  forces  produce  equal  distortions  in  a  body,  these 
forces  must  be  equal  to  one  another.  If  a  man  can  pull  a  spring 
out  two  inches,  and  if  200  Ibs.  must  be  hung  on  the  spring  to 
produce  the  same  distortion,  the  man's  pull  is  equal  to  the 
weight  of  200  Ibs. ;  similarly  the  force  required  to  pull  the  spring 
out  two  inches  is  equal  to  that  which  must  be  exerted  to  raise  a 
weight  of  200  Ibs. ;  and  these  can  be  translated,  when  we  know 
the  local  acceleration  of  gravity,  into  forces  measured  in  absolute 
units.  If  he  can  give  it  a  blow  which  will  compress  it  for  a 
moment  to  the  same  extent  as  the  Weight  of  140  kilogrammes 
placed  on  it  would  do,  the  force  of  his  blow  is  equal  to  the 
weight  of  140  kilogrammes  — that  is,  140,000  x  981  =  137,340000 
dynes.  If  he  can,  by  closing  his  hand  firmly,  distort  a  spring  to 
a  certain  extent,  it  can  easily  be  ascertained  what  amount  of 
weight  acting  on  the  spring  is  capable  of  producing  the  same 
distortion.  This  is  usually  done  beforehand,  and  the  instrument 
is  provided  with  a  graduated  scale  which  indicates  what  amounts 
of  weight  —  at  the  place  where  the  instrument  is  made,  be  it 
remembered  —  correspond  to  the  various  readings  of  the  pointer. 
When  his  flexor  muscles  contract  so  as  to  force  the  pointer  of  the 
dynamometer  to  indicate  84  kilogrammes,  the  distortion  produced 
by  them  is  equal  to  that  which  would  be  produced  by  the  Weight 
of  84  kilos.,  at  Paris  if  the  instrument  have  been  made  there ; 
that  is,  since  G  =  mg,  84,000  grms.  x  981  =  82,404000  dynes. 

Illustrations  of  this  principle  abound.  The  attraction  of 
magnetism  may  be  measured  in  a  similar  way.  Let  a  magnet 
attract  a  piece  of  iron,  which  is  attached  to  a  spiral,  to  such  an 
extent  that  the  spiral  is  lengthened,  say  one  inch,  when  the 
magnet  is  at  a  distance  of  a  tenth  of  an  inch  from  the  iron.  It 
is  found  that,  say,  2  Ibs.  3  oz.  must  be  hung  on  that  spiral  to 
produce  the  same  distortion ;  the  magnetic  attraction  is  equal  to 
the  local  Weight  of  a  mass  of  2  Ibs.  3  oz.  This  is  an  undesirable 
method  from  the  practical  point  of  view,  but  it  shows  how 
magnetic  and  other  attracting  forces  can  be  compared  with  forces 
whose  absolute  amounts  we  know. 

If  an  electromagnet  can  hold  ten  pounds  of  iron,  but  cannot 
support  ten  pounds  and  a  grain,  the  force  of  attraction  is  equal 
to  the  weight  of  ten  pounds ;  for  instead  of  the  magnetic  attrac- 
tion, we  might  have  used,  in  order  to  prevent  the  ten-pound 


in.]  FORCE.  39 

mass  of  iron  from  falling,  another  ten-pound  mass  connected  with 
it  by  a  cord  passed  over  a  pulley. 

If  we  take  a  bar  of  metal,  suspend  it  on  centres  at  each 
end,  fix  it  firmly  at  one  end  so  as  to  prevent  that  end  from 
rotating,  and  hang  a  known  mass  over  the  side  of  that  end 
which  is  free  to  rotate,  we  find  that  the  bar  is  twisted;  this 
effect  is  measurable.  Whatever  other  force  will  produce  the 
same  effect  must  be  equal  to  the  known  Weight  which  caused  it. 
If  the  body  to  be  twisted  be  a  glass  or  silk  fibre,  the  amount  of 
force  required  to  twist  it  is  small.  To  twist  such  a  fibre  through 
a  certain  number  of  degrees,  a  certain  fractional  number  of 
grammes'  weight  must  be  applied  at  unit-distance  from  the  cen- 
tre. If  an  electric  attraction  be  applied  to  a  body  suspended  by 
such  a  silk  fibre,  the  suspended  body  is  attracted,  the  suspending 
fibre  may  be  twisted ;  to  produce  the  observed  torsion  or  twist,  a 
certain  number  of  grammes'  weight  must  be  applied  ;  the  electric 
Attraction  can  be  stated  to  be  equal  to  the  Weight  of  so  many 
grammes,  and  therefore  to  so  many  absolute  Units  of  Force. 

Ruhelage :  Equilibrium-position. —  It  is  often  advantageous  to  meas- 
ure the  force  acting  on  a  displaceable  object,  by  balancing  that  displacing 
force  against  another  force,  so  adjusted  as  to  bring  the  displacement  back 
to  zero  value.  A  magnetic  needle  deflected  by  a  current  is  twisted  back  into 
its  original  position  by  a  twisted  suspending  fibre ;  the  torsion  imparted  to 
the  suspending  thread  is  measurable  and  represents  a  known  number  of 
dynes.  The  force  acting  on  the  needle  is  thus  measured.  The  advantage 
of  this  method  is  that  we  obtain  precisely  what  we  wish,  the  full  force 
exerted  by  the  current  on  the  needle  when  in  its  original  position,  not  the 
force  acting  on  it  in  any  other  position ;  and  we  thus  eliminate  any  disturb- 
ance produced  by  such  variations  in  that  force  as  may  be  due  to  variations 
in  the  position  of  the  suspended  needle  itself. 

The  fourth  method  is  that  of  oscillations.  If  a  magnet  be 
brought  near  another  magnet  it  oscillates  from  side  to  side.  If 
it  be  brought  near  a  stronger  magnet  it  oscillates  more  fre- 
quently, It  can  be  proved  that  the  velocities  produced  vary  as 
the  square  root  of  the  forces  causing  the  oscillations.  Hence 
we  count  the  number  of  oscillations  in  a  given  period  in  two 
cases,  and  the  ratio  of  'their  squares  is  the  ratio  of  the  two  forces. 
If,  for  instance,  a  magnetic  needle  oscillate  fifteen  times  a  minute 
in  the  presence  of  a  magnet  A,  and  sixty  times  in  presence  of  a 
magnet  B  ;  the  forces  acting  in  the  two  cases  are  as  the  square 
of  15  is  to  the  square  of  60,  or  as  1  to  16.  In  this  way  we  are 
able  to  compare  the  forces  acting  under  the  given  conditions, 
but  we  do  not  learn  the  absolute  amount  of  either.  That  must 
be  ascertained  by  one  of  the  methods  previously  discussed. 


CHAPTER  IV. 

WORK   AND   ENERGY. 

Work.  —  When  a  force  "  acts  upon "  a  body,  and  that  body 
moves  in  the  direction  of  the  force,  that  force  is  said  to  Do 
Work,  and  the  work  said  to  be  done  by  it  is  measured  by  the 
product  of  F,  the  force  acting  in  a  certain  direction,  into  s,  the 
space  through  which  the  body  has  moved  in  that  direction ; 

Work  =  W  =  Fs  =  was. 

For  example  :  Steam  exerts  on  the  piston  of  a  cylinder  a  mean  force  or 
pressure  of,  say,  30  Ibs.  per  square  inch ;  the  area  of  the  piston  is,  say,  30 
square  inches  ;  the  whole  pressure  exerted  is  thus  equal  to  the  weight  of  900 
Ibs.  The  piston  is  thrust  through,  say,  16  inches.  The  work  done  is  900 
Ibs.-wt.  x  lj  ft.  =  1200  foot-pounds  at  each  stroke. 

Conversely,  when  a  force  acts  upon  a  body  and  that  body 
moves  or  is  moved  in  a  direction  opposed  to  that  of  the  force, 
that  force  is  said  to  be  Resisted,  and  work  is  said  to  be  done 
against  it ;  and  Fs  =  W ;  the  product  of  the  force  resisted,  into 
the  space  traversed  against  that  force,  represents  the  Work  said 
to  be  done  against  the  force  so  resisted. 

When  a  ten-pound  mass  is  raised  ten  feet  against  gravity,  the  work  done 
against  gravity  is  equal  to  the  product  of  the  space  traversed  into  the  force 
resisted  —  i.e.,  10  ft.  x  wt.  of  10  Ibs.  =  100  foot-pounds.  In  this  case 
Work  =  Fs  as  usual ;  but  F,  the  force  resisted,  is  the  Weight  of  a  mass  ra, 
and  therefore  F  =  G  =  mg ;  consequently  the  work  done,  W=  Fs  —  mgs  =  mgh. 

Suppose  a  man  to  walk  against  a  heavy  gale  of  wind,  the  mean  pressure 
of  which  is  40  Ibs.  per  square  foot.  If  the  surface  presented  to  the  wind- 
pressure  be  virtually  5  sq.  ft.,  the  total  pressure  of  the  wind  will  be  200 
Ibs.,  and  the  effort  of  walking  against  it  will  be  the  same  as  if  the  man 
pulled  a  weight  of  200  Ibs.  out  of  a  pit  by  means  of  a  cord  thrown  over  a 
pulley.  If  the  man  make  his  way  for  a  mile,  he  will  have  resisted  a  mean 
pressure  of  200  Ibs.  through  a  space  of  5280  ft.  He  will,  therefore,  have 
done  1,056,000  foot-pounds  of  work ;  an  amount  of  work  which,  otherwise 
directed,  would  have  sufficed  to  lift  him  up  (his  total  weight  being  supposed 
to  be  150  Ibs.),  to  twice  the  height  of  Snowdon. 

There  is  no  Work  done  against  or  by  the  force  acting  unless 
there  be  actual  Motion.  We  might  imagine  machinery  to  be 

40 


[CHAP,  iv.]  WORK.  4^ 

driven  by  an  avalanche  during  its  fall ;  but  not  before,  and  not 
after.  Gravity  does  no  Work  upon  a  resting  stone:  it  does 
work  upon  a  falling  stone. 

If  Fs  =  1,  we  have  the  Unit  of  Work.  This  is  the  case 
when  F  =  1  and  s  =  1 ;  that  is,  a  unit  of  work  is  done  when  a 
body  acted  on  by  unit  force  moves  through  a  unit  distance  in 
the  direction  of  the  force.  In  C.G.S.  measures  the  unit  of 
work  is  done  by  raising  ^IT  gramme  (mass  whose  weight  at 
Paris  =  1  Dyne)  to  the  vertical  height  of  one  centimetre.  This 
is  the  Erg.  The  erg  is,  however,  a  very  small  unit  of  work, 
and  for  many  purposes  it  is  convenient  to  use  the  Megalerg, 
which  is  equal  to  1,000,000  Ergs  and  would  therefore  be  the 
amount  of  work  done  in  raising  -Mjfy-  =  10-19  grammes  through 
one  metre;  or  the  Ergten,  1010  or  10,000,000,000  Ergs  ;  or  the 
Joule,  107  or  10,000,000  Ergs. 

In  British  measures  32-2  units  of  work  are  done  in  raising  a  pound-mass 
through  one  foot.  Such  units  of  work  are  called  foot-poundals.  British 
engineers  are  in  the  habit  of  using  the  foot-pound  (the  work  spent  in  raising 
one  pound  one  foot)  as  a  unit  of  work.  This  would  be  satisfactory  if  foot- 
pounds were  equal  over  the  whole  earth,  but  g,  the  acceleration  of  gravity, 
varies  from  place  to  place.  Hence  the  foot-pound  is  from  place  to  place  a 
variable  measure,  varying  between  the  Equator  and  the  Poles  by  about  one- 
half  (0-512)  per  cent. ;  and  it  has  to  be  reduced  for  each  place  to  absolute 
units  of  work  by  the  equation  —  Work  =  Force  overcome  x  Space  =  Weight 
x  a=  mgs,  and  the  foot-pound  is  equal  to  g  foot-poundals,  where  g  is  meas- 
ured in  ft. /sec.2  (g  =  32-2  nearly).  The  foot-pound  is  equal  to  13,562,691 
Ergs,  when  g  =  981  cm. /sec.2  The  kilogramme-metre,  or  French  engineers' 
unit  of  work,  is  1000  grins,  xg  x  100  cm.  =  98,100,000  Ergs. 

Any  amount  of  work  may  be  specified  as  the  product  of  two 
numbers,  which  represent  respectively  a  Force  and  a  Displace- 
ment. These  may  vary,  but  if  they  have  the  same  product  the 
amount  of  work  done  is  the  same.  A  pound  raised  100  feet,  100 
pounds  raised  one  foot,  fifty  pounds  raised  two  feet,  four  pounds 
raised  twenty-five  feet,  all  represent  the  same  amount  of  work, 
namely,  100  foot-pounds,  it  being  here  assumed  that  the  force  of 
gravity  is  uniform  within  heights  of  100  feet. 

Since  Work  =  Fs,  it  follows  that  F  =  Work  -*•  s ;  whence  Force  is  the 
number  of  Units  of  Work  done  upon  or  by  a  body  moving  in  a  straight  line, 
divided  by  the  number  of  Units  of  Length  traversed  by  that  body  in  that 
line.  Force  in  a  given  direction  is  therefore  a  rate  at  which  work  is 
observed  to  be  done,  per  unit  not  of  time  but  of  space  traversed  in  that 
direction. 

This  looks  like  a  definition  obtained  by  reasoning  in  a  circle  ;  but  if  it 
be  presented  in  the  equivalent  form  —  Force  is  the  rate  at  which  a  moving 
body  gains  or  loses  either  potential  or  kinetic  Energy  per  Unit  of  Space 


42  WOKK  AND   ENERGY.  [CHAP. 

traversed  —  we  shall  presently  understand  that  it  is  not  a  truism,  for  Energy 
is  a  physical  entity.  In  this  view,  the  Force  in  a  given  direction  is  the 
Energy-Slope  in  that  direction. 

The  Mean  Rate  of  Doing  Work  is  the  whole  Work 
done  in  a  given  time  divided  by  the  Time.  If  an  engine  can 
raise  1,980,000  pounds  vertically  one  foot  in  an  hour,  its  mean 
rate  of  doing  work,  its  Power,  or,  as  Lord  Kelvin  phrases  it,  its 
Activity  (French  puissance),  is  33,000  foot-pounds  per  minute, 
or  550  foot-pounds  per  second.  This  particular  mean  rate  is 
known  by  British  and  American  Engineers  as  a  Horse-power ; 
and  an  engine  of  one  horse-power  can  do  this  amount  of  work. 
A  horse  can,  according  to  General  Morin,  do  26,150  foot-pounds 
per  minute,  and  a  labourer  from  470  (lifting  earth  with  a  spade) 
to  4230  (raising  his  own  weight,  treadmill  exercise)  per  minute. 
The  French  horse-power  (cheval-vapeur)  is  75  kilogram-metres, 
or  7,500,000  #  =  7,357,500,000  Ergs  per  second;  whilst  the  Brit- 
ish horse-power  is  equal  to  7,459,480,050  Ergs  per  second,  when 
#  =  981  cm./sec.2 

If  a  man  weighing  14  stone  run  upstairs  at  such  a  rate  as  to  gain  3 
feet  in  vertical  height  every  second,  his  muscular  system  is  doing  every 
second  the  work  of  carrying  196  Ibs.  up  3  feet,  i.e.,  588  foot-pounds.  If  this 
could  be  kept  up  for  a  minute,  60  x  588  =  35,280  foot-pounds  would  be  done, 
and  the  man  would  be,  in  the  case  supposed,  undergoing  an  exertion  which 
for  the  moment  would  be  much  greater  than  a  horse  can  keep  up,  and  seventy- 
five  times  that  which  a  continuously-toiling  labourer,  lifting  earth  with  a 
spade,  can  sustain ;  and  in  the  most  favourable  circumstances,  a  labourer, 
raising  his  own  weight  merely,  can  only  keep  up  one-eighth  of  this  effort. 

The  Activity,  or  Power,  or  Effective  Horse-Power  of  an  engine  must  be 
distinguished  from  its  Nominal  Horse-Power,  which  is  a  term  based  upon 
certain  dimensions  of  the  cylinder,  and  has  no  well-defined  experimental 
meaning. 

The  Unit  of  Activity  is  frequently  taken  as  one  Watt, 
which  represents  10  Megalergs  per  second.  The  British  horse- 
power is  thus  equal  to  746  Watts  nearly,  the  French  to  735|. 

A  thousand  Watts  are  one  kilowatt.  Inconveniently  enough,  the 
Congres  International  de  Mecanique  Appliquee,  1889,  recommended  as  a 
unit  of  activity  a  <Poncelet'  =  100  kilogramme-metres  per  second  = 
98,100,000  ergs  per  second  =  9-81  Watts. 

Activity  is  also  measured  as  Fv,  Force  x  Velocity  in  the  direction  of 
the  force;  for  v  =  z/t;  and,  therefore,  Activity  =  W/t  =  Fs/t=Fv. 

Energy.  —  When  a  body  weighing  ten  pounds  is  raised  ten 
feet,  and  prevented  by  a  catch  from  falling,  the  work  done  upon 
it  — 100  foot-pounds  —  can  be  recovered  by  permitting  it  to  fall 
upon  a  train  of  mechanism.  If  the  mechanism  were  perfect, 
the  work  would  be  so  restored  that  another  ten-pound  mass 


iv.]  -^  POTENTIAL  ENERGY.  43 

might  be  projected  by  it  to  a  height  of  ten  feet,  a  fifty-pound 
mass  to  a  height  of  two  feet,  and  so  on.  The  fact  that  we  can- 
not obtain  perfect  mechanism  does  not  affect  the  principle.  The 
body  at  a  height  has  therefore  a  power  of  doing  work  equal  to 
the  work  done  upon  it  in  lifting  it.  In  this  case  the  power  of 
doing  work  has  been  conferred  upon  a  body  by  the  separation  of 
it  from  the  earth  against  the  action  of  gravity :  as  it  remains  in 
its  elevated  position,  there  is  a  stress,  or  pull,  or  attraction,  tend- 
ing to  draw  it  down,  and  it  is  only  in  virtue  of  this  stress  that 
it  has  any  power  of  doing  work.  If  the  earth  and  the  elevated 
body  ceased  to  attract  one  another,  the  body  would,  if  liberated, 
not  fall  down,  and  would  not  restore  the  100  ft.-lbs.  of  work 
spent  upon  it.  We  know  that  the  Work  done  in  raising  a 
mass  m  through  a  height  h  against  gravity  is  mgh :  the  energy 
stored  up  in  the  body  is  therefore  equal  to  mgh,  and  is  seen  to 
depend  on  the  mass  of  the  body,  the  height  at  which  it  is  placed, 
and  the  local  accelerative  effect  of  gravity.  Energy  or  power 
of  doing  work,  stored  up  in  this  way,  is  called  Potential  Energy, 
or  Static  Energy,  or  Energy  of  Position,  or  Energy  of 
Stress.  As  an  example  of  Potential  Energy  we  may  take  that 
stored  up  in  a  mill-pond.  The  number  of  units  of  Energy  in 
such  a  pond  may  be  found  by  taking  the  product  of  the  quantity 
of  water  in  it  and  the  average  height  at  which  it  is  placed,  and 
multiplying  that  product  by  the  local  value  of  g.  A  small  quan- 
tity of  water  at  a  great  height  may  obviously  have  the  same 
amount  of  energy  stored  up  in  it  as  a  larger  quantity  at  a  lesser 
height.  If  the  question  be  put,  How  much  work  could  be  got 
by  appropriate  mechanism  from  the  rise  and  fall  of  the  tide  ?  — 
we  consider  (1)  the  total  amount  of  water  carried  into  the  area 
which  can  be  brought  within  the  range  of  the  mechanism,  (2) 
the  average  height  to  which  it  rises,  and  (3)  the  local  value  of  g. 

We  have  also  energy  stored  up  in  such  bodies  as  watch- 
springs.  Work  is  done  upon  them  in  distorting  them,  and  pro- 
ducing a  movement,  not  of  their  mass  as  a  whole,  but  a  relative 
displacement  of  their  parts.  This  work  is  restored  and  utilized 
in  producing  movement  of  the  mechanism  attached.  When  a 
watch-spring  is  distorted  and  held  fast  so  that  the  distortion  or 
strain  persists,  the  whole  mass  remains  in  a  condition  of  Stress, 
and  tends  at  the  first  opportunity  to  restore  the  work  done 
upon  it.  , 

If  we  look  at  our  previous  example  of  the  earth  and  a  stone 
lifted  from  its  surface,  we  see  that  the  phenomenon  is  on  the 


44  WORK  AND   ENERGY.  [CHAP. 

large  scale  one  of  the  same  order.  The  earth  and  the  stone  to- 
gether constitute  a  system  :  when  this  is  deformed  by  pulling  the 
stone  away  from  the  earth,  the  system  tends  to  return  to  its 
original  form,  and  there  is  a  stress  between  the  earth  and  the 
stone,  which  continues  until  the  stone  is  allowed  to  fall  back  to 
the  earth.  If  the  stone  had  been  connected  with  the  earth  by  a 
band  of  indiarubber,  we  would  have  seen  the  indiarubber  to  be 
stretched  or  under  stress,  and  would  easily  see  that  if  the  stone 
were  liberated  it  would  be  pulled  back  towards  the  earth ;  but 
the  question  is,  What  is  under  stress  in  the  actual  case?  for 
there  is  no  visible  connecting  cord  between  the  stone  and 
the  earth.  If  we  could  state  what  this  was,  we  would  be 
able  to  arrive  at  the  cause  of  Gravitation.  As  it  is,  our  know- 
ledge ceases.  That  there  is  some  medium,  and  that  it  may  be 
under  stress,  is  a  theory  necessary  for  the  exposition  of  Electri- 
city, of  Light,  of  Magnetism,  and  of  Heat ;  but  we  are  by  no 
means,  as  yet,  entitled  to  say  that  stress  in  this  medium  is  the 
cause  of  Gravitation. 

Work  may  be  done,  then,  in  altering  the  relative  configura- 
tion of  a  system,  whether  this  consists  of  large  masses  or  of 
smaller  particles.  If  this  system  be  what  is  known  as  a  "Con- 
servative System,"  in  which  a  stress  may  be  established  de- 
pending upon  the  configuration,  and  only  upon  the  configura- 
tion (not  in  any  degree  upon  the  history  of  any  antecedent 
deformations  through  which  the  configuration  in  question  may 
have  been  arrived  at),  the  system  will  tend,  when  work  has 
been  done  upon  it,  to  return  to  its  original  form,  and  to  restore 
the  work  done  upon  it.  If  its  relation  to  surrounding  objects  be 
such  that  it  cannot  so  return,  it  will  be  under  stress,  and  will 
continue  under  stress  until  its  relations  to  surrounding  objects 
have  become  such  as  to  permit  it  liberty  of  restitution ;  then,  at 
the  first  opportunity,  it  will  restore  the  work  done  upon  it. 

The  change  in  its  relations  to  surrounding  objects  necessary  to  render 
this  restitution  possible  may  be  very  small ;  for  example,  a  heavy  mass  may 
be  prevented  from  falling  by  a  very  small  catch,  but  when  the  catch  is  re- 
moved the  body  falls.  The  cause  of  the  body  falling  is  not  simply  the 
release  of  the  catch,  but  also  the  previously  existing  conformation  of  the 
distorted  system. 

Similarly,  the  ingredients  of  Gunpowder  have  a  tendency  to  combine  : 
its  particles  are  chemically  separate,  but  chemically  attract  one  another,  and 
therefore  possess  potential  energy;  the  application  of  a  very  small  amount  of 
heat,  as  by  a  spark,  liberates  these  particles,  which  can  rush  together  and 
form  new  and  stable  compounds,  which  have  no  longer  any  tendency  to  alter 
their  chemical  constitution,  being  no  longer  under  the  same  stress,  having 


iv.]  POTENTIAL  ENERGY.  45 

no  longer  the  same  potential  energy.  As  it  happens  that  in  this  special  case 
the  new  and  stable  compounds  formed  are  mainly  gaseous  at  the  ordinary 
temperature  and  pressure,  the  products  of  combination  occupy  a  much 
larger  bulk  than  the  original  gunpowder,  and  the  result  is  an  explosion. 
The  spark  only  produces  its  own  small  effect;  the  previous  arrangement 
of  the  particles  of  the  powder  is  responsible  for  the  rest. 

Cases  abound  in  which  energy  is  stored  up  in  mechanical  arrangements. 
The  Air-gun  consists  of  a  volume  of  air  which  has  been,  by  work  done 
upon  it,  compressed  into  a  small  bulk,  and  which  tends  to  return  to  its 
original  dimensions.  When  permitted  to  do  so,  it  suddenly  expands,  and 
may  be  made,  in  propelling  bullets,  to  restore  work  done  upon  it.  When  a 
Clock  is  wound  up  by  pulling  up  the  'weights,'  work  is  done  upon  the 
system ;  this  is  restored  by  the  whole  system  returning  to  its  original  form, 
the  weights  descending  to  their  lowest  position.  It  takes  a  definite  number  of 
days  or  hours  to  do  this,  according  to  the  mechanical  arrangements  devised. 
The  work  done  in  bending  a  Bow  is  swiftly  restored  as  the  bow  returns  to 
its  original  form,  and  may  be  spent  in  imparting  motion  to  the  arrow. 

A  Non-conservative  System  is  one  in  which,  when  the 
system  is  deformed,  there  is  no  stress  established  tending  to  re- 
store the  original  arrangement.  Such  a  system  is  exemplified  by 
a  gun  and  bullet.  When  the  bullet  has  left  the  gun,  Newton's 
first  law  applies,  according  to  which  the  bullet  tends  to  go 
straight  on  at  a  uniform  rate,  unless  acted  on  by  impressed 
forces.  The  bullet  forms  a  part  of  two  systems,  one  conserva- 
tive and  the  other  non-conservative ;  its  motion  will  necessarily 
be  that  due  to  its  relations  to  both.  Let  it  be  fired  obliquely 
upwards  :  in  virtue  of  its  separation  from  the  earth,  with  which 
it  forms  a  conservative  system,  a  stress  is  established  which 
brings  it  back  to  some  part  of  the  earth's  surface:  it  does  not, 
in  virtue  of  its  separation  from  the  gun,  tend  to  return  to  the 
barrel  of  the  gun,  but  goes  on  until  it  is  stopped.  The  question, 
What  causes  one  system  to  be  conservative,  another  not  to  be 
so  ?  is  scarcely  to  be  answered  at  present.  The  presumption  is 
that  a  body  if  set  in  motion  will,  according  to  the  first  law  of 
motion,  travel  onwards  in  a  straight  line  and  with  uniform 
velocity,  unless  acted  on  by  impressed  forces  ;  in  other  words, 
that  all  systems  are  non-conservative.  A  shot  fired  vertically 
upwards  should,  according  to  this  law,  pass  on  in  the  same  direc- 
tion without  ceasing;  but  experience  showls  that  it  does  return, 
that  some  impressed  force  does  act  upon  it,  and  this,  which  is 
another  expression  for  the  attraction  of  gravitation,  is  at  present 
not  explained.  Similarly,  the  particles  of  a  distorted  spring 
undoubtedly  form  a  conservative  system ;  stress  is  established 
between  them :  but  the  explanation  of  this  fact  would  imply  a 
knowledge  of  the  constitution  of  those  particles  and  of  their 


46  WORK  AND   ENERGY.  [CHAP. 

actions  upon  one  another,  a  knowledge  which  we  do  not  yet 
possess. 

Kinetic  Energy.  —  Power  of  doing  work  is  possessed  also 
by  bodies  which  are  in  Motion.  If,  for  instance,  a  rifle  bullet 
be  received  on  an  appropriate  mechanism,  the  jolt  suffered  by 
the  instrument  might  be  utilised  in  producing  a  certain  amount 
of  work.  Or  otherwise,  the  bullet,  in  whatever  direction  flying, 
might,  by  a  cord  passed  over  a  pulley,  be  attached  to  a  weight 
which  it  pulled  up.  The  simplest  case  of  this  problem  is,  How 
far  can  a  shot  fired  from  a  rifle  carry  itself  vertically  upwards, 
in  virtue  of  the  power  of  doing  work  possessed  by  it  because  it  is 
in  motion  ?  It  is  known  that  a  body  travelling  upwards  against 
gravity,  and  passing  a  certain  point  with  a  speed  v,  will  rise  to 
a  height  h  =  v2/2g  above  that  point.  The  power  of  doing  work 
possessed  by  the  bullet  in  virtue  of  its  motion  (its  Kinetic 
Energy,  or  Energy  of  Motion,  or  Actual  Energy)  is 
competent,  then,  to  raise  its  own  mass  m  through  a  height 
h  =  v2/2g  against  gravity  whose  local  acceleration  is  g.  The  work 
done  is  mg-h  =  mg-v2/2g  =  ^mv2.  The  Kinetic  Energy,  then,  of 
a  body  moving  in  any  direction  with  speed  v  depends  only  on  its 
Mass  m  and  on  its  Speed  v  —  not  at  all  on  the  local  intensity  of 
gravity ;  and  it  is  independent  of  the  direction  of  the  motion. 

When  the  bullet  arrives  at  the  top  of  its  course  it  has  no 
velocity,  and  therefore  no  kinetic  energy ;  but  it  will  easily  be 
seen  that  if  it  be  caught  when  "  at  the  turn,"  it  can  be  retained 
on  a  ledge,  and  will  there  possess  potential  energy.  This  we 
know  how  to  express  as  mgh.  The  kinetic  energy  which  the 
bullet  has  lost  it  still  retains  under  the  form  of  potential  energy. 
If  it  be  allowed  to  fall,  it  will  lose  its  potential  energy,  and  will 
(in  vacuo)  have  acquired,  in  a  reversed  direction,  the  original 
speed  v  as  it  passes  the  point  of  observation. 

Let  us  suppose  a  body  weighing  10  Ibs.  to  leave  the  ground, 
starting  upwards  with  a  velocity  of  64-4  feet  per  second ; 
let  g  =  32-2  ft./sec.2  Then  the  body  will  ascend  v2/2#,  or 
(64-4)2/2  x  32-2  =  644  ft.  The  body  whose  mass  m  =  lO  Ibs. 
will  rise  644  ft.,  and  if  caught  at  the  instant  when  it  comes 
to  rest  will  have  a  potential  energy  of  644  foot-pounds.  The 
absolute  value  of  this  amount  of  energy  depends  on  the  local 
force  of  gravity,  but  as  g  is  taken  =  32-2,  the  potential  energy 
may  be  expressed  absolutely  as  20,736-8  foot-poundals.  The 
kinetic  energy  which  the  body  possessed  at  the  moment  of 
starting  was  ±mv2  =  £(10  x  (644)2)  =  20,736-8  foot-poundals, 


iv.]  CONSERVATION  OF   ENERGY.  47 

measured  directly  and  irrespectively  of  the  local  force  of  gravity. 
Hence  the  kinetic  energy  lost  by  the  bullet  in  ascending  is  ex- 
actly equal  to  the  potential  energy  gained  by  it.  At  any  inter- 
mediate point,  where  it  has  less  velocity  but  some  potential 
energ}r,  it  will  always  be  found,  in  the  case  supposed,  that  the 
sum  of  the  kinetic  and  potential  energies  is  20,736-8  foot- 
poundals.  The  one  kind  of  energy,  the  potential,  is  transformed 
into  another,  the  kinetic,  and  there  is  in  the  system  (earth  and 
stone)  neither  gain  nor  loss  of  energy  during  the  transformation. 
This  is  the  simplest  case  of  a  widely  applicable  principle,  that 
of  the  Conservation  or  Indestructibility  of  Energy. 

This  principle  is,  that  if  a  system  of  bodies  have  a  certain 
amount  of  energy  in  one  form,  it  must  retain  that  energy  in  one 
form  or  another  unless  it  come  into  such  relations  with  other 
bodies  as,  together  with  them,  to  form  a  larger  system  in  which 
the  energy  becomes  differently  distributed ;  and  if  the  system  be 
so  large  that  there  is  no  other  body  with  which  it  can  enter  into 
such  relations  —  that  is,  if  the  system  which  possesses  the  energy 
be  the  whole  Universe  —  that  system  cannot  gain  or  lose  energy 
by  sharing  with  other  bodies,  and  hence  the  total  amount  of 
Energy  in  the  Universe  is  invariable  and  numerically  constant. 

If  we  take  the  instance  just  discussed,  that  of  the  earth,  the 
bullet,  and  the  gun  pointed  upwards,  these  three  bodies  possessed 
before  the  explosion  a  certain  amount  of  energy,  potential  in  the 
gunpowder:  just  as  the  bullet  left  the  gun,  kinetic  in  the  bullet : 
when  the  bullet  was  detained  at  the  summit  of  its  course, 
potential  between  the  bullet  and  the  earth,  but  always  equal  in 
amount  —  the  same  number  of  foot-poundals.  While  the  kinetic 
energy  was  being  transformed  into  potential,  work  was  being 
done  in  the  conservative  system.  During  this  period  the  bullet 
and  the  earth  were  relatively  moving,  and  the  acceleration 
associated  with  the  transformation  of  one  kind  of  energy  into 
another  is  attributed  to  a  Force  acting  during  that  period.  Force 
is  associated  with  a  variation  in  the  rate  of  change  of  the  con- 
figuration of  a  system  'under  which  the  energy  in  that  system  is 
altered  in  its  distribution  and  form,  and  it  is  said  to  act  only  as 
long  as  that  variation  continues ;  and  every  part  of  a  system 
tends  to  move  so  as  to  get  rid  of  potential  energy  in  the  shortest 
time  by  the  shortest  path. 

Transformations  of  Energy.  —  Energy,  however,  assumes 
other  forms  than  the  two  discussed.  If  the  bullet  in  the  case 
adduced  be  allowed  to  fall  to  the  ground,  it  falls  more  and  more 


48  WORK  AND  ENEKGY.  [CHAP. 

rapidly  until  it  regains  its  original  velocity,  and  therefore  its 
whole  kinetic  energy.  But  this  bullet  may  suddenly  strike  the 
ground  and  lose  all  its  kinetic  energy :  it  has  already  lost  all  its 
potential  energy;  what  has  become  of  the  energy  of  the  system? 
We  find  that  the  bullet  and  the  part  of  the  earth  on  which  it 
has  fallen  are  warmed,  and  we  learn  from  a  wide  induction  of 
similar  cases  that  Heat  is  one  of  the  forms  of  Energy.  It  is 
proved  to  be  so  by  the  observation  that  the  same  amount  of  work, 
if  entirely  spent  in  producing  heat,  will  always  produce  the 
same  amount:  772-55  foot-pounds  of  work  were  found  by  Joule 
to  correspond  to  an  amount  of  heat  capable  of  raising  the 
temperature  of  a  pound  of  water  from  60°  to  61°  F.  The  Heat 
possessed  by  a  body  is  explained  as  being  the  Energy  possessed 
by  it  in  virtue  of  the  motion  of  its  particles.  Just  as  a  swarm 
of  insects  may  remain  nearly  at  the  same  spot  while  each 
individual  insect  is  energetically  bustling  about,  so  a  warm  body 
is  conceived  as  an  aggregation  of  particles  which  are  in  active 
motion  while  the  mass  as  a  whole  has  no  motion.  Heat  is  there- 
fore a  form  of  Kinetic  Energy :  and  the  more  heat  is  imparted  to 
a  body  the  greater  is  the  kinetic  energy  of  each  particle.  If  ra 
represent  the  average  mass  of  the  particles,  and  v  their  average 
velocity,  ^mv2  represents  the  average  kinetic  energy  of  each 
particle ;  and  the  sum  of  all  the  masses  multiplied  by  half  the 
square  of  the  average  velocity  represents  the  intrinsic  kinetic 
energy  of  the  whole  mass.  The  words  "  sum  of "  are  ex- 
pressed by  the  symbol  2.  Hence,  Intrinsic  Kinetic  Energy 
=  ^(Jmv2)=^<2^(m)=^mv2,  where  m  is  the  whole  mass. 

When  a  bullet  possessing  actual  energy  of  motion  impinges 
on  a  target  there  is  a  certain  amount  of  Heat  obtained,  and  the 
bullet  may  be  partly  fused :  there  is  also  a  flash  of  Light  and  a 
certain  amount  of  Sound.  Light  seems  to  be  a  phenomenon  of 
wave-motion  in  that  Ether  whose  existence  throughout  space  is 
apparently  a  necessary  hypothesis ;  so  also  is  Radiant  Heat,  such 
heat  as  streams  to  us  from  the  sun,  or  from  a  fire  across  a  room ; 
and  in  that  Ether,  partly  swinging,  partly  distorted  by  the 
passing  waves,  the  energy  is  partly  kinetic,  partly  potential : 
thus  we  say  that  the  Energy  of  Light  —  or,  briefly,  Light  itself  — 
is  a  distinct  form  of  Energy. 

When  a  tuning-fork  is  made  to  vibrate,  work  is  done  upon 
it  in  giving  it  in  the  first  place  a  distorted  form.  Its  arms  swing 
like  pendulums,  but  their  vibration  gradually  dies  away  and  the 
energy  of  vibration  of  the  fork  becomes  converted  into  the  partly 


iv.]  TRANSFORMATIONS   OF  ENERGY.  49 

kinetic,  partly  potential  energy  of  vibration  of  the  air  —  that  is, 
into  the  Energy  of  Sound ;  and  ultimately  it  is  converted  into 
uniformly-diffused  Heat. 

Energy  may  appear,  then,  as  Energy  of  Mechanical  Position 
or  Motion,  as  Heat,  as  the  Energy  of  Light,  of  Sound,  and  again 
as  that  of  Electrical  or  Magnetic  condition ;  and  a  great  part  of 
our  work  is  to  study  the  modes  in  which  the  various  forms  of 
Energy  are  transformed  and  redistributed,  and  the  Forces  and 
the  phenomena  attributed  to  forces  which  are  associated  with 
these  transformations  and  redistributions. 

A  few  other  examples  of  Transformation  of  Energy  may  here  be  added. 
A  man,  ascending  a  stair,  gains  some  potential  energy :  it  is  found  (Him) 
that  he  is  perceptibly  cooler  for  a  moment.  The  heat  of  his  body  has  been 
partly  transformed  into  potential  energy.  Of  course  the  exertion  of  his 
muscles  and  the  excitement  of  his  circulation  cause  him  to  become  warm 
immediately  afterwards.  When  he  comes  downstairs  he  sacrifices  the 
potential  energy  which  he  had  possessed  when  upstairs  in  virtue  of  his 
elevated  position,  and  which  he  might  conceivably  have  utilised  by  dropping 
himself  out  of  the  window  on  an  appropriate  machine  placed  on  the  pave- 
ment. This  energy  is  not  lost,  for  he  is  (Him)  perceptibly  warmer  at  the 
bottom  of  the  stairs  than  he  had  been  at  the  top.  At  every  step  downstairs 
he  had  arrested  his  own  fall,  and  had  consequently  converted  a  part  of  his 
potential  energy  first  into  kinetic  energy  and  then  into  heat. 

When  a  quantity  of  water  is  decomposed  by  an  electric  current,  the  elec- 
tric current  is  diminished  and  work  is  done  in  separating  a  certain  number 
of  particles  or  atoms  of  oxygen  and  hydrogen.  These  separated  atoms  tend 
to  fall  together  again  and  form  the  stable  compound,  water.  The  mixture 
of  oxygen  and  hydrogen  thus  formed  by  "electrolysis"  possesses  potential 
energy  of  chemical  separation.  When  a  flame  is  applied  to  the  mixture,  a 
process  of  recombination  commences,  and  the  whole  of  this  potential  energy 
is  sacrificed  as  such,  but  appears  in  the  form  of  heat,  light,  and  sound,  and 
may  in  an  appropriate  gas-engine  be  partly  spent  in  doing  mechanical  work. 

The  heat  and  light  produced  by  combustion  and  by  chemical  combina- 
tions in  general  are  forms  of  energy  obtained  by  transformation  of  the 
potential  energy  which  the  particles  had  previously  possessed  in  virtue  of 
their  chemical  separation  and  chemical  affinity.  Under  certain  circumstances 
this  potential  energy  may  not  be  transformed  into  heat  or  light,  but,  as  in 
the  galvanic  battery,  into  the  energy  of  a  current  of  electricity,  which  may 
in  its  turn  be  made  to  do  work,  be  transformed  into  heat,  into  light,  into 
sound,  or  be  spent  in  setting  up  magnetic  condition,  and  so  on. 

When  an  engine  goes  round  without  doing  work  the  steam  remains  hot. 
When  the  engine  does  work  the  steam  is  cooled,  and  the  researches  of  Him 
have  shown  that  the  amount  of  work  done  is  exactly  equivalent  to  the  heat 
which  has  disappeared. 

The  energy  of  an  engine  is  derived  from  the  heat  evolved  by  the  com- 
bustion of  the  coal.  The  coal  of  the  furnace  and  the  oxygen  of  the  air  rush 
together  and  sacrifice  their  energy  of  chemically-separate  position*,  which  was 
originally  obtained  by  the  action  of  the  chlorophyll  in  the  coal-producing 
plants. 


50  WORK  AND  ENERGY.  [CHAP. 

When  a  plant  is  exposed  to  sunlight  it  has  the  power,  by  means  of  the 
chlorophyll  or  colouring  matter  of  the  leaves,  of  breaking  up  carbonic 
dioxide,  CO2,  of  evolving  part  of  its  oxygen  in  the  free  form,  and  of  depositing 
the  carbon  in  a  less  oxidised  form  in  its  own  tissues.  The  work  thus  done 
by  the  plant  in  tearing  asunder  the  constituents  of  CO2  it  is  enabled  to  do  by 
the  energy  supplied  to  it  in  the  form  of  Light  and  Heat  radiated  from  the  Sun. 

The  Sun's  radiant  energy  has  next  to  be  accounted  for.  This  is  not 
derived  from  combustion,  for  the  sun  would  last  but  a  comparatively  short 
time  if  its  energy  were  derived  from  any  such  source  :  its  radiation  of  energy 
seems  to  correspond  to  16,500  horse-power  from  every  square  foot,  and  such 
an  enormous  outflow  would  soon  exhaust  the  store  of  energy  if  the  sun  were 
merely  a  huge  fire :  if  of  coal,  it  could  not  last  much  more  than  about  400 
years.  It  has  been  suggested  that  the  meteorites  which  fall  into  the  sun  in 
great  numbers  are  capable  of  accounting  for  the  sun's  energy  ;  of  the  thick- 
ening of  the  sun  due  to  this  cause  a  very  small  amount  corresponds  to  a  very 
large  amount  of  energy.  Those  meteorites  which  strike  our  own  earth's 
atmosphere  are  retarded  and  greatly  heated  in  their  course  through  the  upper 
regions  of  the  air.  If  they  be  small  enough  they  are  entirely  broken  up, 
and  their  dust,  characteristically  ferruginous,  settles  down  on  the  surface  of 
the  earth,  and  may  be  recognised  in  the  dust  collected  from  some  specially 
favourable  spots,  such  as  glaciers,  roofs,  and  snowy  wastes,  and  the  bottom  of 
the  sea.  The  kinetic  energy  lost  by  a  meteorite  falling  upon  the  earth 
becomes  distributed  between  it  and  the  earth  in  the  system  of  which  the 
meteorite  becomes  a  part,  and  this  contributes  to  the  total  energy  possessed 
by  the  earth  ;  while  its  material  goes  to  increase  the  earth's  mass.  In  this 
way,  Nordenskjold  computes,  the  earth  gains  every  year  at  least  half-a- 
million  tons.  In  the  same  way,  the  meteorites  which  fall  on  the  sun  must 
produce  a  flash  of  light,  some  heat,  and  a  slight  thickening  of  the  sun.  It 
has  also  been  suggested  that  a  very  slight  shrinking  of  the  sun's  mass  would 
evolve  a  large  amount  of  energy,  its  particles  not  being  so  far  from  one 
another  after  this  contraction  in  bulk ;  and  this  view  is  confirmed  by  the 
fact  that  the  sun's  total  heat-radiation  is  greatly  more  than  can  be  accounted 
for  by  any  permissible  demand  on  the  meteorite  theory.  The  Sun  must 
thus  be  considered  as  possessing  a  store  of  Energy,  but  as  having  been  itself 
originally  made  up  by  the  coalescence  of  widely  scattered  material. 

The  question  next  arises,  How  did  the  meteorites  get  their  energy  of 
motion,  or  the  widely  scattered  material  its  potential  energy  ?  This  would 
relegate  us  to  the  consideration  of  the  Universe  as  a  system  of  masses  and 
particles  containing  as  a  whole  a  fixed  quantity  of  energy :  and  this  would 
bring  us  to  the  problem  of  the  origin  of  this  system. 

Availability  of  Energy.  — When  a  certain  amount  of  energy 
has  been  spent  in  rubbing  a  button,  tbe  button  is  perceptibly 
warmed.  The  heat  produced  is  exactly  equal  to  the  work  done 
in  rubbing.  It  is  Jwv2,  where  m  is  the  hot  mass  and  v  the 
average  velocity  of  its  moving  particles.  All  this  we  know.  If 
a  little  time  elapse,  the  button  is  no  longer  perceptibly  warm : 
it  has  shared  its  heat  with  surrounding  objects :  their  particles 
have  been  induced  to  oscillate  more  rapidly.  Heat  has  thus  a 
tendency  to  become  uniformly  diffused.  It  is  then  no  longer 


iv.]  AVAILABILITY   OF  ENERGY.  51 

available  to  man  for  doing  work.  It  ceases  to  be  power  of  doing 
work  as  far  as  he  is  concerned ;  but  none  the  less  do  the  parti- 
cles of  a  hot  body  set  in  motion  the  particles  of  a  cooler  body, 
arid  the  energy  which  has  thus  been  imparted  to  these  they  can 
in  their  turn  share  with  the  particles  of  other  cooler  bodies. 
The  Heat  of  a  hot  body  tends  uniformly  to  diffuse  itself  through- 
out the  whole  material  Universe. 

In  every  Transformation  of  Energy  we  find  that  some 
energy  is  wasted  through  conversion  into  Heat,  the  result,  direct 
or  indirect,  of  friction,  noise,  flashes  of  light,  and  so  on.  This 
heat  is  presently  distributed  pretty  uniformly  among  the  sur- 
rounding objects,  and  can  no  more  be  made  use  of  by  us  for  the 
sake  of  producing  work.  A  large  quantity  of  the  Energy  of 
the  Universe  must  have  already  assumed  this  relatively-useless 
condition,  and  in  the  course  of  time  the  whole  of  the  Energy  in 
the  Universe  will  have  assumed  it.  The  Energy  of  the  Universe 
is  a  constant  amount :  some  of  it  is  available,  some  is  non-avail- 
able :  the  former  is  in  every  phenomenon  somewhat  diminished 
but  never  increased :  the  non-available  energy  is  constantly  in- 
creasing: hence  the  Available  Energy  of  the  Universe  tends 
to  zero. 

Lord  Kelvin  expresses  this  by  saying  that  the  Motivity  (the  proportion 
between  the  theoretically-available  energy  and  the  whole  energy)  of  the 
Universe  tends  to  zero. 

If  with  this  clue  we  trace  back  the  history  of  the  Energy  of 
the  Universe,  we  find,  as  we  go  back,  less  and  less  of  the  total 
Energy  of  the  Universe  to  have  become  non-available.  On  going 
back  far  enough  we  arrive  at  a  definite  period  when  none  of  the 
total  energy  had  become  non-available.  But  in  every  actual 
phenomenon  there  is  always  Dissipation  in  this  way  of  some 
part  of  the  total  energy  of  a  system.  Hence  we  find  that  we 
are  forced  to  realise  a  precise  instant  before  which  there  were  no 
phenomena  such  as  those  with  which  we  are  now  acquainted, 
and  since  which  such  phenomena  as  are  due  to  those  relations  of 
matter  and  energy  which  are  within  our  knowledge  have  been 
occurring:  while  in  the  future  we  have  to  contemplate  a  mo- 
ment at  which  the  whole  physical  universe  will  have  run  itself 
down  like  the  weights  of  a  clock,  and  after  which  an  inert,  uni- 
formly-warm mass  will  represent  the  whole  material  order  of 
things. 

The  only  way  of  escape  from  this  conclusion  is  to  lay  emphasis  on  the 
fact  that  one  part  of  the  total  Energy  of  the  Universe  is  unavailable  to  man, 


52  WORK  AND   ENERGY.  [CHAP. 

and  to  suggest  that  at  some  time  a  state  of  things  may  supervene,  as  a  result 
of  which  the  molecular  motion  which  is  implied  in  a  state  of  uniformly- 
diffused  heat  may  be  so  arranged  and  directed  as  once  more  to  produce  a 
state  of  things  such  that  particles  may  become  aggregated  into  masses,  in 
which  all  the  particles  may  move  on  the  whole  in  the  same  direction.  This 
is  what  Clerk  Maxwell's  "  Demon  "  is  pleasantly  imagined  to  do ;  he  separates 
those  particles  which  he  prevents  from  going  in  one  direction  from  those 
which  he  allows  to  go  in  another,  so  that  ere  long,  without  expending  any 
work,  he  has  the  particles  divided  into  two  groups,  moving  in  opposite 
directions.  This  is  interesting,  but  it  is  not  pretended  that  it  is  any  other 
than  a  speculation. 

"  Conservation  of  Force  "  an  erroneous  phrase.  —  There  is  now  no 
warranty  for  this  expression.  It  was  originally  a  translation  of  the  German 
Erhaltung  der  Kraft,  where  Kraft,  meaning  strength  or  force,  was  used  in 
1847  by  Professor  Helmholtz,  for  want  of  a  better  term,  to  indicate  what  is 
now  rigorously  named  Energie  or  Energy.  Forces  are  of  the  same  order  as 
pressures  exerted,  pounds'  or  grammes'  weight,  resistances  overcome ;  forces 
may  be  represented  by  lines  which  indicate  their  magnitude  and  direction. 
Energy  is  of  the  same  order  as  work  accomplished,  as  pounds'  or  grammes' 
weight  or  resistances  overcome  through  a  certain  number  of  feet  or 
centimetres,  and  it  may  be  represented  by  areas  which  are  independent  of 
direction. 

The  Hydraulic  Press  apparently  creates  Force,  and  if  its  action  be  re- 
versed, Force  disappears ;  but  the  work  done  upon  it  must  be  the  same  as 
the  work  done  by  it,  and  though  there  is  no  Conservation  of  Force,  yet  there 
is  strict  Conservation  of  Energy  in  this  as  in  all  those  other  mechanical  con- 
trivances iii  which  Force  is  altered  in  amount. 

We  have  seen  that  Energy  may  be  represented  by  Fs,  the 
product  of  force  acting  or  resisted  through  space  s;  by  mgh 
where  mass  m  is  raised  through  height  h  against  gravity  whose 
local  acceleration  is  g ;  by  ^mv2  when  a  mass  m  has  a  velocity  v 
imparted  to  it.  We  shall  further  see  that  Energy  may  be  repre- 
sented by  the  product  JQV,  where  Q  is  a  charge  of  Electricity 
and  V  a  numerical  quantity  called  Electric  Potential ;  by  the 
product  fop  of  a  volume  &  of  fluid  forced  into  a  space  against 
an  average  pressure  p  units  of  force  per  unit  of  area  of  the 
bounding  surface  of  the  fluid;  by  the  product  of  a  chemical 
affinity  (which  is  equal  to  the  work  done  in  separating  the  atoms 
of  an  equivalent  of  a  chemical  compound)  into  the  number  of 
electro-chemical  equivalents  which  enter  into  combination  ;  and 
in  other  similar  ways.  These  things  will,  however,  find  their 
explanation  in  due  place. 

Problems. 

1.  Energy  is  power  of  doing  work :  this  depends  on  |mv2;  a  body 
moving  with  a  certain  velocity  v  can  pierce  a  plank  of  thickness  t ;  if  it 
move  with  velocity  vt,  what  thickness  can  it  pierce  ?  —  Ans.  tt  =  £(vy/y)2. 


iv.]  ENERGY.  53 

2.  A  shot  travelling  at  the  rate  of  700  feet  a  second  is  just  able  to  pierce 
a  2-inch   board.     What  velocity    is    required  to  pierce   a   3-inch   board  ? 
— Ans.  700  x  (V3  -*-V2)=  85742  feet  per  second. 

3.  A  shot  travelling  at  a  certain  rate  can  bury  itself  10  feet  in  sand : 
how  far  could  a  shot  travelling  with  double  that  speed  bury  itself? — Ans. 
40  feet. 

4.  If  a  mass  of  154-51  pounds  be  allowed  to  fall  10  feet,  but  in  its  fall 
be  made  to  set  a  train  of  mechanism  in  action,  arid  if  that  mechanism  do 
no  other  work  than  to  stir  up  a  pound  of  water  with  a  paddle,  how  much 
will  the  water  thus  stirred  up  be  warmed ?  —  Ans.  2°  F. 

5.  If  a  locomotive  weighing  5000  kilogrammes  run  at  the  uniform  rate 
of  10  metres  per  second  round  a  circular  railway  whose  radius  is  2  kilo- 
metres, what  will  be  its  kinetic  energy? — Ans.   m  =  5,000,000  grms. ;  v  = 
1000  cm.  per  second;  ^mv*  =  2,500000,000000  Ergs,  or  250  Ergtens.     The 
energy  does  not  depend  on  the  radius  of  the  circle,  for  it  does  not  depend 
on  the  form  of  the  path  traversed,  but  only  on  the  velocity  at  each  instant 
along  that  path ;  kinetic  energy  is  independent  of  direction. 

Graphic  Representation  of  Energy.  —  The  representation 
of  work  by  the  product  Fs  (force  acting  into  the  space  through 
which  it  acts  or  is  resisted)  finds  its  graphical  equivalent  in  the 
representation  of  work  done  as  a  rectangular  Area,  the  product 
of  two  lines,  of  which  one  represents  the  Force  acting  and  the 
other  the  Space  through  which  a  body  has  been  moved.  If  any 
instrument  can  be  devised  which  will  mechanically  describe 
such  an  area,  the  amount  of  work  done  by  a  moving  body  can 
be  recorded;  such  an  instrument  is  a  Dynamometer.  This 
name  is,  as  we  have  already  seen,  applied  to  the  apparatus  in 
which  an  elastic  spring  is  deformed,  the  extent  of  its  deforma- 
tion showing,  by  comparison  with  that  produced  by  a  given 
weight,  the  amount  of  Force  acting  on  the  instrument.  The 
same  name  has,  however,  been  given  to  instruments  designed  to 
record  not  only  the  force  acting  on  the  spring  at  any  given 
instant,  but  also  the  whole  Energy  spent  in  producing  the  def- 
ormation, and  measured  by  a  simultaneous  record  of  the  force 
acting  and  of  the  space  through  which  it  has  acted. 

If  a  distorted  spring  have  a  writing-point  attached  to  it, 
as  the  distortion  of  the  spring  varies  the  pencil  will  move  back- 
wards and  forwards  in-  one  line ;  if  a  piece  of  paper  be  held 
against  the  writing-point  as  it  travels  back  and  fore,  the  tracing 
produced  is  not  instructive,  for  it  is  simply  a  line  traced  over  and 
over.  If  the  paper  be  drawn  past  the  writing-point  at  a  uniform 
rate,  the  line  drawn  is  a  curve,  from  which  may  easily  be  de- 
duced the  mean  value  of  the  deforming  force  during  the  whole 
time  of  observation.  If,  however,  the  paper  be  moved  not  uni- 
formly but  at  a  varying  rate,  proportioned  at  every  instant  to 


54 


WORK  AND  ENERGY. 


[CHAP. 


the  space  passed  through  by  the  moving  body  during  given 
successive  equal  periods  of  time  (that  is,  to  the  rate  of  change 
of  deformation  of  the  spring),  then  there  are  two  factors  re- 
corded in  the  same  tracing  —  first,  the  amount  of  Space  passed 
through  (this  being  indicated  by  the  amount  of  paper  unrolled 
under  the  writing-point)  in  a  given  period  of  time  ;  and  second, 
the  Force  which  has  acted  in  producing  deformation  (this  being 
recorded  by  the  oscillations  of  the  writing-point  attached  to  the 
deformed  spring). 

If  the  writing-point  thus  attached  to  the  spring  be  supposed 
to  draw  the  curve  ABCDEF  of  Fig.  9,  the  various  parts  of  the 
line  give  rise  to  the  following  discussion.  The  line  Oabcd  shows 

the  various  spaces  tra- 
versed by  the  body  set 
in  motion  ;  the  lines 

Fig.9.  aA,  6B,  <?C,  etc.,  show 

the  various  pressures  or 
E  forces  in  action  at  suc- 

cessive instants  of  time. 
The  condition  of  affairs 
is  more  easily  realised 
if  we  consider  a  cylin- 
der, the  steam  in  which 
pushes  a  piston.  Then 
the  expansion  of  the 
steam  is  correlated  to 
the  movement  of  the  piston,  which  may  be  represented  by  dis- 
tances along  the  line  Ox.  Then  ab  may  be  supposed  to  denote 
the  expansion  of  the  steam  as  its  volume  increases  from  Oa 
to  Ob;  a  A.  or  £B  its  pressure;  and  in  that  case  the  work  done 
during  the  increase  of  volume  ab  would  be  represented  by  the 
rectangle  aAB6.  When  the  working  substance  expands  still 
further,  so  that  its  increase  in  volume  is  be,  the  pressure  or 
force  acting  is  again  constant,  and  the  work  done  is  repre- 
sented by  the  rectangle  bC.  Next,  when  the  volume  Oc  be- 
comes Odf,  the  pressure  sinks  from  cC  to  dD  ;  the  average 
value  of  the  pressure  is  ^  (<?C  -f-  c?D),  and  the  work  done  (or 
average  pressure  x  space  cd)  =  ^(<?C  +  dD)  cd.  This  is  the  area 
of  the  figure  cCDd,  which  accordingly  represents  the  work  done 
during  the  increase  cd  of  volume.  The  area  De  is  made  up  of 
numerous  rectangles,  and  if  these  be  made  sufficiently  numerous 
the  line  DE  is  a  curved  line.  The  area  comprised  between  the 


L               B         C                  / 

ih 

r 

\| 

D/ 

IT.] 


MEASUREMENT   OF   ENERGY. 


55 


curved  line  DEF,  the  ordinates  dD  and/F,  and  the  abscissa  df, 
represents  the  work  done  during  expansion  from  the  volume  Od 
to  the  volume  Of.  The  area  #AF/a  represents  the  work  done 
during  expansion  from  volume  Oa  to  volume  Of. 

The  Indicator-Diagram.  —  Since  the  time  of  James  Watt  engineers 
have  been  accustomed  to  make  their  engines  record  their  own  working  by 
the  method  just  discussed.  In  the  Indie  at  or -Diagram,  as  the  curve 
traced  out  is  called,  the  two  factors  which  it  is  desired  to  record  are  the 
Space  traversed,  which  is  measured  by  the  amount  of  movement  of 
the  piston,  and  the  Force  acting,  which  is  the  pressure  of  the  steam  in 
the  cylinder.  The  former  factor,  the  space  moved  through  by  the  piston,  is 
determined  by  making  the  piston  or  any  part  of  the  machinery  upon  which  it 
directly  acts  set  in  operation  the  mechanism  that  unrolls  the  paper  upon 
which  the  record  is  to  be  preserved.  This  paper  is  drawn  over  the  writing- 
point  at  a  rate  depending  on  the  velocity  of  the  piston  :  hence  the  spaces 
traversed  by  the  piston  during  successive  intervals  of  time  are  proportional 
to  the  amount  of  paper  which  is  drawn  under  the  writing-point. 

The  pencil  is  borne  by  the  piston  of  a  small  side  cylinder  attached  to 
the  main  cylinder.  The  steam  is  let  into  this,  and  presses  the  little  piston 
outwards ;  it  presses  it  against  a  spring  until  the  resistance  offered  by  that 
spring  prevents  further  propulsion.  If  the  pressure  were  constant,  the 
little  piston  would  remain  at  the  same  level ;  but  as  the  pressure  of  the 
steam  varies,  the  position  of  the  piston  also  varies,  as  it  lies  between  the  op- 
posing spring  and  steam.  Its  displacement,  then,  is  at  each  instant  propor- 
tional to  the  pressure  of  steam  in  the  cylinder,  which  is  the  second  factor. 
If  the  paper  be  rolled  over  the  pencil-point  when  no  steam  has  access  to  the 
side  cylinder,  a  smooth  line  is  drawn,  the  Line  of  No  Pressure;  if  the 
steam  be  allowed  to  enter  the  side  cylinder,  the  divergences  of  the  line 
then  produced,  from  the  line  of  no  pressure,  measure  the  variations  of  the 
pressure  of  the  steam. 

While  the  machine  is  working,  steam  is  not  allowed  to  enter  the  side 
cylinder  until  the  apparatus  is  ready  to  record.  The  small  piston  is  conse- 
quently at  rest.  While  the  piston  of  the  main  cylinder  is  moving  in  one 


Fig.  10. 


direction,  paper  is  rolled  over  the  pencil-point;  as  the  piston  slackens  speed 
the  paper  also  slackens  in  speed ;  as  the  piston  stops,  the  paper  does  the 
same ;  as  the  piston  travels  in  the  opposite  direction,  the  paper  travels  in  the 
opposite  way  at  a  proportionate  rate.  So  long  as  the  steam  is  not  admitted 
into  the  side  cylinder,  the  paper  travels  backwards  and  forwards  over  the 
pencil,  and  the  same  straight  line  is  traced  and  retraced.  The  steam  is  ad- 


56  WORK  AND   ENERGY.  [CHAP,  iv.] 

mitted  to  the  side  cylinder  for  the  space  of  one  complete  oscillation  of  the 
main  piston,  and  the  pencil-point  itself  travels  in  accordance  with  the  vary- 
ing pressure  of  steam  during  that  period.  The  curve  traced  thereby  is  com- 
posed of  two  parts.  The  one,  ACB,  is  produced  during  expansion.  The 
work  done  by  the  steam  in  the  cylinder  during  its  expansion  is  the  area 
ACEba  if  Oab  be  the  line  of  no  pressure.  When  the  piston  has  finished  its 
stroke,  it  —  and  therefore  the  paper  —  stands  for  an  instant  at  rest.  Then 
the  piston  is  pressed  against  the  steam  either  by  other  steam  or  by  the 
atmosphere,  and  the  paper  is  drawn  backwards.  Work  is  thus  done  against 
the  steam  during  the  backward  stroke,  and  it  is  represented  by  the  area 
EDAob,  where  BDA  is  the  line  recorded. 

The  difference  between  the  areas  ACBJa  and  BDAa&  represents  the 
excess  of  work  done  by  the  steam  over  that  done  against  it :  hence  the  total 
work  done  by  the  engine  is  represented  by  the  area  of  the  surface  ACBD 
traced  out  by  the  pencil  in  the  formation  of  the  so-called  Tndicator-Diagram. 

The  pencil  of  the  Indicator  in  tracing  out  such  a  curve  mechanically 
performs  an  operation  equivalent  to  that  which  the  mathematician  effects 
when  he  sums  up  areas  by  means  of  the  algebraic  processes  of  the  Integral 
Calculus. 

In  some  cases  it  is  sufficient  to  know  the  mean  force  acting. 
In  such  cases  the  space  traversed  being  a  known  quantity,  the 
energy  can  be  determined  if  the  mean  force  alone  be  recorded. 
For  example,  if  the  mean  force  required  to  pull  a  vehicle  be 
found,  it  is  a  very  simple  matter  to  multiply  the  recorded  mean 
force  by  the  space  traversed  in  order  to  find  the  total  amount  of 
energy  expended.  In  Marey's  investigations  (Trav.  du  Laborat., 
1875)  into  the  comparative  total  work  expended  on  a  vehicle, 
according  as  an  elastic  spring  is  or  is  not  placed  between  it  and 
the  draught  animal,  a  capsule  was  so  arranged  that  the  air  in  it 
suffered  irregular  compressions  and  rarefactions,  corresponding 
to  the  irregular  jolts  between  the  animal  and  the  car.  The 
writing-point,  set  in  movement  by  the  correspondingly-irregular 
oscillations  of  one  of  the  walls  of  the  capsule,  which  was 
flexible,  described  irregular  lines  when  there  was  no  elastic  in- 
termediary, and  more  regular  ones,  nearer  the  line  of  no  dis- 
turbance, when  there  was  such  an  intermediary  introduced  ; 
from  these  it  was  found  that  the  mean  forces,  and  therefore  the 
amounts  of  energy  expended  in  the  two  cases  were  in  the  ratio 
of  about  4  to  5,  showing  that  the  use  of  an  elastic  spring  be- 
tween a  draught  animal  and  the  vehicle  which  it  draws  results 
in  an  economy  of  labour  amounting  to  about  25  per  cent. 


CHAPTER  V. 

KINEMATICS. 

To  the  part  of  Science  which  deals  with  Motion,  considered  per 
se  and  without  reference  either  to  the  force  producing  it  or  to 
the  body  moved,  is  given  the  name  of  Kinematics.  The  nature 
of  the  questions  discussed  under  this  title  is  essentially  mathe- 
matical ;  and  though  no  great  acquaintance  with  mathematical 
methods  is  presumed  in  the  reader  of  this  volume,  it  will  be 
necessary  to  assume  in  him  a  certain  amount  of  knowledge  of 
the  most  elementary  geometry  and  algebra. 

GENERAL  PROPOSITIONS. 

Direction.  — •  There  cannot  be  Motion  without  Direction ; 
we  cannot  think  of  a  body  or  a  point  as  moving,  and  yet  not 
moving  in  any  direction.  If  it  move  at  all,  it  must  either  move 
so  as  to  travel  constantly  in  the  same  direction,  in  which  case  it  is 
moving  in  a  straight  line;  or  else  the  direction  of  its  motion 
must  change  as  it  proceeds  from  point  to  point  of  the  path  tra- 
versed, so  that  the  body  travels  in  some  kind  of  curved  line. 

In  the  great  majority  of  those  curves  which  possess  physical 
interest  as  being  those  in  which  bodies  actually  do  move,  it  is 
possible  to  draw  at  any  point  of  the  curve  a  straight  line  known 
as  the  Tangent  to  the  curve  at  that  point.  The  Tangent  to  the 
Circle  at  any  point  is  familiar  enough,  and  is  easily  understood 
to  be  a  straight  line  at  right  angles  to  a  radius  connecting  the 
centre  of  the  circle  with  that  point  of  the  circumference  at 
which  the  tangent  is  to  be  drawn  ;  and  the  characteristic  prop- 
erty of  the  line  as  a  tangent  is  that  it  touches  the  circle  without 
cutting  it.  Tangents  may  in  a  similar  way  be  drawn  to  most 
curves,  so  as  at  any  determined  point  to  touch  but  nottp  cut  the 
curve  unless  the  curve  changes  its  curvature  beyond  the  point 
at  which  the  tangent  touches  it. 

57 


58  KINEMATICS.  [CHAP. 

If  a  circle  be  drawn  on  a  very  large  scale,  and  a  tangent  be  drawn  to  it 
at  any  point  chosen,  it  will  be  found  that  the  larger  the  scale  the  more 
nearly  will  the  circle  appear  to  coincide  with  the  tangent  at  the  point  of 
contact.  This  can  easily  be  seen  by  actually  drawing  such  a  figure.  In  fact, 
if  a  circle  be  drawn  on  a  very  great  scale,  any  very  little  part  of  its  circum- 
ference will  appear  to  be  practically  straight.  Of  course  it  is  not  straight, 
but  by  drawing  the  circle  sufficiently  large,  and  by  diminishing  the  size  of 
the  little  part  of  the  circumference  considered,  the  approximation  to  perfect 
straightness  in  the  little  part  or  "element"  considered  may  be  rendered  as 
close  as  may  be  desired.  Such  a  circle  may,  then,  be  considered  as  a 
polygon,  having  an  infinite  —  greater,  that  is,  than  any  definite  assign- 
able —  number  of  sides,  the  length  of  each  of  which  is  indefinitely  small, 
and  each  of  which  coincides  for  an  infinitesimal  distance  with  the  tangent 
which  is  drawn  past  it. 

What  is  true  of  a  large  circle  is  true  of  a  small  one,  and  hence  motion 
in  a  circle  may  be  considered  as  motion  round  a  polygon  of  an  indefinite 
number  of  sides  ;  whence  the  following  proposition. 

As  a  body  or  point  moves  round  a  circle,  the  Direction  of 
its  motion  is  that  of  the  Tangent  at  each  successive  instant. 
Similarly,  the  direction  of  motion  of  a  body  which  travels  in  any 
other  curve  is,  at  each  successive  instant  of  time,  the  same  as 
the  direction  of  the  Tangent  to  the  Curve  at  the  point  of  the 
curve  momentarily  occupied  by  the  moving  body. 

Velocity.  —  We  have  already  anticipated  some  kinematical 
statements  in  discussing  the  velocity  of  a  moving  body.  This 
was  defined  as  the  distance  passed  over  in  a  unit  of  time  by  a 
body  in  motion  ;  and  if  we  consider,  not  the  moving  body,  but 
the  motion  itself,  we  may  say  that  one  of  the  necessary  proper- 
ties of  pure  Motion  is  Velocity.  It  is  not  possible  to  think  of 
Motion  without  thinking  of  a  corresponding  definite  Rate  of 
motion,  which,  if  there  be  motion  at  all,  cannot  be  zero,  and  on 
the  other  hand  cannot  be  infinite,  so  long  as  Space  and  Time  are 
related  to  Motion  in  the  way  in  which  experience  shows  them  to 
be  ;  and  the  idea  of  Rate  of  Movement  is  as  necessary  a  constit- 
uent of  the  idea  of  Motion  as  is  that  of  Direction. 

Velocity  may  be  uniform  or  variable.  The  measurement 
of  uniform  velocity  is  simple  enough  ;  and  it  has  already  been 
explained  on  what  principle  the  measurement  of  variable  veloc- 
ity is  based.  Whether  the  direction  of  motion  be  constant  or 
variable  —  whether  the  moving  particle  travel  in  a  straight  line 
or  in  a  curve  —  the  principle  involved  is  always  the  same,  namely, 
that  the  velocity  v  of  a  moving  particle  is  the  length  of  path 
traversed  by  it  in  a  unit  of  time,  or  the  length  of  path  which 
would  have  been  traversed  by  it  during  a  unit  of  time  if  the 
speed  had  remained  uniform  during  that  period.  In  the  case  of 


v.]  VELOCITY.  59 

motion  in  curved  paths,  there  arise  subsidiary  mathematical  dif- 
ficulties in  the  estimation  of  the  precise  length  of  the  path 
traversed,  but  these  only  arise  in  the  determination  of  the  value 
of  one  of  the  terms  of  the  formula  v  =  s/t,  and  do  not  affect 
the  validity  of  the  formula  itself. 

Velocity  may  be  otherwise  defined  as  the  relation  of  change  of  position 
to  change  of  time.  If  at  a  certain  instant  of  time  the  moving  point  be  at  a 
distance  B  from  a  fixed  point  chosen  as  a  standard  of  reference,  then,  as  time 
goes  on,  the  position  of  the  body,  and  therefore  the  value  of  a,  changes.  Let 
the  time  during  which  motion  is  going  on  be  &,  a  very  small  element  of 
time,  and  the  corresponding  change  of  position  be  8s,  then  this  relation  of 
the  change  of  position  to  the  change  of  time  may  be  expressed  by  the  frac- 
tion 8s/ 8^,  which,  when  &t  is  chosen  sufficiently  small,  becomes  the  function 
familiar  to  students  of  the  Differential  Calculus  as  ds/dt.  This  change  of  B 
in  accord  with  the  passage  of  time  may  be  very  advantageously  represented 
as  a,  where  the  dot  above  the  letter  indicates  "  the  value  of  the  change  in 
unit  of  time  of  "  the  quantity  expressed  by  the  letter  over  which  the  dot  is 
placed,  when  the  unit  of  time  chosen  is  very  small.  This  is  the  notation 
employed  by  Newton  in  his  Fluxions,  B  being  the  change  or  Fluxion  of  s. 
If  the  velocity  (s)  itself  change,  the  change  of  velocity  in  unit  of  time  — 
which  we  otherwise  know  under  the  name  of  Acceleration  —  would  be  rep- 
resented by  the  symbol  a.  So  if  the  acceleration  (S)  itself  varied,  the 
change  of  acceleration  per  unit  of  time  would  be  represented  by  the  symbol 
B.  Such  a  number  of  dots  as  this  rarely  occurs  in  physical  problems. 

We  may  here  make  use  of  previous  discussions  to  bring  together  some 
of  the  symbols  used  to  express  frequently-recurring  terms. 

Distance  of  particle  from  point  of  reference  =  B. 

Velocity,  v  =  B  =  a/t. 

Acceleration,  a  =  change  of  v  in  unit  of  time  =  v/t  =  v  =  a*. 

Force  F  =  ma  =  mv  =  ma. 

Work  W=  PB  =  maa. 

Rate  of  doing  work  (Lord  Kelvin's  Activity,  Newton's  Actio 
Agentis)  =  W/t  =Fa/t  =  Fv  =  maa. 

[Action  (Maupertuis)  =  vs  or  2(vs)  ;  held  by  him  to  be  always 
a  minimum  in  unguided  motion  of  a  conservative  system : 
shown  by  Hamilton  to  present,  in  unguided  motion  between 
fixed  positions,  either  a  minimum  or  a  maximum  value  or 
else  a  value  little  affected  by  slight  variations  in  the  path 
traversed.] 

Energy  =  $mt;2=  \m  (s)2  =  Kinetic  energy. 

Do.    =  W=  Fs  =  maa  =  ma  =  Potential  energy. 

Dimensions.  —  Distance  of  a  particle  from  a  point  of  reference  may  be 
represented  by  a  straight  line,  which  is  measurable  in  units  of  Length. 
Space  traversed  in  a  straight  line  is  therefore  said  to  be  of  one  dimension 
in  Length,  and  may  be  represented  by  the  symbol  [L].  Velocity  is  a  Length 
(space  traversed)  divided  by  a  Time,  and  may  be  represented  by-the  symbol 
[L/T].  Momentum  is  Mass  x  Velocity;  its  dimensions  are  [ML/T].  Ac- 
celeration, the  velocity  acquired  per  unit  of  time,  is  a  Velocity  divided  by  a 
Time,  and  the  Dimensions  of  Acceleration  are  [L/T]  -4-  [T],  or  [L/T2]. 


60  KINEMATICS.  [CHAP. 

Force  is  a  Mass  x  Acceleration,  and  its  dimensions  are  accordingly 
[ML/T2].  Weight  and  Total  Pressure  have  the  same  dimensions  as  Force. 
Force,/,  and  pressure,  jo,  per  unit  area,  have  dimensions  [M/LT2].  Energy, 
if  we  take  the  expression  \mv^  has  the  dimensions  [M]  [L/T]2  or 
[ML2/T2]  ;  while  if  we  take  the  expression  was,  it  is  found  to  have  the 
dimensions  [M]  [L/T2]  [L]  or  [ML2/  T2],  the  same  result.  The  dimensions 
of  Work  are  the  same  as  those  of  Energy.  Energy  per  unit  volume  has 
dimensions  [M/LT2],  like  /  and  p  above.  Those  of  Activity  are  Work 
done  -  Time  =  [ML2/T3]. 

Examples.  —  1.    How  many  British  units   of  force   is  a  Dyne  equal 
to?     The     Dyne  =  [ML/T2]  =  [Gramme    x    Centimetre  /  Second2]  = 


15432  .. 

British  units. 


7000  x  3048 

2.  Suppose  it  were  affirmed  that  Force  is  Rate  of  gain  or  loss  of  Energy 
as  Time  goes  on ;  test  the  statement.     Using  a  dimensional  equation,  we 
would  have  [F]  =  [W/T]  or  [ML/T2]  =  [ML2/T2]  -  [T],  which  is  obvi- 
ously wrong. 

3.  Test  the  statement  that  Force  is  measured  by  Time-rate  of  Change 
of  Momentum.     Similarly  [F]  =  [Mom.]  -*-  [T],  or  [ML/T2]  =  [ML/T] 
-7-  [T],  which  is  consistent. 

4.  Test  the  assertion  that  Force  is  the  Rate  at  which  a  body  gains  or 
loses   Energy   as  it  traverses  Space.      [F]  =  [Energy /distance  traversed] 
=  [W/L],or  [ML/T2]  =  [ML2/T2]- [L]  =  [ML/T2],  which  is  consistent. 

Simultaneous  Motions.  —  If  a  particle  have  by  any  means 
two  separate  independent  motions  communicated  to  it  simul- 
taneously, each  will  produce  its  own  effect,  and  the  total  move- 
ment of  the  particle  can  be  found  by  any  process  of  summation 
which  may  be  found  mathematically  appropriate.  It  will  always 
be  found  as  the  result  of  experiment  on  bodies  which  may  be 
taken  to  represent  particles,  that  if  the  motions  imparted  to  the 
particles  be  themselves  constant  in  velocity  and  direction,  the 
result  of  their  concurrence  is  a  single  motion  in  a  straight  line 
with  a  single  velocity  and  direction.  The  single  motion  which 
is  produced  as  the  result  of  the  concurrence  of  two  motions  is 
called  their  Resultant,  and  they  with  regard  to  the  resultant 
are  called  its  Components.  If  a  ship  travel  from  west  to  east 
and  a  man  on  board  also  walk  from  west  to  east,  the  speed  of 
the  ship  and  the  speed  of  his  walking  will  have  to  be  added 
together  to  find  the  rate  at  which  he  is  moving  eastwards :  if 
the  ship  travel  from  west  to  east  and  he  walk  along  the  ship  from 
east  to  west,  the  difference  between  his  own  speed  and  that  of 
the  ship  is  the  rate  at  which  he  is  travelling  eastwards.  In  the 
latter  case  the  result  may  be  positive  —  i.e.  he  is  really  going 
eastward ;  negative  —  i.e.  he  is  really  going  westward ;  or  zero, 


PARALLELOGRAM   OF   VELOCITIES. 


61 


Fig.il. 


in  which  case  he  has  no  movement  at  all,  the  ship  carrying  him 
east  just  as  much  as  he  walks  to  the  west,  so  that  he  is  really 
beating  time  in  the  same  place.  If  a  steamer  travel  to  the  east 
and  be  at  the  same  time  carried  to  the  north  by  a  current,  the 
path  traversed  by  the  steamer  will  be  a  line  which  is  the  diag- 
onal of  a  parallelogram  whose  sides  represent  the  eastward  and 
northward  velocities  respectively.  The  steamer  will  describe 
this  diagonal  line  in  the  same  time  as 
it  would  have  taken  to  have  steamed  or 
to  have  drifted  along  one  or  the  other 
side  of  the  parallelogram  if  the  steam- 
ing or  the  drifting  respectively  had  been 
the  only  cause  of  its  movement.  Hence, 
to  find  the  Resultant  of  two  simultane- 
ous Velocities,  the  rule  is :  Construct  a 
parallelogram  whose  adjacent  sides  rep- 
resent in  magnitude  and  direction  the 
velocities  produced,  and  the  Diagonal 
which  lies  between  these  adjacent  sides  c 
represents  the  Resultant  Velocity.  If  the  lines  AB,  AC  (Fig. 
11),  represent  in  direction  and,  on  any  conventional  scale,  the 
magnitude  of  the  velocities  simultaneously  imparted  to  the  par- 
ticle A,  the  particle  A  will  move  along  the  line  AD  to  the  point 
D  in  the  same  time  that,  under  the  influence  of  the  velocity  AB 
alone,  it  would  have  taken  to  reach  B,  or,  under  the  influence 
of  the  velocity  AC,  to  reach  the  point  C.  The  actual  construc- 
tion of  the  diagram  representing  the  resultant  of  any  two 
velocities  is  an  easy  matter :  the  calculation  of  the  value  of  the 
resultant — that  is,  of  the  length  of  the  diagonal  —  involves  a 
little  geometrical  working. 

When  the  two  components  are  at  right  angles  to  one  another,  we  resort 
to  Eucl.  I.  47,  which  shows  that  in  a  right-angled  triangle  ACD  (Fig.  12), 
of  which  the  right  angle  is  at  C,  AD2  =  AC2 
+  CD2.  In  the  parallelogram  Fig.  11  (a)  it  is 
plain  that  AD2  =  AC2  +  CD2 ;  but  CD  =  AB ; 
hence  AD2  =  AC2  +  AB2 ;  or  in  words,  the 
square  of  the  resultant  is  equal  to  the  sum  of 
the  squares  of  the  two  components,  if  these 
be  at  right  angles  to  one  another. 


Fig.i2. 


Again,  if  they  be  not  at  right  angles  to  A 
one  another,  they  must  make  either  an  acute  or  an  obtuse  angle.     In  the 
former  case,  we  resort  to  Eucl.  II.  12,  which  shows  that  if  the  parallelogram 
be  drawn  (Fig.  13)  and  the  side  AC  be  produced  so  far  that  a  line  DE  can 
be  drawn  at  right  angles  to  it  from  the  point  D,  the  equation  AD2  =  AC2 


62 


KINEMATICS. 


[CHAP. 


+  CD2  +  2AC  •  CE  is  true ;  this  enables  us  to  find  the  value  of  AD  if  we 
know  that  of  AC  and  AB  (which  is  equal  to  CD),  and  if  we  can  find  that 
of  CE.  In  the  latter  case,  where  the  angle  BAG  between  the  components 

Fig.13.  n  Fig.H. 


AB  and  AC  is  obtuse  (Fig.  14),  Eucl.  II.  13  shows  us  that  if  we  drop  a  per- 
pendicular DE  from  D  upon  the  base  AC,  the  equation  AD2  =  AB2  +  AC2 
—  2AC  •  EC  holds  good,  and  enables  us  to  find  the  value  of  AD. 


Problems. 

1.  A  person  on  board  a  ship  which  is  going  eastwards  walks  back  and 
fore  at  the  rate  of  4  miles  an  hour  relative  to  the  ship  :  the  ship  is  travelling 
at  the  rate  of  12  miles  an  hour.     What  is  his  eastward  velocity  when  he  is 
walking  forward  ?  what  when  he  is  going  aft  ?  what  is  his  average  eastward 
velocity?  —  A ns.  16  miles  an  hour;  8  miles  an  hour;  12  miles  an  hour. 

2.  A  point  moves  with  velocity  a  eastwards  and  velocity  b  westwards 
simultaneously.     What  is  its  eastward  velocity  ?  —  Ans.  a  —  b. 

3.  Interpret  the   result  if  b  is  greater  than  a.  —  Ans.  The   eastward 
velocity  =  a  —  b  ;  this  is  negative  :  the  velocity  must  therefore  be  westward 
and  =  b  —  a. 

4.  Interpret  the  result  if  b  =  a.  — A  ns.  The  eastward  velocity  =  a  —  b  =  0  ; 
or  the  body  is  at  rest. 

5.  If  in  a  railway  carriage  compartment  a  man  walk  across  at  the  rate  of 
5  miles  an  hour  while  the  train  goes  forward  at  the  rate  of  12  miles  an  hour, 
what  will  have  been  his  real  path  and  velocity  relative  to  the  railway  line 
underneath?  —  A  ns.  In  Fig.  11  (a),  if  AB  =  12  and  AC  =  5  :  the  real  path 
is  AD,  which  has  a  value  of  13. 

6.  To  the  same  particle  are  imparted  a  velocity  of  12  and  one  of  6  feet 
per  second  in  directions  which  stand  to  one  another  at  an  angle  of  60° : 
what  is  the  direction  and  the  amount  of  the  resultant  velocity?  —  Ans.   In 
Fig.  13,  if  AB  represent  the  velocity  of  6  feet  per  second,  and  AC  on  the 
same  scale  that  of  12  feet,  the  angle  BAG  being  one  of  60  °,  then  the  line  AD 
will  indicate  the  direction  of  the  resultant  movement,  and  the  equation 
AD2  =  122  +  62  +  2  x  12  x  CE  (CE  being  seen,  since  the  triangle  CDE  is 
half  an  equilateral  triangle,  to  be  equal  to  half  CD  —  that  is,  to  3),  or 
144  +  36  +  72  =  252,  shows  that  AD  =  V252  =  15-822. 

7.  If   in  the  last  question  the  angle  between  the   directions   of  the 
velocities   had  been  120°,  what  would  the  resultant  -velocity  have  been? 
—  Ans.   In    Fig.   14,   if    AB    be    6    and    AC    12,   the   angle    BAC   being 
120°,   AD«_=  122  +  62  -  2  x  12  x  CE  =  144  +  36  -  72  =  108 ;    whence 
AD  =  V108  =  10-392  feet  per  second. 

8.  What  would  have  been  its  direction  ? 

In  the  same  figure,  14,  the  triangle  ADC  has  its  sides  AC  =  12,  CD  =  6, 
and  the  angle  DCA  =  60°.     By  trigonometry  we  find  that  the  angle  DAC  is 


v.]  TRIANGLE   OF  VELOCITIES.      .  .  63 

35°  16',  and  hence  the  direction  of  the  resultant  motion  is  inclined  to  those  of 
its  components  AC  and  AB  at  the  angles  of  35°  16'  and  84°  44'  respectively. 

Triangle  of  Velocities.  —  If  the  figures  just  made  use  of  be 
reduced  to  their  simplest  necessary  elements,  it  will  be  seen  that 
there  is  no  need  to  describe  a  complete  parallelogram  in  order  to 
find  the  line  which  would  be  its  diagonal.  The  three  sides  of  a 
triangle  are  quite  sufficient  to  express  the  relation  between  two 
component  velocities  and  their  resultant,  and  for  the  determina- 
tion of  the  resultant  of  two  velocities  the  rule  may  be  thus 
stated :  Take  a  starting-point ;  from  it  draw  a  line  representing 
in  magnitude  and  direction  one  of  the  component  velocities ; 
from  the  point  thus  arrived  at  —  that  is  to  say,  the  end  of  the 
line  thus  drawn  —  draw  another  line  similarly  representing  the 
second  component  velocity.  The  third  side  may  now  be  laid 
down,  and  the  problem  is  reduced  to  the  form  which  in  trigonom- 
etry is  simple  enough,  namely  —  Given  two  sides  of  a  triangle 
and  the  angle  between  them,  rig  is. 

to  find  the  third  side,  and  the 
angles  which  it  makes  with 
the  two  sides  given.  In  the 
triangle  ABC  (Fig.  15),  if  the  A 
sides  AB  and  BC  represent  the  component  velocities  in  amount 
and  direction,  AC  in  the  same  way  represents  their  resultant; 
and  it  will  be  observed  that  if  the  sides  of  the  triangle  be  taken 
consecutively  in  the  "cyclical "  order,  AB,  BC,  C  A,  the  direc- 
tion of  the  resultant  is  in  this  diagram  opposed  to  that  of  the 
components. 

Resolution  of  a  Velocity  into  Components.  —  The  con- 
verse proposition  is  one  of  very  general  utility.  In  the  former 
case,  by  Composition,  that  single  resultant  was  found  which 
was  the  effect  of  two  simultaneously-imparted  movements.  A 
single  movement  may,  conversely,  be  considered  as  the  resultant 
of  two  component  movements  which  we  may  wish  to  find.  The 
process  of  finding  them  is  known  as  the  Resolution  of  a  mo- 
tion into  its  components.  A  given  pair  of  velocities  can  only 
have  one  resultant,  for  if  two  sides  of  a  triangle  be  fixed,  there 
is  no  scope  for  variation  in  the  position  or  length  of  the  third 
side  ;  but  if  the  resultant  be  given,  it  can  be  resolved  into  com- 
ponents in  an  indefinite  number  of  ways,  for  there  can  be  an  in- 
finite number  of  triangles  made  by  supplying  two,  sides  when 
only  one  side  is  determinately  fixed.  Hence  the  question  how 
to  resolve  a  velocity  into  its  components,  set  in  this  vague  wayf 


64 


KINEMATICS. 


[CHAP. 


Fig.16. 


never  arises  ;  but  the  question  how  to  resolve  a  velocity  into  its 
components  in  certain  fixed  directions  is  of  constant  occurrence. 
Such  a  question  is  generally  solved  by  construction  in  the  fol- 
lowing way :  —  Let  AB  (Fig.  16)  be  a 
line  indicating  the  direction  and  rate  of 
movement  of  a  particle.  It  is  required 
to  know  what  are  the  corresponding 
components  in  two  directions,  LM,  NO, 
arbitrarily  chosen  or  determined  by  the 
conditions  of  the  problem.  Draw  lines 
from  both  extremities  of  the  line  AB, 
parallel  to  the  directions  assigned.  In 
B  this  way  a  parallelogram  will  be  formed 
in  which  AD,  CB  will  be  parallel  to  LM, 
and  DB,  AC  to  NO.  In  this  parallelo- 
gram AB  represents  the  single  motion 
whose  components  are  to  be  found:  the  length  of  AD  or  CB 
represents  the  proportionate  value  of  the  component  parallel 
to  LM,  and  DB  or  AC  the  proportionate  value  of  that  parallel 
to  NO.  Hence  the  problem  is  solved.  Very  little  practice 
enables  one  to  dispense  with  drawing  a  complete  parallelogram, 
and  to  find  the  components  by  constructing  either  the  triangle 
ABC  or  ABD.  If  numerical  values  are  required,  we  can  find 
them  by  means  of  the  known  angles  which  the  directions  of  LM 
and  NO  make  with  that  of  AB :  the  values  of  these  two  angles, 
together  with  the  numerical  value  of  AB,  give  by  trigonometry 
the  numerical  values  of  AD  and  DB,  which  represent  the  com- 
ponents. If  we  wish  to  resolve  a  single  velocity  into  components 
at  right  angles  to  one  another,  the  process  is  precisely  the  same, 
LM  and  NO  being  drawn  at  right  angles  to  one  another.  A 

modified  form  of  the  problem 
rig.17.  which  we  very  often  encoun- 

ter is  —  Given  a  velocity  in  a 
certain  direction,  what  is  the 
value  of  its  component  in 
another  assigned  direction  ? 
This  is  solved  by  the  follow- 
ing construction  :  —  Let  AB 
(Fig.  17)  represent  the  given 
velocity  and  LM  the  direction 
of  the  required  component  of  AB.  Draw  through  A  a  straight 
line,  AC,  indefinite  in  length,  but  parallel  to  LM.  From  the 


.-c 


TRIANGLE   OF   VELOCITIES. 


65 


other  extremity,  B,  of  the  line  AB,  draw  a  line  BD  at  right 
angles  to  AC,  cutting  it  in  the  point  E.  AE  is  the  component 
required  in  the  direction  parallel  to  LM. 

In  Fig.  17,  AE  =  AB-cos£;  and  BE  =  AB.sin£;  where  £  is  the  angle 
BAE. 

Problems. 

1.  A  velocity  of  30  feet  per  second  :  what  is  the  value  of  its  component 
in  a  direction  which  makes  with  its  own  an  angle  of  60°  ?  —  A  ns.   In  Fig.  17, 
if  the  angle  ABD  be  60°,  the  line  BA  may  represent  the  velocity  given;  BE 
represents  its  component  at  an  angle  of  60°  with  it.     The  triangle  BEA  is 
half  an  equilateral  triangle,  and  BE  is  half  of  BA;  it  represents  therefore  a 
component  velocity  of  15  feet  per  second. 

2.  A  velocity  of  20  feet  per  second  :  what  is  the  value  of  its  component 
whose  direction  makes  with  its  own  an  angle  of  30°?  —  Ans.  In  the  same 
figure,  if  the  angle  EAB  be  30°,  and  AB  represent  the  velocity  of  20  feet  per 
second :  in  such  a  triangle  AE  :  AB  :  :   V3/-4  :  1,  and  the  value  of  the  com- 
ponent is  V3/4  x  20  =  V300  =  17-32  feet  per  second. 

3.  A  velocity  of  60  feet  a  second  :  what  is  the  value  of  its  component 
at  an  angle  of  45°?— Ans.   VF/2  x  60  =  4242  feet  per  second. 

4.  A  velocity  v  in  a  certain  direction :  what  is  its  component  at  right 
angles  to  that  direction  ?  —  A  ns.  It  has  none. 

Composition  of  more  than  Two  Velocities.  —  If  more  than 
two  velocities  be  imparted  to  a  body  the  resultant  is  always,  if 
they  themselves  be  uniform  in  amount  and  direction,  a  single 
uniform  motion  in  a  straight  line.  If  the  several  velocities  im- 
parted be  all  in  the 
same  plane,  their  re-  Pig.is. 

sultant  may  easily  be 
found  by  finding  the  D(  -""" 
resultant  of  any  two  of 
them,  compounding  the 
resultant  thus  obtained 
with  any  other  of  the 
velocities  imparted,  and 
so  on,  till  all  the  veloci- 
ties have  been  taken 
into  consideration,  and 
the  final  resultant  ob- 
tained. Let  the  several 
velocities  which  are  im- 
parted to  a  particle  be  represented  by  the  lines  AB,  AC,  AD, 
AE,  AF  (Fig.  18),  all  in  one  plane.  It  is  required-to  find  their 
resultant.  The  resultant  of  AB  and  AC  is  AL ;  the  resultant 
of  AL  and  AD  is  AM  ;  the  resultant  of  AM  and  AE  is  AN  ; 


66 


KINEMATICS. 


[CHAP. 


that  of  AN  and  AF  is  AO,  the  final  resultant  of  the  five  veloci- 
ties AB,  AC,  AD,  AE,  AF.  It  does  not  matter  in  what  order 
they  are  compounded;  it  may  be  left  as  an  exercise  for  the 
reader  to  show  that  the  same  result  is  always  obtained  whatever 
be  the  order  followed. 

The  Polygon  of  Velocities.  —  If,  in  the  last  diagram,  the 
figure  ABLMNOA  be  traced  out,  it  will  be  seen  that  it  is  a 
polygon  whose  sides  represent  the  various  velocities  and  the 
resultant:  for  these  sides  are  AB,  BL  (  =  AC),  LM  (  =AD), 
MN  (  =  AE),  NO  (  =  AF),  and  AO,  which  represents  the  Re- 
sultant. Hence  the  method  of  finding  the  resultant  of  any 
number  of  forces  in  the  same  plane  may  be  exemplified  as  fol- 
lows :  —  Take  a  starting-point  K  (Fig.  19) ;  from  K  draw  the 

line  KP,  representing 
AB  in  magnitude  and 
direction;  from  P  draw 
PQ,  representing  AC ; 
from  Q  QR,  represent- 
>Q  ing  AD  ;  from  R  RS, 
representing  AE ;  from 
S  ST,  representing  AF; 
then  join  KT.  KT  rep- 
resents the  Resultant 

K  (=AB>  sought.    It  will  be  seen 

that  the  direction  of  the  resultant  is  opposed  to  that  of  the 
other  sides  of  the  polygon  taken  in  cyclical  order.  The  rule, 
then,  for  the  composition  of  a  number  of  velocities  in  the  same 
plane  is  —  Construct  a  polygon  with  lines  representing  them  (it 
being  a  matter  of  indifference  in  what  order  they  are  taken,  or 
whether  they  cross  one  another  or  not),  and  if  there  be  a  side 
missing,  complete  it;  it  will  represent  the  magnitude  of  the 
resultant,  and  its  direction  will  be  opposed  to  that  of  the  other 
constituent  sides,  taken  in  cyclical  order.  If  the  two  points  K 
and  T  coincide,  then  the  line  KT  has  no  value,  there  is  no 
resultant  motion,  and  the  result  of  the  simultaneous  velocities 
is,  in  such  a  case,  a  state  of  rest. 

Reference  to  Axes.  —  It  is  often  as  convenient,  or  more  so,  first  to 
resolve  each  velocity  into  two  components,  which  are  made  parallel  to  arbi- 
trarily chosen  axes.  Let  the  same  velocities,  AB,  AC,  AD,  AE,  AF,  be  sup- 
posed as  in  the  previous  paragraphs.  Through  the  point  A  (Fig.  20)  draw 
axes  of  x  and  y  at  right  angles  to  each  other.  Resolve  each  velocity  into 
its  components  parallel  to  these  axes.  AB  is  resolved  into  Ab  and  Aft; 
AC  into  Ac  and  Ay,  and  so  on.  The  value  of  the  resultant  is  found  after 


V-] 


COMPOSITION   OF   VELOCITIES. 


67 


summing  up  with  reference  to  each  axis  separately.  The  total  result  with 
reference  to  the  axis  of  x  is  (Kb  +  Ac  -  Ad  -  Ae  +  A/),  which  has  a  value, 
say,  +  Ar.  In  the  axis  of  y  the  total  result  is  (  -  Aft  +  Ay  +  AS  -  Ae  -  A<£), 
which  has  the  aggregate  value,  say,  +  Ap.  The  resultant  therefore  is  to  be 
drawn  from  A  to  a  point  R,  which  has  co-ordinates,  x  =  +  Ar,  y  =  +  Ap. 


Fig.2i. 


Velocities  not  in  one  Plane.  —  The  same  essential  principles  apply 
here  as  in  the  preceding  paragraphs.  In  the  case  of  a  railway  train  travel- 
ling at  the  same  time  northwards,  westwards,  and  upwards,  the  motion, 
while  it  may  be  represented  by  a  straight  line,  is  the  resultant  of  three  com- 
ponents at  right  angles  to  each  other.  The  proposition  in  three  dimensions, 
which  corresponds  to  that  known  as  the  parallelogram  of  velocities  in  bidi- 
rnensional  space  (in  a  plane),  is  called  the  parallelepipedon  of  velocities. 
If  the  three  velocities,  Ax,  Ay,  Az  (Fig.  21),  at  right  angles  to  each  other, 
be  compounded,  the  resultant  is  expressed  by  a  line  drawn  from  A  to  the 
opposite  angle  of  that  parallelepipedon  of  which  Ax,  Ay,  Az  measure  the 
length,  breadth,  and  thickness.  If  Ax,  Ay,  Az  be  at  right  angles  to  one 
another,  AR2  =  (Ax2  +  Ay*  +  Az2),  while  if 
they  be  not  at  right  angles  to  one  another,  AR 
is  the  diagonal  of  an  oblique  prism. 

Any  rectilineal  velocity  may  be  resolved 
into  three  components  in  an  indefinite  number 
of  ways,  for  there  may  be  an  infinite  number  of 
prisms  constructed  on  a  given  diagonal  line; 
but  there  can  only  be  one  way  of  resolving  such 
a  movement  into  components  if  these  must  be 
at  right  angles  to  one  another  while  the  direc-  *  x 

tion  of  any  one  of  them  is  given,  or  if  the  directions  of  any  two  of  them 
be  assigned. 

The  Polygon  of  Velocities  also  applies  when  the  component  move- 
ments are  not  restricted  to  one  plane,  for  a  so-called  "  gauche  polygone"  or 


68  KINEMATICS.  [CHAP. 

"skew-polygon,"  may  be  realised,  no  three  of  whose  contiguous  sides  are 
in  the  same  plane ;  the  only  essential  criterion  of  such  a  polygon  is  that  it 
shall  be  continuous  and  closed.  If  such  a  polygon  whose  sides  represent 
velocities  be  realised,  but  be  incomplete  or  "  unclosed,"  the  missing  side 
represents  the  Resultant,  and  the  direction  of  the  resultant  —  opposed  to  that 
of  the  rest  of  the  sides  taken  in  cyclical  order  —  and  its  magnitude  are  found 
in  the  same  way  as  if  the  polygon  had  been  restricted  to  a  plane  surface. 

The  method  of  reference  to  axes,  illustrated  by  Fig.  20,  is  of  special 
use  when  extended  to  tridimensional  space.  Of  a  number  of  velocities  in 
different  directions  in  space,  each  may  be  resolved  into  three  components, 
parallel  to  the  axes  of  x,  of  y,  and  of  z,  and  the  resultant  is  found  after 
summing  up  the  effects  produced  with  reference  to  each  of  these  axes 
respectively. 

Change  of  Velocity.  —  This  phrase  is  sometimes  employed, 
as  when  the  statement  is  made  that  a  certain  velocity  has  been 
changed  into  another,  and  the  question  is  asked,  What  has  been 
the  "Change  of  Velocity?"  Another  way  of  stating  the  same 
is  —  A  known  component  and  an  unknown  one  have  produced  a 
given  resultant:  what  was  the  value  of  the  unknown  component? 
This  is  easily  solved  if  the  direction  of  motion  have  not  changed; 
while  if  the  direction  have  also  changed,  the  question  is  answered 
by  the  aid  of  the  triangle  of  velocities;  the  two  sides  being 
known,  the  third  side  is  easily  found. 

Parallelograms,  etc.,  of  Accelerations. — What  is  true  of 
simultaneous  velocities  imparted  in  general  is  true  of  velocities 
simultaneously  imparted  in  unit  of  time  —  that  is,  of  Accelera- 
tions, and  hence,  if  a  body  receive  two  accelerations,  these  must 
be  compounded  in  exactly  the  same  way  as  two  velocities.  So 
every  one  of  the  geometric  propositions  just  laid  down  with 
reference  to  velocities  finds  its  exact  counterpart  in  a  proposition 
relating  to  accelerations,  and  we  thus  have  such  propositions  as 
the  Parallelogram,  the  Triangle,  the  Polygon,  the  Parallelepipe- 
don  of  Accelerations. 

Acceleration  may  therefore  result  in  mere  change  of  direc- 
tion of  motion : '  for  the  original  velocity  compounded  with  that 
produced  in  a  given  time  by  the  acceleration  may  yield  a  result- 
ant velocity  which  is  the  same  in  amount,  but  not  in  direction, 
as  the  original  velocity:  the  triangle  of  velocities  is  then  an 
isosceles  triangle,  the  two  equal  sides  in  which  represent  the 
original  and  the  resultant  velocities  respectively. 

Problems. 

1.  If  the  same  particle  be  simultaneously  affected  by  a  northward  ve- 
locity of  10  feet  per  second,  an  eastward  of  8,  one  towards  the  S.W.  of  7,  to 


v.]  ACCELEKATION.  (39 

the  W.  of  8,  to  the  S.E.  of  5,  and  to  the  KE.  of  7,  find  the  resultant  move- 
ment, and  show  that  it  does  not  matter  in  what  order  the  components  are 
taken. 

2.  If  the  axes  of  x  and  y  be  drawn  at  right  angles  to  one  another  through 
the  common  point  A ;  if  then  the  point  A  be  supposed  to  be  simultaneously 
affected  by  velocities  represented  by  the  following  lines,  viz.,  (a)  one  drawn 
making  an  angle  of  15°  with  Ax,  and  of  such  a  length  as  to  represent  a  velo- 
city of  10  metres  per  second  ;  (b)  one  making  an  angle  of  45°  with  Ax,  and 
representing  a  speed  of  15  metres  per  second ;  (c)  one  making  an  obtuse 
angle  of  120°  with  Ax,  and  representing  8  metres  per  second ;  and  (d)  one 
at  an  angle  of  195°  with  Ax,  and  representing  a  rate  of  12  metres  per  second. 
Find  the  resultant  velocity  (1)  by  the  polygon,  and  (2)  by  reference  to  axes. 

3.  If  a  body  moving  10  miles  an  hour  northward  come  to  move  at  the 
same  rate   southward,  what  is  the  change  of  velocity?  —  Ans.  20  miles 
an  hour. 

4.  If  a  body  be  moving  with  a  velocity  4  miles  an  hour  northward,  and 
be  after  some  time  found  to  be  moving  at  the  same  rate  eastward,  what  is 
the  change  of  velocity? —  Ans.  4x  V  2,  acting  towards  the  S.E. ;  the  hypo- 
tenuse of  a  right-angled  triangle. 

5.  If  a  body  moving  at  the  rate  of  10  feet  a  second  be  found  after  some 
time  to  be  travelling  at  the  same  rate,  but  in  a  direction  inclined  at  an  angle 
of  60°  to  its  former  one,  what  is  the  change  of  velocity?  —  Ans.  10  feet  per 
second,  making,  with  the  original  component  and  the  resultant,  an  equilateral 
triangle. 

Accelerated  Motion.  —  If  a  body  be  moving,  in  a  straight 
line,  at  a  rate  which  increases  or  decreases  with  the  time, 
its  velocity  is  said  to  be  accelerated.  The  acceleration  is 
said  to  be  positive  when  the  velocity  of  the  motion  is  in- 
creased, negative  when  it  is  diminished.  It  is  measured  by  the 
amount  of  increase  or  decrease  of  the  velocity  per  unit  of  time. 
If  a  particle  be,  at  a  certain  initial  instant,  moving  at  a  rate  V0, 
and  if  its  acceleration  be  ±  a,  in  the  same  straight  line,  then  its 
various  rates  of  motion  are  — 

At  the  initial  instant  .  .  V0. 

At  the  end  of  one  second  .  .  V0  ±  a. 

At  the  end  of  two  seconds  .'  V0  ±  2a. 

At  the  end  of  t  seconds       ,  '.'.  V0  ±  a£. 

Hence  we  arrive  at  a  general  equation  expressing  the  rela- 
tion between  vt  the  velocity  attained  at  the  end  of  t  seconds,  v0 
the  original  velocity,  and  ±  a  the  acceleration,  namely,  — 

v,  =  V0  ±  af,      (1.) 

in  which  the  +  or  the  —  sign  is  used  according  to  the  positive 
or  the  negative  character  of  the  uniform  acceleration  a. 

It  is  supposed  that  the  acceleration  is  uniform^  and  hence 
the  average  velocity  during  any  interval  of  time  is  the  arith- 


70  KINEMATICS.  [CHAP. 

metical  mean  between  the  velocity  v0  at  the  commencement 
and  the  velocity  yt  at  the  end  of  the  interval ;  that  is  to  say, 
it  is  equal  to  half  their  sum  or  J  \ (v0)  +  (v0  ±  a£)  j  =  (v0+  J  a£). 
This  being  the  average  velocity  during  the  interval,  the  space 
traversed  will  be  found  by  multiplying  the  average  velocity  by 
the  time,  and  hence  we  have  (s  being  the  space  traversed)  — 

s  =  |  (To  +  v<)*  =  t  Oo  ±  Ia0  =  V0*  ±  <K.     (2.) 
From    equations   (1)    and  (2)  we    may  eliminate  £,  and   thus 
obtain  a  third  equation  — 

v,2  =  v02±2as,     (3.) 

which  expresses  the  relations  between  the  original  and  acquired 
velocities,  the  space  passed  over,  and  the  acceleration.  Also, 

s  =  v,«  T  *t*/'2  =  (v,2  -  v02)  -*-  2a.     (4.) 

All  elementary  problems  concerning  accelerated  movement 
in  one  direction,  which  give  a  sufficient  number  of  terms  to 
enable  a  conclusion  to  be  arrived  at,  can  be  solved  by  the  aid 
of  these  equations. 

Problems. 

1.  If  the  velocity  at  the  initial  instant  be  10  feet  per  second,  and  the 
acceleration  be  +  2  ft.-per-sec.  per  second,  what  will  be  the  speed  at  the  end 
of  13  seconds?  —  Ans.  Here,  by  equation  (1)  (v, being  the  unknown  term), 

v«  =v0  +  at  =  10  +  (2  x  13)  =  36. 

2.  If  the  acceleration  be  —  2  ft.-per-sec.  per  second,  what  will  be  the 
velocity? — Ans.  vt  =  10  —  (2  x  13)  =  —  16 ;  that  is,  16  feet  per  second,  in 
a  direction  opposed  to  the  original  velocity. 

3.  If  the  terminal  velocity  be  20  feet  per  second,  the  acceleration  be  4 
ft.-per-sec.  per  second,  and  the  initial  velocity  4  feet  per  second,  what  was 
the  time  spent  in  attaining  the  ultimate  speed?  —  Ans.   Here,  by  equation 
(1)  (t  being  the  unknown  term),  20  =  4  +  4f,  whence  t  —  4. 

4.  A  body  travels  with  accelerated  velocity  ;  its  attained  velocity  is  100 
feet  per  second,  its  acceleration  is  10  ft.-per-sec.  per  second,  and  it  has  been 
gaining  speed  for  8  seconds.   What  was  the  initial  velocity  ?  —  A  ns.  By  equa- 
tion (1),  v0  being  the  unknown  term,  100  =  v0  +  (10  x  8),  whence  v0  =  20. 

5.  A  body  falls  from  rest :  its  velocity  increases  by  32-2  ft.-per-sec.  per 
second.     What  will  be  its  speed  at  the  end  of  5  seconds?  —  Ans.  By  equa- 
tion (1),  vt  being  the  unknown  term,  and  v0  =  0,  v,  =  0  +  (32-2  x  5)  =  161 
feet  per  second. 

6.  What  space  will  have  been  traversed,  the  terms  remaining  as  in  the 
last  question?  —  Ans.   By  equation  (2),  s  being  the  unknown  quantity,  and 
v0  =  0,  B  =  0  +  \  (32-2  x  25)  =  402-5  feet. 

7.  What  time  will  a  body  take  to  fall  502-5  feet  if  it  be  thrown  down 
from  a  cliff  at  the  initial  rate  of  20  feet  per  second,  and  if  the  acceleration 
of  a  falling  body  be  32-2  ft.-per-sec.  per  second?  —  ,4ns.  Here,  by  equation 
(2),  t  being  the  unknown  term,  502-5  =  20t  +  16-lf2,  a  quadratic:  whence 
t  =  5  seconds. 


v.]  ACCELERATION.  71 

8.  If  the  initial  velocity  had  been  20  feet  per  second  upwards,  how  long 
would  it  take  to  fall?  —  Ans.  Here  the  acceleration  is  opposed  to  the  original 
velocity;  equation  (2)  becomes  —  502-5=20*— 16-1*2;  whence  t= 6-24  seconds. 

9.  What  speed  is  attained  by  a  falling  body  if  it  start  from  rest  and  fall 
1610  feet?  —  Ans.  Here  vt  is  unknown,  v0  =  0,  and  a  =  32-2.    By  equation 
(3),  v,2  =  0  +  (2  x  32-2  x  1610);  .-.  v,  =  322  feet  per  second. 

10.  If  a  body  start  with  initial  velocity  v0,  and  the  acceleration  be  a, 
what  will  be  the  space  traversed  in  the  first,  in  the  second,  in  the  third,  in 
the  fourth  seconds  respectively ;  and  what  will  be  the  space  traversed  in  4 
seconds? —  Ans.  v0  +  a/2 ;  v0  +  3a/2 ;  v0  +  5a/2  ;  v0  +  7a/2 ;  4  v0+  16a/2. 

Composition  of  uniform  with  accelerated  motion.  —  If 

a  particle  be  affected  with  both  a  uniform  and  an  accelerated 
motion,  and  if  these  be  in  the  same  straight  line,  we  have  simply 
the  problem  dealt  with  by  the  aid  of  the  last  four  equations. 
When,  however,  the  uniform  and  the  accelerated  velocities  are 
not  along  the  same  line,  but  are  in  directions  inclined  to  one 
another,  the  resultant  must  be  found  by  a  geometrical  or  an  alge- 
braical process  of  composition  of  velocities.  If,  for  the  sake  of 
fixing  our  ideas,  we  consider  such  a  motion  as  that  of  a  projectile 
fired  horizontally  from  a  gun  placed  on  a  height,  we  see  that  the 
ball  is  affected  with  two  simultaneous  but  independent  motions, 
the  one  horizontal  and  uniform,  the  other  vertically  downward 
and  accelerated.  If  we  consider  the  positions  reached  by  such 
a  body  in  successive  equal  intervals  of  time,  we  find  that  while 
it  passes  forward,  by  reason  of  its  horizontal  component,  over 
spaces  varying  directly  as  the  time,  the  amount  of  its  vertical 
drop  due  to  the  downward  accelerated  component  is  proportional 
(Equation  2,  p.  70,  where  V0  =  0)  to  the  square  of  the  time 
during  which  it  has  been  in  motion ;  so  that  if  we  separately  find 
its  various  positions  at  the  end  of  successive  small  intervals  of 
time,  we  can  draw  a  line  joining  these  positions,  which  line  we 
find  to  be  a  curve  known  as  a  Parabola  (Fig.  22). 

If,  again,  we  consider  that  the  one  movement  is  in  the  axis 
of  #,  and  is  uniform,  so  that  at  the  end  of  time  t  the  horizontal 
component  motion  has  carried  the  body  along  the  axis  of  x  a  dis- 
tance x  =  vt ;  while  the  vertical  fall  represents  the  distance  y, 
along  the  axis  of  y,  at  which  the  body  is  situated  at  the  end  of 
the  same  time  £,  so  that  according  to  equation  (2)  above, 
y  =  |cU2;  then  we  find*  that  in  time  t  (whatever  this  time  be) 

*  This  is  an  instructive  example  of  a  method  frequently  in  use.  One  considera- 
tion leads  us  to  the  equation  x  =  vt :  another  to  the  equation  y  =  fat2 :  the  question 
is,  what  law  governs  the  relations  of  x  and  y  ?  The  two  equations  are^combined  in 
any  way  so  as  to  represent  x  as  some  multiple  (or  other  "  function  ")  of  ?/,  and  also, 
if  possible,  so  as  to  eliminate  a  letter  common  to  both  equations.  Here  the  value  of 
t  (  =  x/v)  derived  from  the  first  equation  may  be  substituted  for  t  in  the  second,  thus 
making  it  y  —  %a  (x/v)2,  whence  x2-r  y  =  const. 


72 


KINEMATICS. 


[CHAP. 


the  body  moves  to  a  position  such  that  its  vertical  distance  y 
from  the  starting  point  bears  to  x2,  the  square  of  its  horizontal 
distance  from  the  starting  point,  a  constant  ratio,  or  in  symbols 
Fig.22.  kx2  =  y,  which   is    recognised. 

~as  "the  Equation  to  "  a  Parab- 
ola. This  indicates  that  in 
order  to  preserve  the  given 
relation  between  the  values  of 
x  and  y,  the  path  of  the  body 
as  it  moves  from  point  to  point 
must  be  in  a  curve  known  as  a 


\ 


\ 


\ 


\ 


\ 


parabola. 

It  is  stated  that  the  body  will 
move  in  a  parabolic  not  in  the  para- 
bolic path;  this  is  because  there  is 
an  indefinite  number  of  parabolic 
paths  possible,  there  being  an  infinite 
number  of  parabolic  curves  (just  as 
there  may  be  an  infinite  variety  in 
the  forms  of  a  jet  of  water  expelled 
from  a  fire-engine),  which  resemble 
each  other  in  having  some  constant 
proportion  between  the  values  of  the 
one  co-ordinate  and  of  the  square  of 
the  other,  but  which  differ  in  the 
numerical  value  of  that  ratio. 

Degrees  of  Freedom  of  a  Par- 
ticle. —  If  a  particle  be  free  to  move 
in  any  direction  in  space,  it  is  said  to 
have  three  "  degrees  of  freedom,"  be- 
cause it  may  move  in  tridimensional  space;  it  may  move,  e.g.,  (1)  up  or 
down,  (2)  forwards  or  backwards,  or  (3)  to  the  right  or  left;  or  more  gen- 
erally, it  may  move  in  the  direction  of  any  of  the  three  axes  arbitrarily 
chosen  at  right  angles  to  one  another,  by  reference  to  which  wre  agree  to 
specify  any  given  direction  in  space,  or  it  may  move  in  any  other  direction, 
motion  in  which  may  be  considered  as  the  resultant  of  simultaneous  mo- 
tions in  the  three  directions  assumed  as  axes  of  reference.  If  the  particle 
be  restricted  to  a  surface,  it  cannot  move  in  a  direction  at  right  angles  to 
that  surface,  and  is  accordingly  said  to  have  one  degree  of  freedom  less ;  it 
has  now  two  degrees  of  freedom,  for  it  may  travel  along  the  surface  in  two 
main  axial  directions  (e.g.,  (1)  forward  or  backward,  (2)  to  the  right  or  left), 
or  in  any  direction  derived  from  the  combination  of  these.  If  the  particle  be 
restricted  to  two  surfaces,  on  both  of  which  it  must  lie  at  the  same  time,  it  can 
lie  nowhere  but  on  the  line  in  which  these  two  surfaces  cut  one  another, 
and  it  has  now  only  the  one  degree  of  freedom  implied  in  the  possibility  of 
moving  (backwards  or  forwards)  along  this  line.  If  the  particle  be  restricted 
to  three  surfaces  which  cut  one  another  in  a  point,  the  particle  cannot  leave 
that  point  without  leaving  one  or  other  of  the  surfaces ;  its  position  is  defi- 
nitely fixed,  and  it  has  no  degree  of  freedom  to  move  in  any  direction. 


v.]  TRANSLATION.  73 

Translation.  —  If  there  be  a  system  of  separate  particles,  all  of  which 
are  affected  with  equal  and  parallel  velocities,  each  particle  will  move  in  such 
a  way  as  to  retain  its  relative  position  with  regard  to  its  fellow-particles,  and 
the  system  will  move  as  a  whole,  undergoing  no  deformation,  just  as  a  com- 
pany of  soldiers,  all  the  constituent  units  of  which  march  in  the  same  direc- 
tion and  at  the  same  rate,  retains  its  formation.  If  a  straight  line  be  drawn 
between  any  two  of  these  particles  when  the  system  is  in  its  initial  position, 
it  will  be  found  that  the  line  drawn  between  the  same  particles  after  such 
movement  will  always  remain  parallel  to  its  former  position,  and  will  be 
unaltered  in  length.  Motion  in  which  every  such  line  remains  parallel  to  all 
its  previous  positions  is  called  Simple  Translation.  If  we  study  the 
motion  of  such  a  straight  line  or  of  the  particles  between  which  it  lies,  we 
shall  have  complete  knowledge  of  the  positions  of  the  various  particles  of 
the  system,  if  that  system  be  restricted  to  a  plane  surface.  If  the  system 
be  not  restricted  to  a  plane  surface,  then  it  is  possible  that  though  one  line 
and  all  lines  parallel  to  it  may  continue  to  be  parallel  to  their  former  posi- 
tions, the  whole  system  may  have  rotated  round  one  of  these  lines  as  round 
an  axis ;  and  hence  in  this  case  it  is  necessary,  before  the  motion  of  the 
system  can  be  said  to  be  a  motion  of  simple  translation,  that  not  a  line  only, 
but  any  plane  through  the  system  —  or,  which  amounts  to  the  same  thing, 
every  line  in  any  such  plane  — should  retain  parallelism  to  its  initial  position. 

Rigid  Body.  —  This  is  an  ideal,  not  physically  realisable.  A 
rigid  body  may  be  regarded  as  a  system  of  particles  which  may 
move  as  a  whole  with  reference  to  surrounding  objects,  but  in 
which  there  can  be  no  displacement  of  its  particles  with  refer- 
ence to  one  another. 

Centre  of  Figure.  —  There  is  in  the  case  of  every  body  of  any  shape 
whatsoever  some  one  point  occupying  a  definite  position,  which  position  may 
be  described  as  the  average  of  all  the  respective  positions  of  the  several  par- 
ticles of  the  body.  A  body  suspended  in  the  air  somewhere  towards  the 
N.W.  will  have  (generally  within  it)  a  point  which  is  not  only  situated  at  an 
average  distance  to  the  north  of  the  point  of  reference,  but  is  also  at  an  aver- 
age distance  to  the  west  and  at  an  average  height ;  and  this  point  is  the 
Centre  of  Figure.  Not  only  with  respect  to  the  planes  chosen  as  those 
of  reference  is  this  point  the  centre  of  figure,  and  its  distance  from  each  of 
these  planes  the  average  of  the  several  distances  of  all  the  particles,  but  it 
has  this  property  with  reference  to  any  plane  whatsoever. 

The  centre  of  figure  of  a  straight  line  or  linear  body  is  its  middle  point ; 
the  centre  of  figure  of  a  circle  is  its  centre ;  the  centre  of  figure  of  a  sphere, 
of  an  ellipsoid,  of  a  spheroid,  is  equally  obvious ;  that  of  a  hollow  spherical 
shell  is  the  centre  of  the  corresponding  solid  sphere,  and  is  therefore  not 
within  the  substance  of  the  shell ;  that  of  a  parallelogram  is  the  point  at 
which  the  diagonals  cross  one  another.  That  of  any  regular  plane  figure  is 
obtained  by  dividing  it  into  numerous  thin  strips  and  bisecting  these;  by 
joining  the  points  of  bisection  a  line  is  drawn  in  which  the  centre  of  figure 
must  lie.  By  repeating  the  process  another  such  line  may  be  obtained. 
These  two  lines  will  cross  one  another  in  some  point,  and  the  point  where 
they  do  so  is  the  only  point  which  lies  in  both  the  lines,  and  it  is  the  centre 
of  figure.  This  holds  good  only  when  the  lines  thus  containing  the  centre 
of  figure  are  straight ;  if  they  be  not  so  the  construction  fails,  and  we  may 


74  KINEMATICS.  [CHAP. 

modify  one  of  the  experimental  methods  described  under  the  Centre  of 
Gravity,  farther  on. 

The  importance  of  the  Centre  of  Figure  lies  in  this :  that  if  a  rigid 
body  be  subject  to  translation  without  rotation,  the  motion  of  the  body  may 

be  quite  effectively  studied  by  considering 
the  movement  of  the  centre  of  figure,  and, 
on  the  other  hand,  if  a  rigid  body  be  sub- 
ject to  accelerations  whose  resultant  passes 
through  the  centre  of  figure,  the  whole  rigid 
body  will  participate  in  the  movement  of 
g*3  '  its  centre,  and  there  will  be  translation: 

while  if  the  resultant  of  accelerations  do  not  pass  through  the  centre,  there 
will  be  rotation.     It  is  assumed  in  this  that  the  body  is  uniform  in  density. 

Rotation  takes  place  when  a  straight  line  drawn  through  a  moving  body 
or  system  of  particles  does  not  continue  to  be  parallel  to  its  previous  direc- 
tions in  space.  Let  us  suppose  the  moving  system  to  be  restricted  to  a  plane 
surface.  Then  a  determinate  line  AB,  arbitrarily  chosen  in  the  body,  may 
move  so  that  its  ultimate  position  is  A'B'  (Fig.  23).  Obviously  the  line  AB, 
and  with  it  the  system,  has  rotated  round  the  point  O.  Again,  the  relative 
positions  of  the  same  line  may  be  AB  and  A'B'  in  Fig.  24.  In  this  case  a 
point  O  may  be  found,*  round  which  the  line  AB  has  rotated  so  as  to  acquire 
\  its  new  position  A'B'.  If  the  lines 

Fig.24.  t  AB  and  A'B'  be  very  nearly  parallel 

to  one  another,  construction  will  show 
that  the  point  O  is  at  a  great  distance. 

\  v*x^  / /  When  the  lines  AB  and  A'B'  are  per- 

\  Vi*vx        •/  /  fectly  parallel,  the  point  O  is  at  an 

\  **y^**  '  indefinitely  great,   an    infinite   dis- 

\  /     XM  tance.    Thus  we  see  that  though  it  is 

convenient  to  regard  translation  and 
rotation  as  distinct  forms  of  motion, 
yet  translation  may  be  considered  as 
a  limiting  case  of  rotation,  effected 
A*  B  round  an  infinitely  distant  centre. 

Any  translation  of  AB  into  a  parallel  position  A'B'  may  be  resolved 
into  a  succession  of  rotations  first  round  one  extremity,  as  A,  then  to  an  equal 
extent  but  in  an  opposite  direction  round  B :  this  is  easily  verified  by  con- 
struction. If  the  line.  AB  cannot  be  brought  into  coincidence  with  the  line 
A'B'  by  a  single  pair  of  such  rotations,  a  sufficient  number  of  pairs  of 
rotations  will  certainly  effect  this. 

Composition  of  Rotations.  —  If  by  reason  of  a  rotation  round  the 
point  O,  the  line  AB  be  brought  into  the  position  A'B' ;  if  it  be  then  rotated 
round  a  point  O',  so  as  to  assume  the  position  A"B"  :  the  two  rotations  can 
be  compounded  into  one  round  a  point  O",  which  is  found  directly  by  com- 
paring the  initial  and  final  positions  AB  and  A"B",  without  reference  to  the 
intermediate  position  A'B'. 

If  a  solid  body,  of  which  one  point  is  fixed,  move  in  any  way  whatso- 
ever, the  result  is  the  same  as  if  it  had  revolved  round  some  definite  axis 

*  Join  AA' ;  bisect  AA'  in  L ;  draw  LO  at  right  angles  to  AA'.  Join  BB' ;  bisect 
BB'  in  M ;  draw  MO  at  right  angles  to  BB'.  LO  and  MO  will  intersect  in  O  ;  O  is  the 
point  required.  Join  OA,  OA',  OB,  OB';  the  triangles  OAB  and  OA'B'  are  equal  in 
every  respect. 


v.]  COMPOSITION   OF   ROTATIONS.  75 

passing  through  that  point.     Any  movement  under  this  condition  is  equiva- 
lent to  a  single  rotation. 

If  a  body  be  subject  to  two  or  three  simultaneous  rotations  round  axes 
which  meet  in  a  fixed  point,  the  resultant  movement  is  rotation  round  a 
single  axis,  which  is  found  by  a  construction  precisely  the  same  as  that  of  the 
parallelogram  or  the  parallelepipedon  of  velocities  :  the  sides  of  the  figure 
represent  in  direction  the  axes  round  which  the  rotations  occur,  and  in  length 
the  amount  of  angular  velocity  ;*  the  diagonal  obtained  represents  in  the 
same  way  the  axis  and  the  angular  velocity  of  the  resultant  rotation.  Simi- 
larly, a  rotation  round  any  axis  can  be  resolved  into  component  rotations 
round  other  axes  passing  through  a  point  in  the  original  axis. 

The  most  indeterminate  motion  of  a  rigid  body  may  always  be  resolved 
into  the  same  motion  as  that  of  a  screw  in  its  nut,  namely,  a  Rotation  and 
a  Translation.  As  the  body  continues  to  move,  the  axis  of  the  imagi- 
nary screw  may  change  its  direction  in  space  ;  but  when  considered  at  and  for 
the  space  of  any  particular  very  small  instant,  it  may  be  regarded  as  fixed, 
and  is  then  called  the  Instantaneous  Axis.  As  limiting  cases,  the 
translation  may  =  0,  when  there  is  Simple  Rotation  ;  or  the  angular  velocity 
of  rotation  may  =  0,  in  which  case  we  would  have  Simple  Translation.  The 
ultimate  position  attained  may  always  be  reached  by  means  of  a  single  trans- 
lation and  a  single  rotation  round  some  axis. 

Precession.  —  When  a  new  rotation  is  superposed  on  an  existing  one, 
the  axis  of  the  resultant  rotation  is  more  nearly  parallel  to  the  axis  of  the 
superposed  rotation  than  that  of  the  original  rotation  had  been.      F.    24a 
Assume  a  top  (Fig.  24  a)  to  be  spinning  at  any  particular  instant     _ 
round  the  axis  AB.     The  top  tends  to  fall :  that  is,  to  rotate 
round  an  axis  through  A,  say  at  right  angles  to  the  plane  of  the 
paper.     The  axis  AB  therefore  tends  to  work  round,  off  the 
plane  of  the  paper,  so  as  to  become  more  nearly  parallel  to 
the  axis  passing  through  A  at  right  angles  to  the  paper.     The 
result  is,  that  the  axis  AB  pivots  round  the  point  A,  and  B 
describes  a  circle.     This  accounts  for  the  gyration  or  Preces- 
sion of  a  spinning-top.     The  angle  BAG  is  the  Angle  of  Pre-  . 

cession.     If  the  top  be  so  acted  upon  as  to  make  its  movement  A 

of  precession  more  or  less  rapid,  this  is  equivalent  to  the  introduction  of  a 

*  When  a  body  moves  in  a  circular  path,  its  velocity  in  that  path  may  be  measured 
by  the  length  of  the  path  traversed  divided  by  the  time,  as  usual  ;  or  it  may,  in  many 
respects,  more  conveniently  be  expressed  in  terms  of  angular  velocity.  Here  the 
path  is  measured,  not  directly  in  terms  of  its  own  length,  but  with  reference  to  the 
angle  which  it  subtends  and  to  the  length  of  the  radius.  The  unit  angle  or  radian 
is  that  angle  (57°'29578  =  570<17'-44"-8  nearly)  which  is  subtended  by  apart  of  the 
circumference  equal  in  length  to  the  radius.  Hence  the  circumference  =  360°  =  2r 
radians,  IT  being  equal  to  3' 1416.  Unit  angular  velocity  is  that  under  which  a  particle 
travelling  in  a  circle  whose  radius  =  1  would  itself  describe  a  path  =1  —  that  is,  unit 
angular  velocity  is  that  of  a  rotating  body  which  traverses  the  unit  angle  —  in  unit  of 
time.  If  the  radius  be  r  and  the  angle  traversed  be  6,  the  part  of  the  circumfer- 
ence passed  over  is  rd,  and  if  this  be  accomplished  in  time  t,  the  linear  velocity  of  a 
particle  on  the  circumference  is  v  =  r9/t ;  that  of  a  particle  nearer  or  farther  from  the 
centre  is  proportionately  less  or  greater ;  while  the  Angular  Velocity  of  all  the  parti- 
cles of  a  rotating  wheel  is  the  same,  namely,0/£  =w  =  v/r.  The  Dimensions  of  angular 
velocity  are  an  Angle  (=  Arc  -5-  Radius) **a  Time  =  [(L-s-L)]  ^-[T]=<T-i]  ;  and  this 
is  manifestly  correct,  for  angular  velocity  does  not  depend  on  the  size  of  the  circle, 
but  only  on  the  time  taken  to  go  round  it.  Similarly,  angular  acceleration  u>  =  a/r; 
and  it  has  Dimensions  [1/T2]. 


76  KINEMATICS.  [CHAP. 

new  or  third  rotation,  round  AC  ;  if  the  precession  be  made  more  rapid,  the 
resultant  axis  now  lies  between  AB  and  AC,  and  the  top  rises.  If  the  point 
A  be  really  some  little  way  up  the  axis  of  the  top,  so  that  the  point  of  the 
top  tends  to  describe  a  little  circle,  say  on  a  sheet  of  paper,  the  top  gradually 
rises,  if  the  spin  be  rapid  enough,  into  a  vertical  position  and  "sleeps."  The 
reason  of  this  is  that  the  spin  tends  to  make  the  point  of  the  top  travel 
wheel-wise  along  the  paper  at  a  certain  rate  :  the  precession  tends  to  make 
it  describe  a  circle  on  the  paper  at  a  certain  rate :  if  the  former  rate  exceed 
the  latter,  the  point  is  hurried  on  in  its  path  on  the  paper :  the  energy  of  the 
spin  is  partly  converted  into  energy  of  precessional  motion  :  and  this  is  equiv- 
alent to  accelerating  the  precessory  movement :  so  the  top  rises.  In  the  same 
way  pebbles,  egg-shells,  hard-boiled  eggs,  etc.,  rise  up  and  spin  round  their 
longest  axis  if  spun  fast  enough. 

Precession  may,  in  a  freely-suspended  rotating  body,  be  caused  by  an 
unsymmetrical  or  unbalanced  distribution  of  the  rotating  mass  round  the 
actual  instantaneous  axis  of  rotation.  Thus  the  equatorial  protuberance  of 
the  Earth  enables  the  attractions  of  the  Sun  and  Moon  to  exercise  a  tilting 
action  which  results  in  a  precession  whose  period  is  about  26,000  years,  and 
the  angle  of  which  is  23°  30'. 

Nutation.  — Variations  in  the  tilting  forces  which  give  rise  to  Preces- 
sion cause  variations  in  the  speed  and  angle  of  precession.  Thus,  in  the 
case  of  the  Earth,  there  are  three  simultaneous  sets  of  fluctuations  in  the 
Angle  of  Precession :  one  of  nineteen  years'  period,  due  to  varying  angles 
between  the  axis  of  rotation  and  the  moon's  orbit ;  one  of  a  half-year's  period, 
the  sun's  tilting  action  being  zero  at  the  solstices ;  one  of  a  fortnight's  period, 
the  moon's  tilting  action  being  zero  twice  in  the  lunar  month.  These  fluctua- 
tions in  the  angle  of  precession  convert  the  precessional  circle  into  a  wavy 
line :  and  this  phenomenon  is  called  Nutation. 

Degrees  of  Freedom  of  a  Rigid  Body.  —  When  a  rigid  body  is  abso- 
lutely free  to  move  in  any  direction  in  space,  it  is  said  to  have  six  degrees 
of  freedom.  These  are  (1)  three  degrees  of  freedom  of  translation,  like 
those  of  a  simple  particle ;  and  (2)  three  degrees  of  freedom  of  rotation 
round  three  axes  arbitrarily  chosen  at  right  angles  to  one  another.  Any 
such  body  may  move,  for  example,  (1)  upwards  or  downwards,  (2)  to  the  N". 
or  S.,  (3)  to  the  E.  or  W.,  or  it  may  rotate  round  (4)  a  vertical  axis,  (5)  an 
axis  lying  N.  and  S.,  or  (6)  an  axis  lying  E.  and  W.  Any  rotation  not 
round  these  axes,  or  any  translation  not  in  the  direction  of  these  axes,  may 
be  resolved  into  its  components,  round  or  parallel  to  them;  and  as  any 
change  of  position  whatsoever  may  be  produced  by  a  single  translation  and 
a  single  rotation,  any  motion  whatsoever  may  be  effected  by  a  body  which 
has  these  six  degrees  of  freedom. 

If  one  point  in  a  rigid  body  be  fixed,  there  can  be  no  translation,  and 
three  degrees  of  freedom  are  thus  lost ;  the  body  has,  however,  unlimited 
freedom  of  rotation  round  any  axis  passing  through  the  fixed  point,  and  thus 
retains  three  degrees  of  freedom.  If  a  line  in  the  body  be  fixed  in  position, 
there  can  be  no  translation,  and  there  can  be  no  rotation  except  round  this 
fixed  line,  and  so  there  can  be  only  one  degree  of  freedom,  which  corresponds 
to  that  rotation.  If  a  surface  (or,  which  amounts  to  the  same  thing,  if  three 
points)  in  the  body  be  fixed  in  position,  there  can  be  neither  translation  nor 
rotation,  and  the  rigid  body  has  no  freedom. 

If  a  point  in  the  body  be  restricted  to  motion  along  a  given  line,  there 
can  only  be  one  translation,  but  there  may  be  any  rotation,  and  so  the  rigid 


v.]  DEGREES  OF  FREEDOM.  77 

body  has  four  degrees  of  freedom.  When  a  given  line  in  the  body  must 
coincide  with  some  part  of  a  line  assigned  in  space,  there  can  be  only  one 
translation  —  that  along  the  line  assigned,  and  one  rotation  —  that  round 
the  line ;  and  here  we  find  the  rigid  body  to  have  two  degrees  of  freedom. 
If  a  point  in  the  body  be  restricted  to  a  given  surface,  the  only  motion  which 
is  impossible  is  translation  in  a  direction  at  right  angles  to  the  surface,  and 
hence  the  body  has  in  this  case  five  degrees  of  freedom.  If  a  line  in  the 
body  be  restricted  to  a  given  surface,  one  translation  is  impossible,  as  in 
the  previous  instance,  and  there  are  two  rotations  possible,  the  one  round 
the  line  which  is  restricted  to  the  surface,  and  the  other  round  an  axis  at 
right  angles  to  the  surface :  in  this  case  there  are  accordingly  four  degrees 
of  freedom.  If  three  points  in  a  body  be  restricted  to  a  surface,  there  can 
be  rotation  round  an  axis  at  right  angles  to  the  surface,  and  there  can  be 
translation  in  any  direction  along  the  surface  but  not  away  from  it,  so  that 
in  this  case  we  have  three  degrees  of  freedom. 

Strain.  —  When  a  body  is  not  rigid,  its  particles  may  so 
move  with  reference  to  one  another  that  their  displacement  pro- 
duces deformation,  and  such  relative  motion  of  the  particles  of 
which  a  body  is  made  up  is  called  a  Strain  of  the  body. 

Suppose  a  circular  plate  to  be  expanded  uniformly,  as  a  disc  of  iron  is 
when  heated ;  the  radius  will  enlarge  in  the  ratio  of  (say)  1  to  a ;  the  area  of 
the  plate  increases  in  the  ratio  1 :  a2.  The  linear  expansion  is  the  difference 
between  the  initial  and  the  final  length  of  the  radius,  i.e.  r(a  —  1)  where  r 
is  the  original  radius,  and  is  hence  proportional  to  (a  — 1).  If  the  body  have 
contracted,  a  is  less  than  1,  and  a  —  1  is  negative ;  hence  the  linear  expan- 
sion is  negative.  The  superficial  expansion  is  the  difference  between  the 
areas  before  and  after  the  strain,  viz.  {?r(ar)2— Trr2}  =  7rr2(a2  —  1)  ;*  hence 
the  superficial  expansion  is  proportional  to  (a2—  1).  A  square  similarly 
affected  has  its  sides  and  area  increased  or  diminished  in  the  same  ratios : 
so  would  a  parallelogram  or  any  other  plane  figure,  if  the  linear  expansion 
were  the  same  in  all  directions.  Again,  suppose  a  globular  body  to  be  thus 
uniformly  expanded ;  it  increases  in  size  and  becomes  a  larger  globe  :  if  its 
radius  increase  in  the  ratio  1 :  a,  its  bulk  will  increase  in  the  ratio  1  :  a8,  and 
its  cubical  expansion  will  be  proportional  to  a8  —  !.  Cubes,  parallelepipeda, 
and  all  other  solid  figures,  would  under  the  same  circumstances  become 
larger  or  smaller  cubes,  parallelepipeda,  etc.,  whose  sides  and  bulk  would 
bear  similar  ratios  to  their  original  dimensions. 

Suppose  a  square  to  be  unequally  dilated  or  contracted  along  axes  parallel 
to  its  sides,  the  square  will  become  a  parallelogram.  A  circle  will  thus 
become  an  ellipse ;  an  ellipse  will  become  an  ellipse  of  another  form.  As  a 
circle  is  an  ellipse  of  a  particular  form  whose  length  (its  major  axis)  is  equal 
to  its  breadth  (its  minor,  axis),  any  ellipse  may  be  converted  by  a  strain  into 
a  circle,  if  its  axes  be  in  due  proportion  lengthened  or  shortened.  If  the 
expansions  along  the  two  rectangular  axes  be  in  the  ratios  1 :  a  and  1 : 6, 
the  area  of  the  resultant  parallelogram,  ellipse,  or  circle,  will  be  to  that  of 
the  original  figure  in  the  ratio  ab :  1. 

If  the  body  be  a  cube  and  be  unequally  expanded  in  directions  parallel 
to  its  sides,  it  becomes  an  unequal-sided  parallelepipedon.  If  the  several 

*  TT  _  3-14159 ....  the  ratio  of  the  circumference  of  a  circle  to  its  diameter.  The 
area  of  a  circle  =  "7854  X  diar.2  =  3'1416  X  rad.2  =  Trr2. 


78 


KINEMATICS. 


[CHAP. 


sides  expand  in  the  respective  ratios  1 :  a,  1:5,  and  1 :  c,  the  bulk  of  the 
parallelepipedon  will  bear  to  that  of  the  cube  the  ratio  abc :  1.  A  sphere 
strained  in  the  same  way  will  become  an  ellipsoid,  a  so-called  "  strain-ellip- 
soid " :  any  ellipsoid  will  become  an  ellipsoid  of  a  different  form,  and  may 
become  that  particular  kind  of  ellipsoid  known  as  a  sphere,  an  ellipsoid 
whose  three  axes  are  equal  to  one  another. 

It  will  be  seen  on  drawing  any  of  the  figures  just  described  that  any  two 
parallel  lines  drawn  through  the  body  in  its  original  form  will  be  parallel 
after  the  strain.  In  this  kind  of  strain,  called  Homogeneous  Strain, 
there  are  always  three  axes,  which  were  at  right  angles  to  one  another  in 
the  body  in  its  original  form,  and  which  continue  to  be  so  after  the  strain. 
These  are  the  axes  of  the  strain-ellipsoid  into  which  an  imaginary  sphere 
existing  in  the  body  would  be  transformed  by  the  strain. 

Shear.  —  If  a  body  be  so  distorted  that  one  plane  passing 
through  it  is  fixed  while  others  move  past  this  at  rates  pro- 


Fig.25. 


A' 


P1 


portional  to  their  dis- 
B'  tances    from    it ;  —  if, 
^      for  example,  the  body 
ABCD  (Fig.  25)  be  so 
distorted  that  CD  re- 
mains  in    its    original 
c  position,  while  the  line 

AB  travels  into  the  position  A'B',  the  body  is  being  sheared,  or  is 
undergoing  the  kind  of  deformation  or  strain  known  as  a  Shear. 

The  greater  the  distance  of  any  plane  in  the  body  from  the  plane  passing 
through  CD,  the  greater  will  be  its  displacement  from  its  original  position ; 
and  a  body  so  sheared  we  may  conceive  as  made  up  of  an  indefinite  number 
of  indefinitely  thin  layers  which  relatively  move  by  slipping  over  one 
another.  A  shear  is  measured  by  the  amount  of  relative  motion  between 
two  non-distorted  planes  which  are  situated  at  a  unit  distance  from  one 
another,  and  which  remain  parallel.  In  Fig.  25  the  amount  of  the  shear  of 
ABCD  is  the  ratio  of  the  displacement  A  A'  or  PP'  to  PC,  the  shortest  possible 
line  drawn  between  the  two  parallel  planes  AB  and  CD,  and  vertical  to  them 
both ;  that  is,  the  Shear  is  equal  to  PP'/PC  =  tan  PCP'=  tan  0. 

Circular  Motion.  —  If  a  body  move  in  a  circular  path,  as,  for  example, 
a  stone  whirled  in  a  sling,  its  motion  at  every  instant  is  compounded  of  a 
tangential  motion  and  a  motion  towards  the  centre. 
If  it  pass  the  point  A  in  Fig.  26  with  such  a  veloc- 
ity that  it  would  in  a  unit  of  time,  if  not  drawn 
towards  the  centre,  have  reached  the  point  C,  and 
if  it  be  at  the  end  of  that  interval  found  at  the 
point  D,  it  is  evident  that  the  acceleration  towards 
the  centre  must  have  produced  in  unit  of  time  the 
change  of  position  represented  by  the  line  CD.  In 
other  words,  the  whirling  sling-stone  is  constantly 
being  drawn  in  from  the  tangential  path,  which,  in 
virtue  of  its  inertia,  it  would  at  every  instant  natu- 
rally take ;  and  a  planet  in  its  orbit  is  constantly  falling  towards  the  sun, 
but  does  not  proceed  straight  towards  it,  for  the  resultant  of  its  tangential 


Fig.26. 


CIRCULAR  MOTION. 


79 


and  its  centripetal  motions  is  an  elliptical  path  which  is  approximately 
circular. 

If  the  line  CD  be  prolonged  through  the  centre  to  the  point  E,  Eucl. 
III.  36  shows  that  CD-CE  =  AC2.  If  v  represent  the  velocity  of  the  body 
in  the  direction  AC,  v  the  average  velocity  in  the  direction  CD,  and  r  the 
radius  of  the  circle,  we  thus  find  v  (2r  +  v)  =  v2 ;  or  2rv  +  v2  =  v2.  (i). 

If  the  unit  of  time  taken  be  sufficiently  small,  the  square  of  the  small 
quantity  v  will  be  so  small  as  to  be  negligible,  and  the  above  equation 
will  become  2vr  =  v2.  (ii).  But  v  is  the  average  velocity  of  fall  towards 
the  centre  O  during  the  instant  in  question,  and  hence  the  velocity  at  the 
end  of  the  interval  is  2v ;  this  is  the  velocity  acquired  in  unit  of  time,  and 
hence  the  acceleration  towards  the  centre  is  a  =  2v.  Hence  the  equation 
(ii)  may  be  written  ar  =  v2  or  a  =  v2/r ;  the  Acceleration  towards  the 
centre  of  the  circular  path  in  which  a  body  is  moving  is  numerically  equal 
to  the  Square  of  its  Tangential  Velocity  v  at  any  instant  divided  by  the 
Radius  of  curvature.  If  a  body  be  travelling  in  any  other  curve,  there  can 
at  every  instant  be  found  a  circle,  a  part  of  the  circumference  of  which  coin- 
cides, to  an  indefinite  approximation,  with  the  curve  at  the  instant. 

Curvature.  —  Any  curve  may  be  considered  as  made  up  of  successive 
elements,  each  of  which  approximately  coincides  with  a  part  of  the  circum- 

Fie.27. 


ference  of  a  particular  "  osculating  "  circle,  which  may  always  be  found. 
For  each  element  of  the  curve,  the  radius  of  the  corresponding  osculating 
circle  —  whose  circumference  coincides  with  that  element,  and  which  would 
have  the  same  tangent  —  is  called  the  instantaneous  radius  of  curvature  ; 
and  as  the  curve  passes*  from  point  to  point  the  osculating  circle  may  be 
changed  in  respect  of  its  radius  or  its  centre.  Thus  in  an  ellipse,  near  the 
extremity  of  the  major  axis,  the  osculating  circle  of  curvature  is  smaller  than 
it  is  near  the  end  of  the  minor  axis,  as  is  shown  in  Fig.  27.  Accordingly, 
if  a  body  move  in  a  curved  path,  its  acceleration  at  every  instant  towards  the 
instantaneous  centre  of  curvature  is  numerically  equal  to  the  square  of  the 
instantaneous  tangential  velocity  divided  by  the  instantaneous  radius  of 
curvature. 

But  in  a  curve,  the  Curvature  is  the  angle  through  which  the  tangent 
sweeps  round  per  unit  of  length  of  the  curve,  and  this  varies  inversely  as  the 


80 


KINEMATICS. 


[CHAP. 


radius,  as  may  be  seen  on  comparing  the  circles  in  Fig.  28.  The  radius  of  I 
is  twice  that  of  II :  the  length  AA'  =  Cc'  is  supposed  to  be  a  unit  of  length. 
In  I  the  tangent  AB  has  swept  round  into  the  position  A'B'  through  an 
angle  0 :  the  tangent  CD  has  swept  through  twice  as  great  an  angle,  the 
length  of  circular  path  traversed  being  the  same  :  wherefore  the  curvature 
(as  above  defined)  of  the  circle  II  is  twice  that  of  the  circle  I,  and  that  of 
any  circle  is  inversely  as  the  radius :  and  since  curvature  and  acceleration 

A  \ 


towards  the  instantaneous  centre  both  vary  inversely  as  the  radius,  they  are 
proportional  to  one  another,  and  therefore  the  acceleration  of  a  body  moving 
in  a  curved  path  is  directed  towards  the  instantaneous  centre  of  curvature, 
and  is  equal  to  the  product  of  the  square  of  the  instantaneous  tangential 
velocity  into  the  curvature.  a  =  v2/r;  l/r=c;  .-.  a  =  t>2c,  where  c  is  the 
curvature.  Hence  a  comet  turning  sharply  round  the  sun,  the  curvature  of 
its  path  being  very  great,  has  a  very  great  acceleration  inwards. 


SIMPLE  HARMONIC  MOTION  AND  WAVE-MOTION. 

Motion  in  a  circle  may  be  practically  effected  by  a  heavy 
ball  suspended  by  a  string,  and  set  to  swing  in  a  circular  path. 
A  pendulum  set  to  swing  in  this  way  goes  by  the  name  of  the 
"  Conical  Pendulum."  If  the  path  of  the  bob  of  the  so-called 
conical  pendulum  be  looked  at  from  above,  it  appears  circular : 
if  looked  at  from  a  point  somewhat  to  one  side,  it  appears 
elliptical :  as  the  eye  approaches  the  level  of  the  plane  in  which 
the  bob  travels,  its  path  appears  to  be  an  ellipse  comparatively 
long  and  narrow ;  and  as  the  eye  is  placed  exactly  on  a  level  with 
that  plane,  the  bob  appears  to  travel  backwards  and  forwards 


SIMPLE   HARMONIC   MOTION. 


81 


Figf.29. 


in  a  straight  line.  In  a  similar  way,  the  satellites  of  Jupiter, 
which  travel  round  that  planet  pretty  nearly  in  the  plane  of  the 
Ecliptic,*  and  therefore  astronomically  on  a  level  with  ourselves, 
seem  to  travel  backwards  and  forwards  in  lines  nearly  straight. 
The  bob  of  the  conical  pendulum  and  the  satellites  of  Jupiter 
appear  to  move  very  slowly  at  the  end  of  their  apparently  linear 
courses.  This  is  because  the  moving  body  is  really  travelling 
either  towards  the  eye  of  the  observer  or  away  from  it  at  the  time 
when  it  appears  to  be  at  the  end  of  its  swing.  When  it  is  travel- 
ling right  across  the  field  of  view,  when  it  is  in  the  middle  of  its 
apparently  linear  path,  it  seems  to  travel  rapidly.  Just  in  the 
same  way  a  railway  train  seems  to  be  moving  very  much  faster 
when  it  runs  right  across  the  field  of  view  than  when  it  is  coming 
or  going  round  a  curve,  and  is  seen  not  broadside  but  end-on. 

If  we  represent  the  circle  in  which  the  body  is  moving  by 
the  circle  QAR,  and  its  apparent  linear  path  by  the  line  QR,  and 
if  we  represent  a  certain  number  of  positions  of  the  body  in  the 
circle  by  the  points  A,  B,  C,  D, 
etc.,  we  may  define  the  appar- 
ent motion  of  the  body  in  the 
straight  line  QR  by  finding  the 
points  a,  5,  <?,  etc.,  to  which 
the  respective  positions  of  the 
body  in  the  Circle  of  Refer- 
ence correspond.  This  is  done 
by  drawing  lines  Aa,  B6,  etc., 
at  right  angles  to  the  line  QR. 
It  will  be  easily  seen  that  if 
QA,  AB,  BC,  Co,  etc.,  be  equal 
to  one  another,  the  correspond- 
ing lines  drawn  along  QR  are  longer  near  the  centre  of  that 
line  than  near  its  ends ;  and  these  represent  the  spaces  appar- 
ently traversed  in  equal  intervals  of  time. 

A  representation  of  the  relative  values  of  these  lines  Qa,  ab,  etc.,  is 
obtained  as  follows.  If  the  line  QR  be  taken  as  the  axis  of  x,  the  line  OA 
may  be  supposed  to  sweep  round  into  the  successive  positions  OB,  OC,  Oo, 
etc.  As  it  does  so,  it  forms  an  increasing  angle  with  the  line  OQ.  Then 
the  lengths  of  the  lines  Oa,  Oft,  Oc,  etc.,  bear  to  one  another  the  ratios  of 
the  cosines  of  the  angles  QOA,  QOB,  etc. 

*The  bodies  which  make  up  the  solar  system  may  be  said  in  a  rough  way  to  be 
situated  in  a  plane  fixed  in  space,  and  called  the  Ecliptic,  from  which  they  do  not 
very  widely  depart.  Objects  moving  in  their  respective  orbits  in  this  plane  may 
appear  to  pass  and  obscure  —  i.e.  eclipse — one  another,  like  so  many  ships  at  sea. 

G 


82  KINEMATICS.  [CHAP. 

These  angles  will,  if  the  corresponding  motion  in  the  circle  of  reference 
be  uniform,  depend  directly  on  the  time.  If  the  angle  swept  through  in  unit 
of  time  be  to,  that  swept  through  in  time  t  is  tot:  hence,  if  the  starting-point 
in  time  be  that  instant  at  which  the  body  is  at  the  point  Q,  the  apparent 
distance  x  of  the  body  from  the  point  O  will  be  proportional  to  cos  to,  and,  if 
a  be  the  radius,  will  be  equal  to  a  cos  to.  Hence  x  =  a  cos  to.  When  to  =  0, 
since  cosO°  =  l,  x  =  a :  when  to  =  90°,  since  cos  90°  =  0,  x  =  0:  when 
to  =  180°,  since  cos  180°  =  -1,  x=-a:  when  to  =  270°,  x  =  0:  when 
to  =  360°  =  2?r,  x  =  a.  As  the  radius  continues  to  sweep  round,  the  values 
of  x  repeat  themselves. 

Such  a  motion  as  that  apparently  executed  backwards  and 
forwards  along  the  line  QR  is  called  Simple  Harmonic  Motion 
or  S.H.M.  Such  motion  must  be  studied  with  great  care,  for 
actual  instances  of  it  occur  throughout  the  phenomena  of  Optics 
and  Acoustics,  and  of  many  other  parts  of  physics. 

The  length  OQ  or  OR  of  the  swing  from  the  median  posi- 
tion O  is  called  the  Amplitude,  a,  of  the  S.H.M.  Simple  Har- 
monic motion,  then,  is  motion  which  is  a  periodic*  function  of 
the  time  (i.e.  repeats  itself  at  regular  intervals),  which  is 
effected  backwards  and  forwards  along  a  line,  and  which  may  be 
studied  by  comparison  with  uniform  motion  round  a  circle  of 
reference,  of  which  the  line  of  S.H.M.  is  the  diameter,  and  of 
which  accordingly  the  Amplitude  of  S.H.M.  measures  the 
Radius. 

The  Period,  T,  of  a  S.H.M.  is  the  interval  of  time  which 
elapses  between  the  passage  of  the  moving  particle  over  a  cer- 
tain point  and  the  next  passage  of  the  same  particle  over  the 
same  point  in  the  same  direction.  This  corresponds  to  the  time 
during  which  one  complete  revolution  would  be  effected  round 
the  circle  of  reference  ;  T  =  2?r/a). 

It  is  understood  that  when  the  moving  body  appears  to 
travel  from  left  to  right,  its  motion  is  positive  ;  when  it  moves 
from  right  to  left,  its  motion  is  in  the  negative  direction.  When 
the  particle  is  at  Q  in  Fig.  29,  it  is  said  to  be  in  its  position  of 
greatest  positive  elongation:  when  at  R,  it  is  in  its  position 
of  greatest  negative  elongation. 

At  any  instant  the  position  of  the  particle  executing  the 
S.H.  Motion  may  be  stated  in  terms  of  the  Phase  of  the  S.H.M. 
at  that  instant,  —  the  Phase  being  the  interval  -of  time,  the  frac- 

*  If  x  vary  when  y  varies,  as,  for  instance,  if  x  =  ay,  or  if  x2  =  y3  +ay2  +  by  +c, 
or  x  —  log  y,  or  if  in  any  other  way  whatsoever  the  value  of  x  depend  on  that  oiy,  x 
is  said  to  be  a  function  of  y  ;  and  if  x  recur  to  the  same  value  while  y  goes  on  uni- 
formly increasing  or  diminishing,  x  is  said  to  be  a  periodic  function  of  y  :  if  x  =  cos  y, 
or  =  tan  y,  or  =  sin  y,  etc.,  as  y  goes  on  increasing,  x  recurs  to  the  same  values,  for 
cos  y  =  cos  (2?r  +  y)  =  cos  (4?r  +  y)  =  cos  (Gir  +  y)  =  cos  (Sir  +  y),  etc. 


v.]  SIMPLE   HARMONIC   MOTION.  #3 

tion  of  a  period,  which  has  elapsed  since  the  particle  last  passed 
through  O,  the  middle  point  of  its  course,  in  the  direction 
reckoned  as  positive. 

The  phase  of  a  S.H.M.,  at  any  instant,  may  also  be  stated  by  specifying, 
for  that  instant,  the  corresponding  angle  swept  round,  in  the  circle  of  refer- 
ence, past  the  point  o ;  and  the  difference  of  phase  between  two  S.H.M.'s 
may  be  stated  by  specifying  the  difference  between  two  such  angles,  taken 
simultaneously. 

If  the  starting-point  in  time  be,  not  the  instant  at  which  the  particle  was 
at  the  point  Q  in  the  circle  of  reference  (Fig.  29),  but  so  many  units  of  time 
after  that  instant  that  the  angle  traversed  is  not  t<a  but  (t<a  +  e),  then  the 
displacement,  or  distance  from  O,  along  the  axis  of  x  is  x  =  a  cos  (i<a  -f  e}. 
This  term  e  is  called  the  Epoch. 

Acceleration  in  S.H.M.  proportional  to  Displacement. — 

In  Fig.  29  the  moving  particle,  when  it  describes  a  circular  path, 
does  so  under  the  influence  of  an  acceleration  v2/r  towards  the 
centre.  This  may  be  resolved,  when  the  particle  is  at  any  point 
A,  into  (-cos  AOQ-fl2/r)  parallel  to  QR,  and  (-sin  AOQ-v2/r) 
at  right  angles  to  it.  The  former  component  is  alone  effective  in 
reference  to  a  body  moving  in  S.H.M.  in  the  line  QR,  and,  being 
always  towards  the  centre,  it  is  alternately  in  the  same  and  in  an 
opposite  direction  to  the  movement  of  the  body.  But  this  accel- 
eration towards  the  centre  =(—  cos  AOQ  •  v2/r)=(O#./r) .  (_  v2/r) 
=  Oa«(  —  v2/r2)  =  Oa  x  a  constant  negative  number,  for  v  (i.e. 
the  maximum  velocity  in  the  line  QR  =  the  uniform  speed  v 
in  the  circle)  and  r  are  constant.  The  Acceleration  when  the 
particle  is  at  any  point  a  in  its  S.H.M.  is  thus  proportional  and 
also  opposite  to  Oa  =  x,  the  Displacement  from  O. 

Again,  the  constant  number  (v2/r2)  is  equal  to  the  square  of 
v/r  =  a>,  the  angular  velocity  in  the  circle  of  reference.  Therefore 

m  _     /acceleration   at  any  point* 
^displacement  at  that  point. 

The  acceleration,  a  =  x,  at  any  instant  being  proportional  and  opposite  to 
the  displacement  x,  we  have  x  =  —  n2x,  where  n2  is  a  factor  always  positive. 
This  is  a  Differential  Equation  :  the  solution  is,  that  at  the  end  of  any  time 
t,  the  displacement  x,  =  a  cos  n£,  where  a  is  the  maximum  value  of  x.  This 
agrees  with  the  previous  equation  x  =  a  cos  ta>,  on  the  footing  that  n2  =  <o2. 

The  Frequency  of  a  S.H.M.  is  the  number  of  periods  per  second,  the 
number  of  revolutions  round  the  circle  of  reference,  or  of  complete  to-and- 
fro  oscillations  per  second.  This  is  n  =  1/T  =  <o/27r  =n/2ir. 

Isochronous  S.H.M.'S.  —  Since  o>  is  the  angular  velocity 
(page  75,  note),  arid  since  the  time  taken  to  execute  one  com- 
plete revolution  round  the  circle  of  reference  is  T  =  27r/o>,  then 
if  w,  the  angular  velocity  in  the  circle  of  reference,  be  constant, 


84 


KINEMATICS. 


[CHAP. 


the  time  T  —  that  is,  the  period  of  the  S.H.M.  —  is  independent 
of  the  amplitude ;  for  the  amplitude  does  not  enter  into  that 
formula  which  expresses  the  value  of  T,  namely,  T  =  ZTT/CO.  This 
criterion,  the  constancy  of  o>,  is  satisfied  if  the  quotient 

acceleration  , 

^ — = —          -  be   a   constant  number.     In  other  words,   if   the 

displacement 

acceleration  with  which  a  particle  tends  to  return  to  its  median 
position  bear  a  fixed  proportion  to  the  displacement,  the  particle 
will  execute  a  S.H.M.  whose  period  is  independent  of  the  ampli- 
tude of  oscillation.  This  proposition  is  one  of  high  importance 
in  the  theory  of  the  Pendulum,  of  Elastic  bodies,  of  Sound,  of 
Heat,  and  of  Light. 

Projection  of  a  S.H.M.  always  an  Apparent  S.H.M.  —  It 

Fig.30. 
A 


is  understood  that  when 
a  line  AB  is  looked  at 
from  the  position  c  in 
Fig.  30,  that  line  ap- 
pears to  be  shortened,  and 
to  assume  the  length  DE, 
and  the  line  DE,  or  de, 
at  right  angles  to  cO,  is 
called  a  Projection  of 
AB.  There  may  be  as 
many  projections  as  there 
are  possible  directions  of 
the  line  Oc.  When  the 
eye  is  placed  somewhere 
d  in  the  line  Ocr  the  line 
AB  does  not  appear  to  be 
shortened,  and  the  projec- 
tion ab  of  the  line  AB  is 
equal  to  that  line  itself; 
when  the  direction  of 

sight  has  become  AB<?/;,  the  projection  of  the  line  AB,  thus  seen 

end-on,  is  a  point  merely. 

If  the  position  of  the  point  of  view  be  intermediate  between  these 
extremes,  as,  for  example,  at  c,  the  projection  ed  is  to  the  original  AB  as  the 
cosine  of  the  angle  AOD  in  the  figure  is  to  unity. 

If  now  a  body  execute  S.H.M.  in  the  line  AB,  the  diagram 
will  show  that,  if  regarded  from  c,  it  will  appear  to  perform  a 
S.H.M.  corresponding  in  period  and  phase,  though  not  in  ampli- 


HARMONIC   CURVE. 


85 


tude,  in  the  line  DE ;  or,  in  other  words,  the  projection  of  a 
S.H.M.  is  itself  an  apparent  S.H.M. 

Harmonic  Curve.  — If  a  S.H.M.  in  one  line  be  compounded 
with  a  uniform  motion  in  a  direction  at  right  angles  to  that  line, 
the  resultant  path  may  be  found  by  the  following  construction. 
Let  A  and  B  (Fig.  31)  be  the  points  of  greatest  elongation  of  the 
particle.  Let  the  particle  be  also  made  at  the  same  time  to  travel 
uniformly  from  left  to  right.  Draw  ACB,  the  circle  of  reference. 
On  it  lay  off  (say)  sixteen  equidistant  points,  D,  E,  F,  G,  etc. ; 
lines  drawn  through  these  at  right  angles  to  the  line  AB  deter- 
mine the  points  on  that  line  which  define  the  positions  of  the 
particle,  so  far  as  these  positions  are  determined  by  the  S.H.M., 
at  equal  intervals  of  ^  of  the  period.  These  lines  being  drawn 
as  in  the  figure,  other  lines  may  be  drawn  parallel  to  AB  and 
cutting  the  axis  OX  at  equal  intervals,  each  of  which  represents 

Fig.31. 


the  amount  of  motion  from  left  to  right  during  -^  of  the  period 
of  the  S.H.M.  The  previous  examples  of  composition  of  simul- 
taneous motions  will  show  us  that  the  successive  positions  after 
successive  intervals  (T*g  of  a  period  in  this  case)  will  be  found  by 
marking  off  points,  such  as  those  indicated  in  the  diagram,  the 
distances  of  which  along  the  axis  of  x  represent  the  displace- 
ments due  to  the  uniform  motion,  and  whose  distances  along  the 
axis  of  y  represent  the  displacements  due  to  the  S.H.M.  If 
these  points  be  joined  they  give  rise  to  a  very  characteristic 
curve,  the  Curve  of  Sines,  or  the  Harmonic  Curve. 

The  geometrical  property  of  this  curve  evidently  is  that,  of  any  point  in 
it  the  Abscissa  (along  OX)  is  proportional  to  the  time,  while  the  Ordinates 
(or  distances  from  the  axis  of  x)  are  proportional  to  the  sines  of  angles, 
which  are  themselves  proportional  to  the  time.  The  ordinates,  therefore, 
pass  through  positive  and  negative  values  alternately,  while  the  abscissae 
uniformly  alter  in  value. 

Take  the  closed  figure  bounded  by  any  one  half-period  portion  of  the 
curve  of  sines  lying  entirely  above  or  entirely  below  the  base  line  OX, 
Fig.  31,  together  with  the  corresponding  portion  of  the  base-line  cut  off  by 
it.  The  area  of  that  figure  is  2/-jr  times  the  rectangle  between  the  amplitude 
OA  and  the  portion  of  the  base-line  so  cut  off.  During  such  a  half -period, 
therefore,  the  average  value  of  the  ordinates  is  2 /IT  x  the  maximum 
value  OA.  , 

There  may  be  several  curves  of  sines,  differing  from  one 
another  in  the  amplitude  of  the  S.H.M.,  or  in  the  rapidity  of 


86  KINEMATICS.  [CHAP. 

the  uniform  motion  in  the  direction  of  the  axis  of  x.  Evidently, 
if  the  amplitude  of  the  S.H.M.  be  greater  or  less,  the  undulations 
of  the  Harmonic  Curve  will  be  deeper  or  shallower ;  while,  if 
the  motion  along  the  axis  of  x  be  slower  or  quicker,  the  undu- 
lations of  the  resultant  curve  will  be  closer  together  or  farther 
apart. 

When  a  pendulum  is  set  to  swing,  its  oscillatory  motion  is 
visibly  quickest  at  the  middle  of  its  course  and  slackens  towards 
each  end  of  it ;  so  that  the  motion  of  a  pendulum  is  very  much 
like  S.H.M.,  and  hence,  if  a  pendulum  be  made  to  carry  sand  and 
to  drop  it  as  it  travels,  it  will  deposit  a  trace  which  is  much 
thicker  at  each  end  of  its  course,  where  its  bob  is  moving  slowly, 
than  it  is  at  the  middle  where  its  course  is  rapid.  If  the  pendu- 
lum be  made  to  oscillate,  while  the  frame  which  supports  it  is  at 
the  same  time  made  to  travel  in  a  direction  at  right  angles  to 
the  plane  of  oscillation  of  the  bob,  or  —  what  amounts  to  the 
same  thing  —  if  the  surface  on  which  the  sand  is  received  be 
made  to  travel  under  the  oscillating  pendulum,  which  is  sus- 
pended from  a  fixed  support,  the  sand  is  deposited  in  a  curve 
which  can  hardly  be  distinguished  from  the  Harmonic  Curve. 
But  any  motion  which,  when  so  compounded  with  a  uniform  rec- 
tilinear motion,  produces  the  characteristic  Curve  of  Sines  must 
itself  be  a  S.H.M.,  and  hence  the  motion  of  the  pendulum, 
projected  on  the  plane  which  receives  the  sand,  is  approximately 
a  S.H.M. 

That  the  trace  left  by  the  falling  sand  does  not  with  perfect  exactitude 
coincide  with  any  harmonic  curve  is  due  to  the  fact  that  though  the  motion 
of  the  bob  in  its  curved  path  is  nearly  S.H.M.,  the  necessary  divergence 
between  that  curved  path  and  the  flat  plane  on  which  the  sand  is  deposited 
expresses  itself  as  a  slight  distortion  of  the  resultant  trace.  If  the  arc  of 
oscillation  be  small,  this  distortion  is  so  very  small  that  most  of  the  proper- 
ties of  Simple  Harmonic  Motion  can  be  practically  demonstrated  by  the  use 
of  pendulums  which  record  their  own  movement  in  some  such  way  as  that 
mentioned. 

The  trace  left  by  such  a  moving  body  does  not  present,  in 
the  parts  corresponding  to  the  greatest  positive  or  negative 
elongation,  so  steep  an  ascent  or  descent  as  it  does  when  it 
crosses  the  axis  OX  (Fig.  31) ;  this  indicates  .that  the  body 
moving  in  S.H.M.  is  moving  more  rapidly  at  the  centre  of  its 
course  than  at  its  ends. 

Composition  of  Simple  Harmonic  Motions.  —  If  the  same 
body  be  subjected  to  two  different  S.H.M. 's,  the  problem  of  their 
composition  may  in  general  be  solved  with  great  ease  by  the  use 


COMPOSITION  OF   SIMPLE   HARMONIC   MOTIONS. 


87 


Fig.32. 


\ 


b\ 


of  the  respective  circles  of  reference.     (1.)  Let  the  two  motions 
be  equal  and   in  the  same  direction:    the  resultant  will   be  a 
S.H.M.  of  double  amplitude.    (2.)  Let  the  two  motions  be  equal 
and  in  the  same  line,  but  differ  from  one  another  in  phase  by 
half  a  period:  the  resultant  will  be  Rest.     (3.)  Let  the  two 
S.H.M.'s  be  equal,  at  right  angles  to  one  another  (AB  and  CD, 
Fig.  32),  and  in  the  same  phase,  so  that  when  the  moving  parti- 
cle is  at  O  it  is  moving  in  a  positive  direction  with  reference  to 
both  axes:    its  real  course  will   be  in  a  line  PP',  making  an 
angle  of  45°  with  both  AB  and  CD.     (4.)  If  it  be  half  a  period 
behind  in  one  of  the  S.H.M.'s,  so  as  to  be  moving  in  the  + 
direction  (from  O  to  D)  with  reference  to  one  axis,  and  nega- 
tively (from  O  to  B)  with  ref- 
erence to  the  other,  the  resul-  Q 
tant  will  be  S.H.M.  in  the  line 
Q'Q.     (5.)  If  the  one  S.H.M. 
be  a  quarter  period  behind  the 
other,  so  that  while  the  moving 
particle   is   at   the    middle    of 
(one,  say)  its  vertical  oscilla-  c 
tion,   it   is    only   just   leaving 
the  point  of  greatest  negative 
elongation,  in  respect    to   the 
other  —  its   horizontal    oscilla- 
tion—  its  motion  will  be  com- 
pounded of  one  forwards,  from  p 
C  towards  O,  and  one  upwards,  past  O  towards  A;  the  result  will 
be  that  the  motion  of  the  body  will  be  restricted  to  the  circum- 
ference of  the  circle  DBC,  and  the  body  will  move  round  that 
circle  in  the  direction  CA.    Similarly  (6),  if  the  horizontal  move- 
ment be  in  advance  of  the  vertical  by  a  quarter  period,  so  as  to 
be  already  bringing  the  body  back  from  its  position  of  greatest 
positive  elongation  while  it  is  still  moving  vertically  upwards 
past  O  towards  A,  the  body  will  travel  in  the  circle  DAC  in  the 
direction  DA.     Hence  we  have  the  very  important  proposition 
that  motion  in  a  circular  path  may  be  considered  to  be  made  up 
of  two  S.H.M.'s,  the  one  a  quarter  of  a  period  in  advance  of  or 
behind  the  other,  according  to  the  direction  in  which  the  body 
travels  in  the  circle. 

Though  the  conical  pendulum  shows  this  when,  its  motion 
is  watched,  perhaps  the  simple  piece  of  mechanism  drawn  in 
Fig.  33  may  make  it  even  plainer.  The  circular  plate  ACB  has 


88 


KINEMATICS. 


[CHAP. 


a  pin  D  set  in  it.  This  pin  works  in  a  sliding  piece  within  a 
slot  in  the  frame  EF.  The  frame  EF  is  connected  with  two 
sliding  bars,  which  run  between  the 
guides  G,  so  that  lateral  motion  is  im- 
possible. Let  the  circular  plate  ACB 
be  rotated  uniformly:  the  frame  EF 
Fig.33.  will  be  moved  upwards  and  downwards 
alternately,  while  the  pin  D  will  move 
in  the  slot  from  right  to  left  and  from 
left  to  right  alternately.  It  will  easily 
be  seen  on  making  a  model,  or  on  imagi- 
ning the  diagram  to  act,  that  the  oscilla- 
tions of  D  in  its  slot,  and  those  by  which 
it  produces  alternating  ("  reciprocat- 
B  ing ")  motion  of  EF,  do  not  agree  in 
phase,  but  differ  by  a  quarter  of  a 
period,  the  one  being  at  the  middle 
when  the  other  is  at  the  end  of  its 
course. 

The  circular  motion  of  the  pin  D  is, 
therefore,  compounded  of  two  S.H.M/s, 
of  which  one  is  easily  conveyed  to  the 
frame  EF,  while  the  other  cannot,  be- 
cause of  the  arrangement  of  the  guides 
G,  be  so  conveyed.  The  apparent  con- 
version of  the  circular  motion  of  the  disc 
ACB  into  the  reciprocating  motion  of 
the  sliding-bars  is  in  reality,  then,  due  to  the  suppression  of  one 
of  its  simple  harmonic  components  ;  and  the  motion  of  the  slid- 
ing-bars is  exactly  S.H.M.  if  D  rotate  uniformly. 

Circular  transformed  into  Reciprocating  Motion.  —  In  accordance 
with  this  principle,  mechanism  intended  to  transform  rotatory  into  recipro- 
cating motion  is  in  reality  mechanism  which,  with  more  or  less  completeness, 
suppresses  one  of  the  S.H.M.'s  of  each  particle  of  a  rotating  body.  The  most 
usual  device  is  that  of  a  crank ;  this  may  be  seen  in  one  form  or  another  in 
almost  every  piece  of  machinery  worked  by  steam-power.  In  Fig.  34  the 
wheel  A  is  rotated  almost  uniformly,  and  the  crank  B  is  turned  round  along 
with  it ;  attached  to  the  crank  B  by  a  joint  at  C  is  the  rod  D,  which  is,  in 
its  turn,  attached  by  the  joint  E  to  the  rod  F ;  this  run's  between  the  guides 
G.  Here  the  motion  of  the  bar  F  between  the  guides  is  only  approximately 
Simple  Harmonic,  but  approximates  more  and  more  nearly  to  that  condition 
the  longer  the  bar  D  and  the  shorter  the  crank  B,  or,  in  other  words,  the 
less  F  and  D  together  diverge  from  a  straight  line.  The  rod  F  may  be  made 
to  work  a  pump-handle,  a  saw,  or  any  such  contrivance  whose  use  requires 
reciprocating  motion. 


v-3 


CIRCULAR   MOTION. 


89 


Conversion  of  Reciprocating  into  Circular  Motion.  — If  in  Fig. 
34  the  rod  F  be  supposed  to  be  pushed  towards  the  crank  B,  then  D  will  be 
pushed  over  towards  C,  and  the  wheel  will  turn  until  E,  O,  and  C  are  in  the 
same  straight  line.  No  further  pushing  will  make  the  wheel  A  turn  any 
farther ;  neither  will  pulling,  when  the  crank  is  in  this  position,  have  any 
effect.  If  in  the  same  figure  the  rod  F  be  pulled  instead  of  pushed,  the 
points  E,  C,  and  O  will  come  to  be  in  the  same  straight  line ;  and  in  this 


Fig.34. 


position,  again,  neither  pulling  nor  pushing  will  have  any  effect  in  making 
the  wheel  turn.  There  are  therefore  two  positions  or  dead  points  in  which 
a  piston  cannot  by  means  of  a  crank  set  a  wheel  in  motion.  If,  however, 
the  wheel  A  be  heavy,  or,  better  still,  if  it  be  connected  with  a  heavy  fly- 
wheel, when  it  arrives  at  the  dead  points  its  Inertia,  or  that  of  the  flywheel, 
manifests  itself  by  the  wheel  A  rotating  past  these  unfavourable  positions 
into  others  in  which  the  reciprocating  movement  of  F  can  act  effectively ;  the 
wheel  A  is  thus  set  in  continuous  motion.  An  example  of  this  is  furnished 
by  the  treadle  and  flywheel  of  a  lathe  or  of  a  sewing-machine. 

In  the  marine-engine,  since  there  can  be  no  flywheel  on  board  ship,  some 
other  contrivance  is  necessary.  Two  cylinders,  and  therefore  two  reciprocat- 
ing pistons,  both  acting  on  the  same  wheelwork,  are  so  arranged  that  when 
the  one  crank  is  at  its  dead  points  the  other  is  at  the  middle  of  its  course, 
and  therefore  at  its  position  of  greatest  advantage ;  or  there  may  be  three, 
at  a  mutual  angle  of  120°. 

Composition  of  Simple  Harmonic  Motions  (resumed). — 
Let  the  two  S.H.M.'s  differ  by  some  other  fraction  of  a  period 
than  the  half  or  the  quarter,  as,  for  instance,  the  twelfth:  if 
they  be  equal  in  amplitude  they  both  have  the  same  or  equal 
circles  of  reference.  Let  the  circumference  of  these  be  divided 
into  twenty-four  equal  parts,  as  in  Fig.  35,  in  which  only  a  part 
of  each  circle  of  reference  is  shown. 

If,  now,  the  particle  be  at  the  middle  of  its  course  with  refer- 
ence to  the  S.H.M.  along  the  axis  BA,  and  if  it  be  at  the  same 
time  ^2  of  a  period  behind  (so  as  not  yet  to  have  arrived  at  the 
central  point)  in  its  execution  of  the  S.H.M.  referred  to  the  axis 
CD,  the  point  at  which  it  must  be  situated  in  order  to  satisfy 
both  these  conditions  must  be  N,  G.  When  y1^  period  has 
elapsed,  it  will  have  advanced  to  the  middle  of  its  horizontal 
course,  but  will  have  moved  vertically  as  far  as  the  point  M,  H ; 
at  the  end  of  another  -^  it  will  be  at  the  point  L,  J,  then  at 


90 


KINEMATICS. 


[CHAP. 


A,  K  ;  then  it  still  advances  horizontally  to  the  limit  of  its 
course,  but  returns  along  AB  to  L,  thus  reaching  the  point  L,  D  : 
then  it  returns  to  M,  K,  and  so  on,  and  in  this  way  it  describes  an 
ellipse.  When  the  difference  of  phase  is  less,  the  point  N,  G  is 
nearer  to  O,  and  the  ellipse  is  narrower  ;  when  there  is  no  differ- 
ence of  phase,  the  point  N,  G  coincides  with  O,  and  this  ellipse 
is  a  straight  line,  as  has  been  already  learned  (Fig.  32).  When, 
on  the  other  hand,  the  difference  of  phase  is  greater,  the  point 
N,  G  is  farther  from  O,  and  the  ellipse  widens  out  until,  when 


Fig.35. 


A,K 


^ 


Q,o 


N,G 


Q,G 


H,P 


N,J 


the  difference  of  phase  is  ^  period,  the  point  N,  G  is  opposite  to 
C,  and  the  ellipse  is  a  perfect  circle. 

When  the  amplitudes  are  not  equal,  the  circles  of  reference 
will  not  be  equal.  If  the  two  S.H.M.'s  be  in  AB  and  CD,  the  cor- 
responding construction  is  shown  in  Fig.  36.  If  they  be  in  the 
same  phase,  the  resultant  is  S.H.M.  in  the  line  R'OR ;  if  they 
differ  in  phase  by  half  a  period,  the  resultant,  is  S.H.M.  in  the 
line  QOQ' ;  if  the  difference  of  phase  be  J  period,  the  path  is  the 
ellipse  ADBC,  traversed  in  the  direction  BC  if  the  S.H.M.  in 
AB  be  J  period  in  advance,  and  in  the  direction  BD  if  it  be  £ 
period  in  arrear.  If  the  difference  of  phase  be  any  other  frac- 
tion of  a  period,  the  resultant  will  be  motion  in  some  other 


COMPOSITION  OF  SIMPLE   HARMONIC   MOTIONS. 


91 


ellipse  contained  within  the  same  bounding  rectangle 
QRQ'R'.     The  construction  is  the  same  as  that  of  Fig.  35. 

Composition  of  S.H.M.'s  of  different  period.  — The 
same  method  with  little  modification  may  be  here  employed. 
The  respective  circles  of  reference  are  drawn  and  are  divided  into 
arcs  corresponding  to  equal  intervals  of  time.  The 
lines  representing  the  S.H.M.'s  are  divided  in  accordance  with 
the  now  well-known  construction,  and  the  positions  of  the  body 
traced  out  accordingly. 

Fig.36. 


\ 


\ 


\ 


In  Fig.  37  the  periods  are  as  two  to  three,  the  period  of 
the  vertical  S.H.M.  being  the  shorter :  the  ranges  of  oscillation 
are  represented  by  tjie  lengths  of  AB,  CD  respectively.  The 
respective  circles  of  reference  are  drawn:  they  are  equally 
divided  into  arcs  corresponding  to  intervals  of  time  arbitrarily 
chosen,  say  the  sixteenth  part  of  the  period  of  the  more  rapid 
oscillation  in  AB,  this  being  the  fa  of  the  period  of  the  slower 
oscillation  in  CD.  The  arcs  AB  and  CD  having  been  thus  divided 
into  segments  corresponding  to  equal  intervals  of  tiirfe,  the  usual 
construction  enables  us  to  trace  out,  point  by  point,  the  path  of 
the  body,  which  — if  we  assume  the  body  to  be  in  the  position 


92 


KINEMATICS. 


[CHAP. 


of  greatest  positive  elongation  with  respect  to  both  S.H.M.'s 
simultaneously,  and  therefore  to  touch  the  point  R,  there  being 
there  no  difference  of  phase  —  we  find  to  be  a  looped  curve 
over  which  the  body  travels  backwards  and  forwards  without 
quitting  it. 

In  Fig.  38  the  period  of  the  S.H.M.  in  CD  is  -£  of  that  in 
AB.  The  construction  is  essentially  the  same.  The  arcs  cut  off 
on  the  circle  AB  must  subtend  angles  at  their  centre  |  of  those 
similarly  subtended  by  the  arcs  on  CD.  Hence  the  circle  AB 

Fig.37. 


\ 


\ 


\ 


V 


\ 


has  been  divided  into  arcs,  each  of  which  represents  one-twen- 
tieth of  the  circumference,  while  the  circle  CD  has  been  divided 
into  16.  These  curves  may  assume  a  variety  of  forms  depend- 
ing on  variations  in  the  relative  lengths  of  AB  and  of  CD. 

Composition  of  S.H.M.'s  differing  in  Phase. — If  the 
'two  S.H.M.'s  differ  in  phase,  and  if  those  chosen  as  illustra- 
tions be,  for  convenience  of  reference,  the  same  as  those  of 
Figs.  37  and  38,  the  same  circles  may  be  dra.wn  and  divided 
in  the  same  way.  If  the  difference  of  phase  between  the  two 
S.H.M.'s  correspond  to  such  a  fraction  of  the  period  of  the 
more  rapid  oscillation  as  may  be  represented  by  the  arc  DE 


COMPOSITION  OF   SIMPLE   HARMONIC   MOTIONS. 


93 


(whether  this  be  an  aliquot  part  of  the  circumference  or  not), 
the  body  cannot  be  at  the  extremity  of  both  its  S.H.M.'s  at  one 
time,  and  when  it  is  opposite  the  point  A  it  will  simultaneously 
be  opposite  not  the  point  D  but  the  point  E.  Figs.  39  and  40 
indicate  the  modifications  undergone  by  the  resultant  curves  of 
Figs.  37  and  38  in  consequence  of  such  differences  of  phase. 

These  resultant  curves  vary  considerably  in  form,  according 
to  the  amount  of  difference  of  phase  of  the  component  S.H.M.'s. 
Fig.  41  shows,  for  example,  a  series  of  modifications  of  the  curve 

Fig.88. 


of  the  ratio  1 :  2,  in  which  the  more  rapid  oscillation  is  in  advance 
by  periods  which  successively  differ  from  one  another  by  one- 
eighth  of  the  period  of  the  more  rapid  oscillation. 

A  S.H.M.  in  a  third  dimension  may  be  compounded  with  two  in  a  plane. 

Composition  of  non-commensurable  S.H.M.'s.  —  The  pe- 
riods in  all  the  cases  already  considered  have  been  commen- 
surable, i.e.  they  have  borne  to  one  another  ratios  expressible 
in  whole  numbers,  and  consequently,  after  a  certain  number  of 
oscillations,  the  moving  body  has  returned  to  the  starting-point, 
and  the  path  has  been  a  closed  curve  which  the  body  has  trav- 
ersed repeatedly.  If,  however,  the  periods  be  not  commeusur- 


94 


KINEMATICS. 


[CHAP. 


able,  the  body  cannot  return  to  the  starting-point  after  any 
definite  number  of  oscillations,  and  the  path  never  becomes  a 
closed  curve. 

Composition  of  S.H.M.'s  whose  periods  approximate  to 
an  aliquot  ratio.  —  If  the  periods  of  the  two  component  S.H.M.'s 


be,  for  example,  very  nearly  as  one  to  two,  but  not  exactly  so, 
the  resultant  curve  may  be,  at  a  given  moment,  practically  the 
same  as  that  of  (a)  in  Fig.  41.  The  moving  body  cannot,  how- 
ever, continue  to  maintain  this  parabolic  path,  for  the  want  of 
exact  aliquot  proportion  of  the  two  periods  causes  one  of  the 
two  S.H.M.'s  to  pass  in  advance  of  the  other,  which,  as  it  were, 
lags  behind,  and  thus  it  establishes  an  increasing  difference  of 


COMPOSITION   OF   SIMPLE   HABMONIC   MOTIONS. 


95 


phase.  When  this  accumulated  difference  of  phase  amounts 
to  £  the  period  of  the  more  rapid  oscillation,  the  path  described 
is  approximately  that  of  (6)  in  Fig.  41.  In  this  way,  by  continu- 
ous modification,  the  curve  passes  successively  through  all  the 
forms  shown  in  Fig.  41. 


If  the  respective  periods  be  as  10  : 21,  their  ratio  is  approximately  1 : 2, 
but  not  exactly  so ;  when  the  slower  S.H.M.  has  been  effected  5  times,  the 
quicker  has  been  effected  not  10  but  10£  times,  and  consequently  the  quicker 
motion  is  a  |  period  in  advance,  and  the  form  of  the  path  has  been  modified 
from  nearly  that  of  (a)  to  nearly  that  of  (e)  in  Fig.  41.  When  10  of  the 
slower  S.H.M.'s  have  been  effected,  and  of  course  in  the  same  time  21  of  the 
more  rapid  ones,  the  path  resumes  for  an  instant  its  original  £orm  (a). 

If  the  periods  were  as  1000 :  2001,  in  the  same  way  it  will  be  seen  that 
the  path  regains  its  original  form,  when  1000  of  the  more  slowly-performed 
S.H.M.'s  have  been  executed. 


96 


KINEMATICS. 


[CHAP. 


Hence  the  less  the  proportionate  divergence  from  the  simple 
aliquot  ratio  to  which  the  actual  ratio  approximates,  the  greater 
the  number  of  oscillations  that  must  be  performed,  and  hence  the 
longer  the  time  that  must  elapse  before  the  original  form  of  the 
path  recurs,  as  it  will  do,  approximately  if  the  periods  be  non- 
commensurable,  perfectly  if  they  be  commensurable. 


Fig.  41. 


Resolution  of  S.H.M.  into  two  rectangular  components.  — 

We  have  seen  that  two  S.H.M.'s  at  right  angles  to  one  another, 
and  having  the  same  period  and  phase,  may  be  compounded  into 
Fig.42.  a  single  S.H.M.  by  a  construc- 

A  Vt  tion  precisely  the  same  as  that 

of  the  rectangular  parallelo- 
gram of  velocities.  Con- 
versely, just  as  a  velocity  may 
be  resolved  into  two  compo- 
nent velocities  in  any  two  di- 
rections at  right  angles  to  one 
B  another,  so  may  any  S.H.M. 
be  resolved  into  two  S.H.M.'s  in  any  two  directions  at  right 
angles  to  one  another.  If  in  Fig.  42  the  S.H.M.  be  in  AB,  it 
may  be  resolved  into  two,  in  xtx  and  y ty  respectively. 

Any  number  of  S.H.M.'s,  in  any  directions,  may  be  resolved  into  their 
components  in  three  rectangular  axes,  and  these  may  then  be  compounded. 

Composition  of  any  S.H.M.  with  a  uniform  movement  in  any 
direction.  —  If  we  wish  to  compound  a  S.H.M.,  which  is  effected  in  a  line 
neither  parallel  nor  at  right  angles  to  the  axis  Ore,  with  a  uniform  motion 
in  the  direction  Ox,  we  must  first  break  the  S.H.M.  up  into  its  components 
in  the  respective  directions  Ox  and  yyt.  We  may  then  compound  the  com- 
ponent S.H.M.  in  the  direction  yy,  with  the  uniform  movement  in  the  axis  of 


v.]  COMPOSITION  OF  SIMPLE   HARMONIC   MOTIONS.  97 

x,  thus  producing  (as  was  done  in  Fig.  31)  the  Harmonic  Curve  or  Curve 
of  Sines,  indicated  by  a  dotted  line  in  Fig.  43.  We  may  then  from  point  to 
point  compound  this  harmonic  curve  with  the  component  S.H.M.  in  xtx  by 
determining  point  after  point  in  advance  of  or  behind  the  dotted  harmonic 
curve  to  an  extent  corresponding  to  the  displacement  produced  by  that  com- 

Fig.48. 


BN 

ponent.  The  resultant  path,  indicated  by  the  thick  dotted  line,  is  com- 
pounded, then,  of  a  uniform  motion  in  the  axis  of  x,  a  S.H.M.  in  the  same 
axis,  and  another  S.H.M.  of  the  same  period  and  phase  in  a  line  at  right 
angles  to  that  axis.  The  form  of  the  resultant  curve  varies  according  to  the 
speed  of  that  uniform  motion  which  is  compounded  with  the  oblique  S.H.M. 

Composition  of  two  S.H.M.'s  in  the  same  line. —  If  two 
S.H.M.'s  in  the  same  line  be  compounded,  the  resultant  motion 
will  also  be  in  the  same  line,  and  it  is  best  studied  by  reference 
to  the  harmonic  curve.  Let  two  S.H.M.'s,  which  have  the  same 
periods  and  phases,  and  which  are  in  the  same  straight  line  AB, 
have  the  amplitudes  OA  and  OC,  and  let  a  corresponding  Har- 
monic Curve  be  traced  for  each.  Then  the  corresponding  curve 
produced  by  the  superposition  of  these  two  motions  may  be 
traced  from  point  to  point  by  adding  the  displacements  sepa- 
rately indicated  by  the  harmonic  curves.  This  resultant  is 
found  to  be  a  Harmonic  Curve,  and  on  careful  drawing  to 

•"\ 


scale  it  may  be  shown  absolutely  to  coincide  with  the  Curve  of 
Sines  derived  from  a  S.H.M.  in  a  line  whose  direction  is  the 
same  as  that  of  AB,  and  whose  amplitude  is  equal  to  the  sum  of 
OA  and  OC.  On  the  other  hand,  when  these  two  S.H.M.'s  are 
in  opposite  phases,  differing  by  half  a  period,  so  that  while  one 
raises  the  body  above  the  point  O,  the  other  depresses  it  below 
that  point,  the  resultant  curve  is  also  found  to  be  a  Harmonic 
Curve  corresponding  to  a  S.H.M.  whose  amplitude  is  equal  to 


98  KINEMATICS.  [CHAP. 

the  difference  between  the  amplitudes  of  the  two  components. 
Hence  two  S.H.M.'s  of  the  same  period  and  in  the  same  straight 
line  will,  when  compounded,  produce  a  single  S.H.M.  of  the 
same  period  and  in  the  same  line,  whose  amplitude  is  the  sum  of 
the  amplitudes  of  the  components  if  they  agree  in  phase,  and 
their  difference  if  their  phases  be  opposed.  Manifestly,  if  the 
two  component  S.H.M.'s  be  equal  to  one  another,  the  resultant 
will  be,  in  amplitude,  double  of  either  of  them  if  they  agree  in 
phase,  and  will  be  zero  —  that  is,  the  body  will  be  at  rest  —  if 
they  be  opposed  in  phase,  the  corresponding  harmonic  curve 
being  in  this  latter  case  reduced  to  a  straight  line. 

If  the  phases  be  neither  in  exact  accord  nor  in  exact  opposition,  the 
resultant  curve  is  still  Harmonic,  but  the  amplitude  is  found  by  a  construc- 
tion the  same  as  that  of  the  parallelogram  of  velocities  :  lines  representing 
the  two  component  amplitudes  are  laid  down  at  an  angle  representing  the 
difference  of  phase,  and  the  diagonal  thus  found  represents,  in  length,  the  am- 
plitude sought.  Any  number  of  S.H.M.'s  of  the  same  period  and  in  the 
same  straight  line,  but  differing  to  any  extent  in  amplitude  and  phase,  may 
be  similarly  compounded,  by  a  construction  like  that  of  the  Polygon  of 
Velocities. 

If  the  two  S.H.M.'s  have  different  Periods,  the  result  is 
more  complex.  Let  the  periods  of  the  S.H.M.'s  bear  the  ratio 
3 :  8.  Then  the  Harmonic  Curve  corresponding  to  the  more 
rapid  S.H.M.  will  present  eight  undulations  for  every  three  of 
the  less  rapid  ones.  These  are  drawn  in  Fig.  45  (I).  On  add- 
ing the  displacements  represented  by  these  curves,  the  resultant 
may  be  traced  from  point  to  point,  and  is  found  to  form  a 
comparatively  complex  curve.  Obviously  there  may  be  an 
indefinite  number  of  forms  of  this  resulting  curve,  for  the  ratio 
of  the  amplitudes  may  vary  indefinitely. 

In  Fig.  45  the  curves  show  the  change  produced  in  a  com- 
pound harmonic  curve  by  a  difference  of  phase  in  the  component 
S.H.M.'s.  The  curve  (1)  is  that  corresponding  to  the  composi- 
tion of  two  S.H.M.'s  whose  periods  are  as  3  :  8,  whose  amplitudes 
are  as  there  shown,  and  whose  phases  at  the  point  A  coincide. 
In  the  next,  the  more  rapid  S.H.M.  is  seen  to  be,  in  respect  of 
its  phase,  in  arrear  by  an  interval  of  time  represented  by  the 
length  of  the  line  AB,  and  the  superposition  of  the  two  curves 
now  produces  a  resultant  slightly  differing  from  its  predecessor, 
but,  on  the  whole,  similar  to  it.  A  similar  construction  pro- 
duces the  succeeding  curves,  in  which  the  differences  of  phase 
correspond  to  the  respective  intervals  AC,  AD,  etc.  It  will  be 
plain  that  if  the  differences  of  phase  be  intermediate  between 


COMPOSITION   OF  SIMPLE   HARMONIC   MOTIONS. 


99 


those  chosen  in  the  figure,  there  may  be  drawn  any  number  of 
resultant  curves  intermediate  in   form   between   those  shown. 


u 


— ."N  '-- 

V        /    \  \   / 


The  time  taken  by  the  moving  body  to  go  once  through    its 
periodic  movement,  or,  in  other  words,  the  period  of  the  result- 


100  KINEMATICS.  [CHAP. 

ant  complex  harmonic  motion,  is  unaltered  by  variations  in  the 
amplitude  and  phase  of  the  component  S.H.M.'s,  and  depends 
only  on  their  relative  periods. 

If  in  Fig.  45  the  difference  of  phase  were  not  constant  but 
continuously  increased,  the  curve  would  successively  assume  all 
the  forms  there  shown,  and  .it  would  naturally  pass  through  all 
the  possible  intermediate  forms,  returning  at  intervals  to  the 
form  (I). 

Beats.  —  If  two  harmonic  curves  be  compounded,  of  which 
one  corresponds  to  a  more  rapid  vibration  than  the  other,  the 
periods  being  approximately  equal,  the  resultant  curve  will  be 
one  which  at  any  one  spot  approximates  in  form  to  the  curve  of 
sines,  but  alternately  waxes  and  wanes  in  amplitude.  If  the 
respective  periods  be  as  2000 :  2001,  the  quicker  oscillation  gains 
2~oVo  Peri°d  on  the  slower  at  each  complete  S.H.  movement,  and 
at  the  end  of  1000  of  the  slower  S.H.M.'s  the  quicker  is  in  com- 
plete disaccord  with  the  slower ;  then,  if  the  amplitudes  of  the 
two  oscillations  be  equal,  the  particle  affected  is  at  rest ;  there- 
after the  quicker  oscillation  comes  more  and  more  completely 
into  renewed  accord  with  the  slower,  and  at  the  end  of  2000  of 
the  slower  oscillations  or  2001  of  the  more  rapid,  the  amplitude 
of  the  compound  vibration  is  equal  to  the  sum  of  those  of  the 
components.  This  is  the  cause  of  beats  in  music.  If  the  periods 
approximate  to  any  other  whole-number  ratio  than  that  of 
equality,  similar  phenomena  occur ;  at  any  one  instant  the  curve 
resembles  the  corresponding  compound  harmonic  curve,  but 
alternately  waxes  and  wanes ;  the  deficiency  of  amplitude 
occurring  once  for  each  complete  oscillation  gained  by  the  more 
rapidly  vibrating  body.  Thus  oscillations  whose  frequencies  are 
500  and  751  per  second  give  one  beat  or  period  of  relatively 
small  amplitude  during  each  751  of  the  more  rapid  vibrations  — 
that  is,  for  every  occasion  on  which  it  gains  one  oscillation  on 
that  number,  750,  which  would  make  the  ratio  of  frequencies 
exactly  the  ratio  2:3.  If  the  oscillations  had  been  500  and  750| 
per  second,  there  would  have  been  a  beat  every  two  seconds. 

If  the  periods  of  the  S.H.M.'s  be  non-commensurable,  the 
resultant  curve  approximates  in  form  to  that  of  the  nearest 
commensurable  ratio,  and  successively  assumes  forms  nearly 
resembling  those  assumed  by  that  curve  when  the  difference 
between  the  phases  of  its  components  gradually  changes. 

As  a  particular  case  of  the  composition  of  harmonic  motions  we  may 
take  the  following  problem,  which  is  of  importance  in  the  Theory  of  Light. 


COMPOSITION   OF   SIMPLE   HARMONIC   MOTIONS. 


101 


A  particle  is  acted  upon  by  two  simultaneous  circular  vibrations.  These, 
considered  singly,  act  in  opposite  directions,  as  in  Fig.  46.  Let  them  be 
identical  in  period  and  in  amplitude.  Let  them  be  resolved  into  compo- 
nents at  right  angles  to  one 
another,  which  lie  respec- 
tively in  the  lines  AB  and 
CD.  Let  the  S.H.M.'s  in 
AB  coincide  in  phase,  while 
those  in  CD  differ  in  phase 
by  half  a  period.  In  such 
a  case  the  one  circular  vibra-  c  | 
tion  resembles  the  other,  as 
an  object  resembles  its  im- 
age in  a  mirror  situated  in 
a  plane  parallel  to  AB ; 
and  the  S.H.M.'s  in  CD 
will  neutralise  one  another, 
while  those  in  AB  will  reinforce  one  another ;  so  that  the  result  will  be  a 
S.H.M.  in  the  plane  AB,  and  of  double  amplitude. 

If,  however,  the  components  in  CD  do  not  differ  in  phase  by  half  a 
period,  the  same  considerations  of  symmetry  do  not  apply  in  reference  to 
the  plane  of  AB  ;  and  the  resultant  motion  cannot  be  a  S.H.M.  in  that 
plane.  If  the  circular  motions  be  identical  in  period  and  amplitude,  there 
must  be  some  plane  in  re- 
spect to  which  the  circular  Fig.47.  / 
motions  are  symmetrical,  as 
in  Fig.  47 ;  the  resultant 
motion  being  a  S.H.M.  of 
identical  period  parallel  to 
that  plane  and  of  doable 
amplitude. 

If  the  two  circular  mo- 
tions differ  in  period,  they 
will  continuously  differ  in 
relative  phase,  and  the  resultant  will  be  a  S.H.M.  of  double  amplitude  effected 
in  a  plane  which  is  constantly  changing  —  a  plane  at  right  angles  to  that  of 
the  paper,  and  rotating  round  O,  the  point  of  rest  in  Figs.  46  and  47. 

Hence  a  S.H.M.  can  always  be  considered  as  compounded  of  two  circular 
oscillations ;  and  if  one  of  these  be  retarded  or  accelerated,  whether  sud- 
denly or  continuously,  the  plane  of  the  S.H.M.  will  be  rotated,  suddenly 
or  continuously. 

Composition  of  several  S.H.M.'s  in  the  same  Line. — 

This  may  be  geometrically  effected  by  the  same  method  as  that 
employed  in  the  construction  of  the  curves  of  Fig.  45,  viz.,  by 
drawing  separately  the  Harmonic  Curves  corresponding  to  each 
S.H.M.,  and  adding  from  point  to  point  all  the  respective  dis- 
placements indicated  by  each  of  these  curves.  Fig.,  48  shows 
the  harmonic  curves  corresponding  to  five  S.H.M.'s,  each  of 
which  is  drawn  so  as  to  represent  its  proper  phase,  period,  and 


102 


KINEMATICS. 


[CHAP. 


amplitude   relatively   to   the   others.     The   resultant  curve  is 

periodic  —  that  is,  the  complex 
form  is  repeated  at  regular  inter- 
vals —  if  the  periods  of  the  com- 
ponent S.H.M.'s  be  commensur- 
able ;  it  cannot  be  if  they  are 
not  so. 

In  all  cases  of  a  body  affected 
by  several  simultaneous  S.H.M.'s, 
in  which  the  component  S.H.M.'s 
have  been  in  the  same  line,  the 
real  resultant  motion  of  the  par- 
ticle may  be  studied  by  finding 
the  complex  harmonic  curve  pro- 
duced by  compounding  these 
S.H.M.'s  with  a  uniform  move- 
ment, and  on  this  curve  laying 
a  card  in  which  a  slit  is  cut 
which  is  laid  at  right  angles  to 
the  axis  of  the  curve.  The  card 
is  then  moved  uniformly  along 
this  axis,  the  slit  being  kept  at 
right  angles  to  it.  At  any  one 
moment  only  one  point  of  the 
resultant  curve  can  be  seen  in 
the  slit,  if  that  slit  be  made  nar- 
row enough.  As  the  card  is 
moved  along,  this  point  appears 
to  move  up  and  down  in  the  slit 
with  greater  or  less  regularity, 
and  the  way  in  which  it  so  moves 
is  the  way  in  which  the  body 
really  moves  when  affected  with 
the  given  simultaneous  S.H.M.'s 
in  the  same  line. 

Fourier's  Theorem.  —  The 
great  variety  in  the  forms  of  the 
resultant  curves  drawn  to  illus- 
trate the  previous  discussion  will 
prepare  the  reader  to  accept  the 
positive  statement  that  by  properly  choosing  a  number  of  har- 
monic curves,  their  amplitudes,  their  periods,  and  their  phases, 


v.]  FOURIER'S  THEOREM.  103 

and  by  compounding  these,  any  Periodic  Curve  of  any  com- 
plexity may  be  built  up,  provided  that  the  curve  required  never 
goes  off  to  an  infinite  distance  from  the  axis.  Conversely,  any 
complex  periodic  motion  must  be  compounded  of  and  may  be 
resolved  into  a  definite  number  of  S.H.M.'s  of  definite  periods, 
definite  amplitudes,  and  definite  phases.  In  order,  however,  that 
such  motion  may  be  periodic  —  that  is,  that  the  complex  resul- 
tant motion  may  accurately  repeat  itself  at  regular  intervals  — 
it  is  necessary  that  the  periods  of  the  component  S.H.M.'s  should 
be  exactly  commensurate ;  for  if  they  were  not  so,  the  resultant 
motion  could  not  exactly  repeat  itself,  and  would  not  be  peri- 
odic. Granted,  however,  the  periodicity  of  the  complex  motion 
and  the  limitation  mentioned  above  as  to  the  form  of  its  curve, 
Fourier's  Theorem  states  that  any  such  motion  is  compounded 
of  a  definite  number  of  commensurate  S.H.M.'s ;  and  this  is  true 
not  only  of  motion  represented  by  the  curve,  but  also,  with 
wider  interpretation,  of  any  phenomenon  which  the  curve 
may  represent. 

Tide  Calculating  Machine.  —  A  number  of  wheels  may  have  pre- 
arranged velocities  imparted  to  them  by  being  separately  connected  with 
cranks,  of  which  a  number  are  actuated  by  the  same  clockwork,  through  the 
intervention  of  toothed  wheels.  Thus  the  mechanism  of  Fig.  33  may  be 
multiplied  and  any  number  of  S.H.M.'s  simultaneously  produced.  If  the 
extremities  of  the  oscillating  rods  be  connected  by  a  tense  and  flexible  cord, 
that  cord  will  be  drawn  upon  or  tightened  to  an  extent  depending  at  each 
instant  upon  the  position  of  the  wheels.  One  extremity  of  this  cord  being 
fixed  to  an  immovable  point,  the  other  may  be  connected  with  a  spring,  and 
the  varying  distortions  of  that  spring  will  indicate  the  varying  tensions  of 
the  cord.  To  some  point  of  the  spring  a  pencil  or  pen  may  be  attached, 
and  under  this  writing-point  a  piece  of  paper  may  be  unrolled  at  a  rate  pro- 
portioned to  the  velocity  of  'the  clockwork.  When  the  mechanism  is  set  in 
motion  there  is  recorded  upon  the  unrolling  paper  a  curve  which  represents 
the  summation  of  all  the  S.H.M.'s  which  are  being  executed.  The  pre- 
liminary adjustment  consists  in  adjusting  for  each  wheel  the  position  of  the 
pin  which  works  in  the  slot  —  this  regulating  the  amplitude,  and  also  in 
modifying  the  angular  position  of  each  crank  so  as  to  represent  the  appro- 
priate epoch  of  each  S.H.M.  at  the  moment  of  starting.  A  machine  of  this 
kind,  after  preliminary  adjustment,  enables  a  curve  to  be  drawn  which  repre- 
sents the  height  of  the  tide  for  every  moment  of  a  considerable  period, 
such  as  a  year,  without  the  aid  of  further  calculation  than  that  involved  in 
deducing  from  astronomical  considerations,  and  from  the  tidal  record  of  a 
place,  a  knowledge  of  the  component  S.H.M.'s,  their  respective  periods, 
amplitudes,  and  epochs. 

Oscillatory  Movement  of  Systems  of  Particles.  —  In 
Fig.  49,  A,  B,  C,  D,  etc.,  represent  a  number  of  particles  in  a 
linear  series  at  equal  distances.  In  the  lower  part  of  the  figure 


104  KINEMATICS.  [CHAP. 

the  same  particles  are  seen,  each  describing  a  circular  path  in 
the  plane  of  the  paper. 

The  point  B  is  represented  as  being  -J-  period  behind  A,  C 
an  equal  amount  behind  B,  and  so  on.  If  a  line  be  drawn 
through  the  positions  of  A,  B,  C,  D,  etc.,  at  a  given  instant,  it 
will  be  seen  that  the  system  of  particles  has  assumed  for  that 
instant  the  form  of  a  line  more  or  less  resembling  the  curve  of 
sines.  If  such  an  interval  of  time  be  now  supposed  to  elapse 
that  each  particle  has  moved  forward  through,  say,  \  of  the  cir- 
cumference of  its  circular  path,  a  line  similarly  drawn  through 
the  then  position  of  the  particles  will  present  exactly  the  same 
form,  but  it  will  lie,  as  shown  by  the  dotted  line  in  Fig.  49,  at  a 
little  distance  (=  BD)  from  its  previous  position.  A  similar 
result  is  obtained  after  the  lapse  of  any  other  interval  of  time. 
So  long  as  the  particles  continue  to  move  in  their  respective  cir- 
cular paths,  so  long  will  there  apparently  be  a  Travelling  of 
a  Wave-Form  along  the  system  of  particles.  Particles  whose 

Fig.49. 

JVBCDEF6HI         i'St'S'J,?^*81!/ 

position  is  intermediate  to  those  shown  as  equidistant  will  be 
found  to  occupy  intermediate  positions  on  the  same  line,  which 
presents  no  abrupt  angles  but  is  a  continuous  curve  ;  and  so,  if 
we  roughly  represent  a  linear  system  of  particles  by  a  chain  or 
cord,  we  can  understand  how  it  is  possible  for  a  wave-form  to 
run  along  such  a  cord  while  its  component  material  particles 
never  separate  themselves  by  more  than  a  certain  distance  from 
the  mean  positions  round  which  they  oscillate. 

Wave-Length.  —  Such  a  wave-form  is  seen  to  consist  of  a 
successive  series  of  parts  which  resemble  one  another.  In  Fig. 
49  the  particle  B  and  the  particle  J  are  seen  to  be  at  the  same 
time  in  their  position  of  maximum  displacement  in  the  same 
direction,  and  the  form  assumed  by  the  system  between  B  and  J 
is  repeated  be}^ond  J  and  behind  B.  This  distance  between  B 
and  J  is  called  the  Wave -length,  the  distance  between  a  point 
on  one  wave  and  a  similarly  situated  point  on  the  next  wave. 
The  point  of  maximum  displacement  in  one  direction  may  be 
called,  from  the  analogy  of  waves  on  the  surface  of  the  sea,  the 
Crest  of  the  wave ;  the  point  of  maximum  displacement  in  the 
opposite  direction  may  in  the  same  way  be  called  the  Trough, 


v.]  WAVES.  105 

and  the  wave-length  may  be  defined  as  the  distance  between 
crest  and  crest,  or  that  between  trough  and  trough.  Each  wave 
is  in  this  case  like  its  successor  and  its  predecessor,  and  an 
observer  stationed  at  a  fixed  point  near  the  cord  would  perceive 
a  succession  of  similar  waves  passing  him.  When  the  wave- 
length is  great,  a  smaller  number  of  waves  will  pass  him  in  a 
given  time  than  when  the  wave-length  is  small ;  twice  the  wave- 
length, half  as  many  waves  pass ;  half  the  wave-length,  twice  as 
many  waves ;  thus  the  number  of  waves  arriving  at  a  given 
point  in  a  given  time  is  inversely  proportional  to  the  wave- 
length. 

Velocity  of  Propagation.  —  If  the  particles  perform  their 
individual  revolutions  in,  say,  half  the  time  supposed  to  be 
taken  by  those  represented  in  Fig.  49,  and  if  the  amount  by 
which  B  lags  behind  A  be  still  the  same  fraction  (-|)  of  the  time 
taken  by  each  particle  to  perform  a  complete  revolution,  short- 
ened though  that  be,  the  form  assumed  by  .the  cord  will  be  the 
same  as  in  Fig.  49 ;  the  wave-length  will  be  the  same,  but  the 
wave  will  travel  twice  as  fast.  The  velocity  of  propagation 
of  the  wave  would  vary  directly  as  the  velocity  of  oscillation  of 
each  particle. 

If,  however,  the  retardation  of  B  behind  A,  of  C  behind  B, 
and  so  forth,  be  independent  of  the  rapidity  of  motion  of  these 
particles, — if,  that  is  to  say,  the  retardation  be  for  a  period  of 
time  which  depends  only  on  the  distance  between  the  particles, 
then,  whatever  the  rate  of  oscillation  of  the  particles,  the  wave 
will  travel  at  the  same  rate,  but  the  wave-length  may  vary,  and 
the  form  of  the  wave  may  vary  with  it.  This  is  illustrated  by 
Fig.  50,  in  which  the  particles  are  represented  as  moving,  for 


example's  sake,  twice  as  fast  as  those  shown  in  Fig.  49 ;  while 
the  interval  of  time  by  which  B  is  delayed  in  its  path  as  com- 
pared with  A  is  exactly  the  same  in  the  two  figures,  and  there- 
fore in  Fig.  50  twice -as  large  a  fraction  of  the  circumference 
expresses  the  difference  of  phase  between  A  and  B.  In  Fig.  49 
the  difference  of  phase  between  A  and  B  is  assumed  to'be  repre- 
sented by  a  difference  of  45°  in  their  positions  on  their  respec- 
tive circles ;  in  Fig.  50,  the  movement  being  twice  as  rapid,  and 


106  KINEMATICS.  [CHAP. 

B  being  retarded  by  the  same  interval  of  time  as  in  Fig.  49,  A 
must  assume  a  position  twice  as  far  in  advance  of  B,  i.e.  90°. 

The  results  shown  in  this  case  by  drawing  the  wave  in 
angular  outline,  are  —  (1)  that  there  are  twice  as  many  waves, 
the  wave-length  being  half  as  great  as  in  the  previous  figure ; 
and  (2)  that  the  speed  of  travelling  of  the  wave-form  is  the 
same  in  both  cases ;  for  if  a  sufficient  time  be  supposed  to  have 
elapsed  to  permit  each  particle  to  have  performed  half  a  revolu- 
tion (this  corresponding  to  the  time  allowed  in  Fig.  49  for  the 
accomplishment  of  a  quarter  revolution),  and  if  the  then  posi- 
tion of  the  chain  of  particles  be  investigated,  the  wave-form  will 
be  found  to  have  travelled  forward  through  a  space  which  is  the 
same  as  in  Fig.  49.  If,  then,  the  relative  retardation  of  the  par- 
ticles be  independent  of  the  speed  of  the  particles,  the  rate  of 
propagation  of  the  wave-form  will  be  constant,  and  the  only 
effect  of  a  change  in  the  rate  of  the  oscillation  of  the  particles 
will  be  a  change  in  the  wave-length  and  in  the  corresponding 
curved  form  assumed  by  the  chain,  and  in  the  number  of  waves 
which  pass  any  given  point  during  a  given  interval  of  time. 

The  circular  form  is  not  a  necessary  attribute  of  the  path  of 
each  particle  :  the  path  may  be  elliptical  with  a  similar  result, 
the  difference  being  one  in  the  form  of  the  resultant  wave. 

The  two  limiting  cases  are  of  great  interest.  These  are 
(1)  the  case  in  which  the  ellipse  is  reduced  to  a  straight  line  at 
right  angles  to  the  chain  of  particles ;  and  (2)  that  in  which  it 
is  reduced  to  a  straight  line  in  the  same  direction  as  that  chain. 
The  former  gives  rise  to  Transversal  vibrations  ;  the  latter  to 
Longitudinal  vibrations. 

Transversal  Vibrations.  —  Each  particle  effects  a  S.H.M. 
in  a  direction  at  right  angles  to  the  chain  of  particles.  In  Fig. 
51  are  shown  (a)  the  series  of  particles  unaffected  by  vibra- 
tion ;  (6)  the  same  particles  affected  by  transversal  vibrations, 

Flg.51. 


executing  S.H.M.'s,  the  phases  of  which  differ-to  an  equal  extent 
in  equidistant  particles.  The  form  assumed  by  the  system  of 
particles  in  this  case  is  exactly  the  curve  of  sines.  It  is  an 
easy  matter  to  show  in  this  case,  as  in  those  previously  dis- 
cussed, that  as  the  particles  perform  their  several  S.H.M.'s  the 
wave-form  travels  along  the  cord. 


v.]  -  TRANSVERSAL   VIBRATIONS.  107 

Composition  of  Transversal  Vibrations.  —  It  is  quite  pos- 
sible for  an  indefinite  cord  or  series  of  particles,  such  as  the  one 
considered  in  these  paragraphs,  to  have  several  wave-motions 
running  along  it  simultaneously,  each  producing  its  own  effect, 
and  the  total  effect  of  their  united  action  should  be  traceable  by 
some  process  of  composition  analogous  to  our  previous  composi- 
tion of  simultaneous  movements.  There  are  two  main  cases  to 
be  considered — (1)  that  in  which  the  vibrations  are  in  the 
same  plane,  and  (2)  that  in  which  they  are  not  in  the  same 
plane. 

Transversal  Vibrations  in  the  same  plane :  their  Com- 
position. —  Since  the  effect  of  transversal  vibrations  on  an 
indefinite  straight  cord  is  to  cause  it  to  assume  the  form  of  the 
curve  of  sines,  and  since  each  vibration  acts  independently  in 
this  sense,  the  effect  of  compounding  such  movements  is  reduced 
to  exactly  the  same  problem  as  has  been  considered  on  pages 
97-103,  and  there  illustrated.  If,  for  instance,  we  refer  to  Fig. 
48,  the  resultant  wave  on  a  perfectly  flexible  and  extensible  cord 
on  which  the  five  wave-systems  there  represented  were  simul- 
taneously travelling,  would,  for  the  instant  at  which  the  phases 
happened  to  be  as  there  shown,  assume  the  form  there  drawn; 
but  when  that  wave  had  travelled  a  little  way  along  the  cord, 
the  relative  phases  of  the  component  transversal  vibrations 
would  have  altered,  and  the  wave  would  thus  continuously  alter 
its  form  from  instant  to  instant,  returning,  however,  nearly  to 
the  form  shown  in  Fig.  48,  as  often  as  the  same  coincidence  of 
phases  recurred. 

Their  Resolution.  —  If  a  changing  wave-form  run  in  this 
way  along  a  cord,  and  if  the  same  form  recur  at  regular  intervals, 
the  wave-form  passing  at  every  recurrence  through  the  same 
changes,  then  Fourier's  theorem  applies,  and  the  most  complex 
phenomenon  of  this  kind  may  be  analysed  or  resolved  into  a 
number  of  separate  waves,  whose  periods  are  commensurable, 
running  simultaneously  along  the  cord. 

Transversal  Vibrations  not  in  the  same  plane :  their  Com- 
position. —  Here  we  have  to  consider  two  cases  —  (1)  that  in 
which  the  vibrations  are  at  right  angles  to  one  another,  and  (2) 
that  in  which  they  are  not  so.  The  latter  case  differs  from  the 
former  only  in  the  form  of  the  curve  described  by  each  particle. 

Let  the  simultaneous  vibrations  be  in  planes  at  right  angles 
to  one  another.  The  motion  of  each  particle  is  confined  to  a 
plane  at  right  angles  to  the  line  of  the  cord.  In  the  plane  in 


108  KINEMATICS.  [CHAP. 

which  it  moves,  each  particle  describes  paths  such  as  those 
exemplified  in  Figs.  35-40.  In  these  figures  OA  may  be  taken 
as  representing  the  amplitude  and  direction  of  the  S.H.M.  in 
one  plane,  and  OD  as  representing  the  amplitude  and  direction 
of  the  S.H.M.  in  the  plane  at  right  angles  to  it.  If  the  periods 
of  the  component  vibrations  be  not  the  same,  the  different  par- 
ticles of  the  cord  will  be  in  different  phases  of  S.H.M.,  and  the 
form  of  their  respective  paths  will  differ ;  and  as  the  compound 
wave  runs  along  the  cord,  the  path  of  each  particle  will  pass 
successively,  and  in  the  way  exemplified  in  Fig.  41,  through  all 
those  forms  which  are  possible  as  the  result  of  the  rectangular 
composition  of  S.H.M.'s  whose  periods  are  those  belonging  to 
the  wave-motions  which  are  to  be  compounded. 

If  the  periods  of  the  vibrations  to  be  compounded  be  the 
same,  the  points  at  which  the  respective  curves  of  sines  cross  the 
line  of  mean  positions  will  be  the  same  for  each  vibration. 
Hence,  if  one  particle  describe  an  ellipse  or  circle,  all  the 
particles  which  are  in  motion  describe  circles  or  ellipses  similar 
in  form,  though  differing  in  size  or  in  the  direction  of  motion ; 
while,  as  the  wave  runs  along,  any  given  particle  will  describe 
an  ellipse  or  circle  which  alternately  enlarges,  diminishes, 
vanishes,  and  reappears,  but  in  the  reversed  direction,  first 
enlarging  and  then  diminishing,  and  then  vanishes  to  reappear, 
recommencing  the  cycle  in  the  original  direction. 

This  kind  of  movement  may  be  roughly  realised  by  taking  a  rope,  fixing 
it  at  one  end,  and  rapidly  rotating  the  hand  which  holds  the  free  end.  The 
rope  assumes,  and  may  by  practice  and  dexterity  be  caused  to  retain,  a 
form  in  which  there  are  a  certain  number  of  fixed  points,  the  number  of 
which  may  increase  with  the  rapidity  of  movement  of  the  hand.  On  each 
side  of  these  steady  points  the  particles  of  the  rope  are  describing  circles  or 
ellipses  in  opposite  directions.  If  this  condition  —  instead  of  being  steady 
in  its  position  —  travelled  along  a  rope  of  indefinite  length,  it  will,  on 
consideration,  become  plain  how  a  particle  would  rotate  first  in  one  direc- 
tion and  then  in  the  other,  and  how,  while  rotating  in  each  direction,  the 
extent  of  motion  is  at  first  small,  increases  to  a  maximum,  and  then  wanes 
away  till  it  vanishes,  again  to  reappear. 

If  the  rectangular  vibrations  which  have  to  be  compounded 
be  more  than  two  in  number,  the  problem  of  finding  the  approx- 
imate path  at  any  instant  is  precisely  that  of  compounding  sev- 
eral S.H.M.'s.  Point  by  point  the  independent  displacements 
produced  by  each  S.H.M.,  whatever  the  plane  of  that  S.H.M., 
must  be  added  together  and  the  resultant  points  joined. 

Resolution  of  Transversal  Vibrations  in  general.  —  In  the 
paragraph  illustrated  by  Fig.  42,  it  has  been  seen  that  any  S.H.M. 


v.]  TRANSVERSAL  VIBRATIONS.  109 

may  be  resolved  into  any  two  others  in  directions  at  any  angles 
to  each  other.  The  only  case  of  other  than  theoretical  interest 
in  its  application  to  the  doctrine  of  transversal  vibrations  is  that 
in  which  the  S.H.M.  is  resolved  into  two  components  at  right 
angles  to  each  other,  these  components  then  being  of  the  same 
period  and  phase,  and  their  amplitudes  and  directions  being 
represented  by  the  sides  of  a  rectangle  of  which  the  diagonal 
may,  in  the  same  respects,  represent  the  original  S.H.M. 

If  a  simple  transversal  vibration  in  one  plane  were  prevented 
from  taking  place  in  a  direction  at  right  angles  to  that  plane, 
such  prevention  would  be  superfluous,  and  the  vibration  would 
not  be  affected.  If  it  were  prevented  from  taking  place  in  the 
plane  in  which  it  is  actually  occurring,  obviously  the  vibration 
would  cease.  If,  again,  it  were  hindered  by  some  cause  which 
prevented  any  movement  in  a  given  plane,  inclined  to  the  plane 
of  vibration  at  an  angle  intermediate  to  these  extremes,  then  a 
reference  to  Fig.  42  will  enable  us  to  see  that  if  a  vibration  in 
a  plane  passing  through  AB  be  prevented  from  effecting  any 
vibratory  movement  in  a  plane  passing  through  yy  y,  the  result 
is  as  if  the  vibration  in  the  plane  AB  had  been  broken  up  into 
two  components  of  the  same  phase  and  period  as  the  original 
one,  and  executed  in  the  planes  xxt  and  yyt  respectively,  and 
the  latter  of  these  then  extinguished  ;  there  is  thus  left  only  a 
vibration  in  the  plane  xxr  the  amplitude  of  which,  as  compared 
with  that  of  the  original  vibration  in  the  plane  passing  through 
AB,  depends  on  the  angle  between  AB  and  xxt,  being  propor- 
tional to  the  cosine  of  that  angle. 

Let  now  the  plane  yyt  rotate  round  the  centre  O  ;  when  its  position  is  at 
right  angles  to  AB,  xx,  sweeps  round  so  as  to  coincide  with  AB,  and  there  is 
neither  diminution  of  the  vibration  in  amplitude  nor  change  in  its  direction  ; 
when  yy,  coincides  with  AB,  xx,  is  reduced  to  nothing,  and  the  vibration  is 
completely  stopped  :  between  these  limits  there  is  an  indefinite  number  of 
positions  of  yyt,  and  an  indefinite  number  of  corresponding  values  of  ampli- 
tude of  the  resultant  vibration  in  xxt,  as  that  plane  sweeps  round  at  right 
angles  to  yyr 

What  is  thus  true  of  one  transversal  vibration  is  true  of 
each  of  all  those  which  may  be  running  simultaneously  along 
the  cord  ;  and  as  the  effect  of  inhibiting  vibration  parallel  to 
the  plane  of  yyt  (Fig.  42)  is  to  restrict  it  to  that  of  xxt,  what- 
ever the  original  direction  of  AB,  so,  if  a  number  of  wave- 
motions  affect  the  same  cord  simultaneously,  and  if  the  cord  be 
restricted  from  executing  any  vibrations  whatsoever  parallel  to 
a  certain  plane  yy  r  the  result  will  be  a  more  or  less  complex 


HO  KINEMATICS.  [CHAP. 

vibration  restricted  to  the  plane  xxt  at  right  angles  to  yy r  If 
we  can  suppose  such  a  cord,  along  which  a  number  of  waves  are 
running,  to  be  passed  through  a  slot  in  a  thick  wall,  so  that  all 
vibrations  in  a  direction  at  right  angles  to  this  slot  are  com- 
pletety  prevented,  and  if  then  the  same  cord  be  passed  through 
another  such  slot  placed  at  right  angles  to  the  first,  all  vibra- 
tion whatsoever  will  be  prevented  in  the  part  of  the  cord  which 
lies  beyond  the  second  slot,  and  that  part  of  the  cord  will  be 
at  rest.  If,  however,  the  second  slot  be  inclined  at  any  other 
than  a  right  angle  to  the  first,  a  certain  amount  of  vibration 
will  pass  through,  which  will  be  executed  in  the  plane  of  the 
second  slot,  with  an  amplitude  which  will  be  proportional  to 
the  cosine  of  the  angle  between  the  two  slots ;  in  this  way  the 
more  nearly  the  two  slots  coincide  in  direction,  the  greater 
the  amplitude  of  that  vibration  which  affects  the  cord  beyond 
the  double  obstruction. 

Longitudinal  Vibrations. — The  other  limiting  case  of  vibra- 
tions of  a  linear  series  of  particles,  spoken  of  on  page  106,  was 
that  in  which  each  particle  performed  a  S.H.M.  in  the  direction 
of  the  line  of  particles.  The  result  is  represented  in  Fig.  52,  in 


which  a  indicates  the  primitive  position  of  the  particles  when  at 
rest;  and  5,  <?,  c?,  their  positions  after  equal  intervals  of  time, 
when  oscillating  in  this  way.  Each  particle  has  its  amplitude 
of  vibration,  may  be  in  a  certain  phase  and  have  a  certain  period 
of  oscillation :  the  wave-motion  runs  along  the  vibrating  cord : 
several  wave-motions  of  this  kind  may  simultaneously  affect  it ; 
and  a  complex  longitudinal  wave-motion  may  be  analysed  into 
simple  wave-motions,  as  in  the  preceding  paragraphs  we  have 
seen  that  we  may  analyse  a  complex  transversal  wave  into  its 
components.  The  study  of  this  kind  of  vibration  is,  however, 
greatly  facilitated  by  giving  an  arbitrary  representation  to  the 
form  of  the  wave.  If  the  disturbance  represented  in  Fig.  52  b 
be  indicated  by  drawing  lines  at  right  angles  to  the  line  of  prim- 
itive position,  each  of  these  lines  having  a  vertical  height, 
positive  or  negative,  equal  to  the  horizontal  displacement 
of  the  corresponding  particle,  forwards  or  backwards  in  the  line 


v.]  LONGITUDINAL   VIBRATIONS.  HI 

of  the  cord;  by  joining  the  extremities  of  these  ordinates  we 
shall  produce,  as  has  been  done  in  Fig.  52  e,  a  curved  line  which 
is  the  curve  of  sines,  or  harmonic  curve,  simple  or  compound, 
with  which  we  are  already  familiar.  The  interpretation  of  such 
a  curved  line  would,  however,  be  different  in  the  case  of  a 
longitudinal  vibration  from  that  of  the  same  form  in  the  case  of 
transversal  vibrations.  In  the  latter  case,  the  curve  represents  the 
actual  form  assumed  by  the  cord:  in  the  former  case,  that  of  the 
longitudinal  vibrations,  it  only  indicates  from  point  to  point 
the  extent  of  the  departure  of  the  corresponding  particle  from 
its  primitive  position.  In  a  longitudinal  wave,  there  are,  at  the 
successive  points  of  zero  displacement,  where  particles  occupy 
their  original  positions,  alternate  maxima  of  crowding  together 
and  of  separation  of  the  particles,  or  alternate  maxima  and 
minima  of  density;  while  midway  between  these  are  points 
of  maximum  displacement,  at  which  the  density  remains 
unchanged. 

We  have  hitherto  supposed  the  vibration  to  be  permanent, 
and  to  be  kept  up  by  a  continuous  and  periodic  movement  of 
each  of  the  particles  of  a  linear  body.  Let  it  be  now  supposed 
that  there  is  some  relation  between  the  particles  of  this  linear 
body,  such  that,  when  one  particle  is  displaced,  it  executes  a 
S.H.M.,  and,  in  some  way  exerting  Force  upon  them,  induces  its 
neighbours  to  commence  executing  similar  S.H.M.'s,  following 
its  own  at  intervals  of  time,  and  therefore  with  differences  of 
phase,  corresponding  to  their  respective  distances  from  it.  The 
result  will  be  as  shown  in  Fig.  53.  There  the  cord  is  first  seen 
undisturbed ;  then  the  particle  A  being  disturbed  moves  in 
S.H.M.,  and  sets  the  following  particles  B,  C,  etc.,  in  motion. 
On  comparing  b  and  c  it  will  be  seen  that  the  Wave-Form 
travels  along  or  is  propagated,  the  "Wave -front"  travelling 
onwards  with  a  velocity  equal  to  that  of  the  permanent  wave 
described  in  the  preceding  paragraphs. 

If  the  disturbed  particle  A  do  not  oscillate  continuously, 
but  travel  once  merely  through  a  complete  S.H.M.,  the  figure 
53/  shows  that  a  single  wave  is  propagated,  leaving  at  rest  the 
part  of  the  cord  which  it  has  traversed,  and  continually  dis- 
placing fresh  particles  if  the  cord  be  of  indefinite  length.  If 
the  motion  of  each  particle  be  not  completely  extinguished  after 
the  execution  of  one  exact  S.H.M.,  but  dwindle  ^away  with 
diminishing  amplitude,  the  wave  is  not  single,  but  is  followed 
by  a  certain  number  of  shallower  waves,  which  presently  die 


112        •  KINEMATICS.  [CHAP. 

away.  If  the  wave-motion  be  kept  up  by  a  continuous  supply 
of  energy,  there  may  be  a  continuous  succession  of  waves,  equi- 
distant and  following  one  another  at  equal  intervals  of  time. 
The  distance  between  any  point  in  one  wave  and  a  precisely 
similar  point  in  its  predecessor  or  successor  is  the  wave-length  of 
the  wave ;  the  distance  traversed  by  a  given  wave  during  one 
second  is  the  velocity  of  propagation  of  the  wave ;  and  this 
velocity,  divided  by  the  length  of  each  wave,  plainly  gives  the 
number  of  waves  which  pass  a  given  spot  during  a  second  of 
time,  or,  in  other  words,  the  Frequency  of  the  undulation; 
while  the  reciprocal  of  this  number  gives  the  length  of  time 

Fig.58. 
ABCDEFGH  IJ 


a 


taken  by  a  single  complete  wave  in  passing  a  given  spot,  and 
therefore  denotes  the  period  of  one  complete  oscillation. 

If  X  be  the  wave-length,  v  the  velocity  of  propagation,  and  n  the  fre- 
quency or  number  of  waves  per  second,  T  the  period  of  each  wave, 


In  these  cases  of  propagation  of  wave-motion  along  a  linear 
body  the  wave-front  implicates  only  one  particle,  and  its  form 
is  accordingly  a  single  point.  The  amplitude  of  the  wave 
as  it  travels  along  will  be  constant,  if  no  energy  be  lost  by 
the  way. 

Waves  on  a  surface.  —  On  a  surface  there  mav  run,  from 
a  starting-point,  waves  of  compression  and  rarefaction  in  the 
plane  of  the  surface,  or  waves  of  vibration  transverse  to  that 
plane.  If  one  point  be  disturbed  the  disturbance  is  propagated 


WAVES  ON  A   SURFACE. 


113 


in  all  directions.  If  it  be  equally  so  in  all  directions,  the 
wave-front  will  be  circular;  if  the  material  be  such  that  the 
velocity  of  propagation  in  one  direction  differs  from  that  at  right 
angles  to  it,  the  result  will  be  an  elliptical  wave-front.  In 
Fig.  54  is  shown  a  wave  in  a  membrane  (an  ideal  solid  which 
has  length  and  breadth,  but  indefinitely  small  thickness),  whose 
structure  is  such  that  the  disturbance  is  propagated  equally  in 
all  directions ;  and  at  Z,  m,  n,  three  points  are  shown  which 
themselves  act  as  centres  of  disturbance,  and  the  wave-front  is 
propagated  to  /',  m\  n1,  the  whole  still  retaining  its  circular 
form.  In  the  same  way  each  point  on  an  elliptical  wave-front 
may  act  as  the  centre  of  an  elliptical  disturbance ;  the  propaga- 
tion of  an  elliptical  wave  is  thus  kept  up. 

In  Fig.  54  it  will  be  seen  that  the  lines  K',  mm',  nnr,  are 
parts  of  radii  of  the  circles;  and  these  lines  are  hence  at  right 
angles  to  the  cir-  Fig. 54. 

cles  at  Z,  m,  H, 
and  also  at  Z',  m', 
n'.  The  normal 
(i.e.  a  line  per- 
pendicular to  the 
tangent)  to  the 
wave  -  front  at 
any  point  at  any 
instant  is  also 
normal  to  the 
correspond  ing 
part  of  the  wave- 
front  at  any  suc- 
ceeding instant, 
if  the  medium 
be  isotropic,  i.e.  if  the  velocity  of  propagation  be  equal  in  all 
directions.  If,  knowing  the  form  of  the  wave-front  at  any 
instant,  we  desire  to  .learn  its  form  after  any  given  interval 
of  time,  this  may  be  found  by  drawing  normals  to  the  original 
wave-front  equal  to  each  other,  and  of  lengths  corresponding  to 
the  time  indicated,  and  by  joining  their  extremities.  In  this 
case  we  simply  obtain  a  circle  surrounding  a  circle,  but  we  shall 
soon  come  upon  cases  in  which  the  result  is  not  so  extremely 
simple. 

When  the  initial  disturbance  is  single,  the  wave  which  is 
produced  is  also  single.  The  energy  imparted  to  the  system  by 


114  KINEMATICS.  [CHAP. 

the  single  disturbance  remains,  a  fixed  quantity.  As  the  cir- 
cular wave  progressively  increases,  it  acts  upon  material  whose 
mass  increases  with  the  radius ;  the  energy  imparted  to  each 
particle  varies  inversely  as  the  radius  ;  and  the  amplitude  of 
movement  of  each  particle  in  its  S.H.M.  varies  inversely  as  the 
square  root  of  the  radius.  In  this  way,  the  farther  the  wave 
has  travelled  from  its  centre  of  disturbance,  the  shallower  it 
becomes. 

At  two  instants  of  time,  the  respective  radii  of  the  circular  wave  are 
r  and  r/,  and  the  maximum  velocities  of  the  particles  affected  v  and  vr  Let 
m  be  the  amount  of  mass  affected  per  unit  of  length  of  the  wave-front. 
The  whole  mass  set  in  motion  is,  at  the  two  successive  instants,  2irr  •  m  and 
27rr,  •  m.  But  the  energy  is  constant,  and  2-nrm  •  0*/2==2irr/m  •  vf/2.  Whence 
r/r/^r/2/^2,  and  vtfv—  Vr/r,. 

But  the  amplitude  of  a  S.H.M.  is  proportional  to  the  velocity  with 
which  the  particle  passes  its  mean  position  ;  for  the  velocity  in  the  circle  of 
reference,  which  is  equal  to  the  velocity  at  the  middle  point  of  the  S.H.M., 
is  (2?rx  radius  of  circle  -=-  P e r i o d)  =  (27T/T  x  amplitude)  ;  whence  the 
amplitude  =  y-  T/2?r,  and  therefore  varies  as  that  velocity.  The  amplitudes 
therefore  vary  inversely  as  the  square  roots  of  the  radius* of  the  wave. 

Waves  propagated  in  a  tridimensional  substance.  —  The 

solid  figure  whose  surface  is  everywhere  at  equal  distances  from 
its  centre  is  a  globe  or  sphere.  If  a  disturbance  at  a  point  be 
propagated  with  equal  velocities  in  all  directions  in  space,  the 
form  of  the  wave-front  will  be  spherical.  If  the  velocities  be 
unequal,  the  wave-front  will  be  ellipsoidal  or  spheroidal.  As 
the  distance  from  the  centre  increases,  the  amplitudes  of  oscilla- 
tion of  the  particles  will  diminish,  for  they  vary  inversely  as 
the  distance  from  the  centre  of  disturbance. 

When  the  radius  of  the  spherical  wave  changes  from  r  to  r,,  the  mass 
set  in  motion  becomes  greater  in  the  proportion  of  r2  to  r/2;  but  the  energy, 
proportional  to  27rrV2,  or  to  27rr/2y/2,  remains  constant.  Therefore  r2y2  = 
rfvf,  and  vt/v  =  r/rn  or  the  velocity  of  the  particles  varies  inversely  as  the 
radius  of  the  wave-front.  But  the  amplitude  varies  as  the  velocity,  and 
therefore  varies  inversely  as  the  radius  or  the  distance  of  the  wave-front 
from  the  centre  of  disturbance  ;  and  the  energy  of  motion  of  each  particle  — 
that  is,  the  Intensity  of  its  vibration  —  varies  as  the  square  of  the  ampli- 
tude, and  therefore  inversely  as  the  square  of  the  radius.  This  corresponds 
to  the  statement  that  the  intensity  of  Light  varies  inversely  as  the  square  of 
the  distance  from  the  illuminating  point. 

Concentric  Waves.  —  In  all  these  cases,  if  the  primitive 
disturbance  be  repeated  at  regular  intervals,  the  wave  system 
will  assume  the  form  of  equidistant  and  concentric  waves  of 
circular,  elliptical,  spherical,  ellipsoidal,  spheroidal  form,  as  the 
case  may  be.  If  the  primitive  disturbance  be  repeated  at  irreg- 


CONCENTRIC    WAVES. 


115 


ular  intervals,  the  waves  will  still  be  concentric  but  not  equi- 
distant, and  they  will  arrive  at  any  point  in  an  order  of  irregular 
sequence  exactly  reproducing  the  irregularity  of  the  central 
disturbance. 

If  the  central  particle  be  affected  by  complex  periodic  dis- 
turbances, these  will  be,  as  regards  the  period  and  the  phase  as 
well  as  the  relative,  but  not  the  absolute,  amplitude  of  every 
component  motion,  faithfully  reproduced  in  the  motion  of  any 
particle  affected  by  the  resultant  complex  wave-motion ;  and  this 
motion  of  such  a  particle  may  in  many  experimental  instances 
be  taken  cognisance  of  by  an  observer. 

Direction  of  the  wave-front.  —  When  a  wave  is  said  to  be 
at  a  certain  time  and  place  travelling  in  a  certain  Direction,  in 
an  isotropic  medium,  it  is  meant  that  the  normal  to  the  wave- 
front,  a  straight  line  drawn  at  right  angles  to  the  wave-front — 
i.e.  at  right  angles  to  its  tangent  or  tangent-plane — takes  the 
direction  said  to  be  that  of  the  wave  itself.  In  Fig.  54  the  lines 
qr  and  st  indicate  the  directions  of  the  circular  wave  at  the 
points  q  and  s. 

Flat  wave-front. — The  nearer  the  centre  of  disturbance, 
the  more  marked  the  convexity  of  the  wave-front ;  the  farther 


the  centre,  the  flatter  the  wave-front:  when  the  centre  of  dis- 
turbance is  very  far,  the  wave-front  may  for  any  small  area  be 
regarded  as  approximately  plane,  just  as  any  small  portion  of 
the  surface  of  a  very  large  sphere  may  be. 

If,  again,  all  the  points  of  a  plane  surface  act  as  centres  of 
disturbance,  then  in  the  immediate  proximity  there  may  be  a  flat- 
fronted  wave.  If  the  disturbed  surface  be  represented  in  section 
by  AB  in  Fig.  55,  the  wave-front  will  be  flat  opposite^its  centre. 

Modifications  in  the  Form  of  a  flat  Wave-Front.  —  Through  the 
upper  and  cooler  layers  of  the  atmosphere,  waves  travel  with  less  rapidity 


116 


KINEMATICS. 


[CHAP. 


than  they  do  through  the  lower.  A  flat  wave-front  is  thus  distorted ;  its 
upper  part  travels  with  the  least  velocity,  and  the  wave-front  conies  to  con- 
verge upwards.  Points  on  a  level  with  the  point  of  disturbance  may  remain 
unaffected,  for  the  wave-front  is,  in  the  main,  restricted  to  its  normals,  and 
the  sound  ascends.  If,  however,  there  be  a  movement  (such  as  that  due  to 
wind)  in  which  the  upper  strata  move  more  rapidly  than  the  lower,  it  is  not 
difficult  to  see  that  the  wave-front  may  come  to  bear  down,  and  that  sound- 
waves may  thus  appear  to  travel  with  the  wind,  and  to  be  best  heard  at  cer- 
tain distances,  which  depend  upon  the  speed  of  the  wind. 

Wave  passing  through  an  aperture.  —  If  a  flat  wave 
impinge  upon  an  obstacle  containing  an  orifice,  it  will,  in  part, 
be  propagated  through  that  orifice.  If  the  orifice  lead  into  the 


Fig.56. 


lumen  of  a  cylin- 
drical tube  whose 
diameter  is  the 
same  as  that  of 
the  orifice,  no  lat- 
eral expansion  of 
the  wave-front  is 
possible  in  that 
tube.  If  there  be 
no  such  tube,there 
may  or  there  may 
not  be  expansion 
of  the  wave  be- 
yond the  orifice. 
Such  expansion, 
if  it  take  place  at 
all,  will  take  place  in  the  way  shown  in  Fig.  56.  The  disturbed 
particles  in  the  aperture  act  as  centres  of  disturbance  to  those 
lying  beyond. 

There  is  a  curious  proposition,  the  nature  of  the  proof  of 
which  will  be  indicated  farther  on,  that  if  the  aperture  through 
wThich  a  wave  passes  be  small  in  comparison  with  the  wave-length^ 
there  will  be  expansion  of  the  wave-front,  such  as  that  shown 
in  Fig.  56;  but  that  if  the  aperture  be  wide  in  comparison  with 
the  wave-length,  the  wave  will  only  travel  in  the  direction  of  all 
the  lines  drawn  normal  to  that  part  of  it  which  passes  through 
the  aperture,  the  wave  therefore  travelling  with  a  correspond- 
ingly limited  amount  of  expansion  or  none  at  all;  wThile  for 
conditions  intermediate,  there  will  be  a  certain  amount  of  expan- 
sion beyond  the  limitation  indicated  by  the  normals. 

When  a  wave-motion  passes  through  an  aperture  relatively  wide,  then 
the  cases  are  three :  — 


WAVES  THROUGH  AN  APERTURE. 


117 


\  \ 


(a.)  The  wave-front  may  be  flat,  as  in  Fig.  57  a,  in  which  case  it  does 
not  expand:  this  is  the  condition  of  a  "parallel  beam  "of  light. 

(6.)  It  may  be  convex, 

as   in   Fig.  576,  in  which  Fig.57. 

case  the  wave -front  ex- 
pands, being  limited  by 
the  normals  pq,  rs.  This 
is  the  condition  of  a  "di- 
vergent beam  "of  light. 

(c.)  It  may  in  some 
cases  be  concave,  in  which 
case  the  wave-front  first 
contracts  and  then  expands: 
this  is  the  condition  of  a 
"convergent  beam"  or 
"convergent  pencil  of 
rays"  passing  through  a 
"Focus." 

When  a  wave  -  front 
passes  through  a  focus,  ex- 
actly or  approximately,  the 
energy,  constant  in  amount, 
is  distributed  over  a  com- 
paratively small  field,  and 
the  intensity  of  disturbance 
is,  at  the  focus,  correspond- 
ingly great. 

Reflexion  of  Lin- 
ear Waves.  —  If  a  lin- 
ear longitudinal  wave 
of  compression  be  inci- 
dent, or  impinge,  on  an 
obstacle  so  firm  that 
the  first  particle  of  it  at  which  the  wave  arrives  does  not  move, 
there  is  then  produced  between  this  first  particle  of  the  obstacle 
and  the  nearest  particle  of  the  vibrating  cord  (which  we  shall 
represent  as  particle  i)  a  Compression,  which  results  in  particle 
i  rebounding  at  a  rate  equal  to  that  with  which  it  struck  the 
obstacle — that  is,  with  velocity  vt — but  in  the  opposite  direc- 
tion, and  in  its  then  meeting  the  next  particle,  ii,  as  it  comes  up 
with  velocity  vn. 

We  may  here  borrow  a  proposition  from  the  theory  of 
Elasticity,  which  shows  that  if  two  equal  elastic  bodies  meet  one 
another  and  rebound,  they  will  do  so  with  exchanged  velocities. 

Particle  i,  meeting  particle  ii,  exchanges  velocities  with  it ; 
i  acquires  velocity  vu  and  return?,  towards  the  obstacle ;  ii 
acquires  vr  Particle  i  strikes  the  obstacle  with  velocity  vlt  and 


118 


KINEMATICS. 


[CHAP. 


rebounds;  in  the  meantime  particle  ii  has  acquired  velocity  vtn 
by  exchange  with  particle  Hi.  When  particles  i  and  ii  again 
meet,  i  is  impelled  towards  the  obstacle  with  velocity  v//y,  and 
the  backward  velocity  vlt  is  imparted  to  particle  ii.  So  on :  the 
particle  i  successively  strikes  and  rebounds  from  the  obstacle 
with  each  successive  velocity,  vn  vu,  vm,  vltn,  etc. ;  at  the  same 
time  the  backward  speed  vt  is  transferred  successively  to  all  the 
particles  ii.  Hi,  iv,  etc.,  and  is  followed  by  the  successive  veloci- 
ties vn,  vni,  vlllti  etc.  The  consequence  is,  that  just  as  the  end 
of  the  wave  is  being  dashed  against  the  obstacle,  a  wave-front 

exactly  like  the  origi- 
nal one  is  travelling 
away  from  the  obstacle, 
at  the  distance  of  one 
wave-length.  The  "  re- 
flected wave"  has  trav- 
elled through  the  incident 
wave,  and  then,  becoming 
clear  of  it,  travels  alone, 
equal  to  the  incident  wave 
in  wave-length,  in  period, 
in  phase,  and  in  amplitude, 
but  opposed  in  direction. 


Fig.58 


direction  at  the  surface  of  impact. 


In  Fig.  58  a  single  wave  is 
shown,  running  along  a  cord 
against  a  fixed  obstacle  AB. 
Within  the  obstacle  thin  lines 
show  the  course  which  the 
wave  would  have  taken,  had 
the  cord  not  been  interrupted. 
To  the  left  of  AB,  light  dotted 
lines  indicate  the  course  of 
the  reflected  wave.  The  light 
dotted  lines  are  seen  to  be 
of  exactly  the  same  form  as 
the  thin  lines  within  AB,  but 
turned  sharply  in  the  reverse 
The  reflected  wave  is  then  a  direct 


continuation  of  the  incident  wave  in  all  but  direction. 

If  of  incident  waves  there  be  a  succession  i  simple,  complex, 
regular,  irregular,  all  these  peculiarities  will  be  faithfully  repro- 
duced in  the  reflected  waves. 

What  has  been  said  of  a  wave  longitudinal  and  commenc- 
ing with  a  compression  may  be  easily  modified  so  as  to  become 


REFLEXION  OF   WAVES. 


119 


generally  applicable  to  the  explanation  of  any  kind  of  linear 
undulatory  disturbance.  If  the  obstacle  stand  fast,  it  is  a  matter 
of  indifference  whether  it  be  composed  of  matter  whose  particles 
lie  more  or  less  closely  together. 

Special  consideration  of  the  reflexion  of  waves  traversing 
space  of  two  dimensions  may  be  omitted. 

Reflexion  of  a  plane  wave-front  at  a  plane  surface. — If  a 
plane  wave-front  meet  a  plane  surface,  any  section  through  the 
wave  and  surface  will  present  a  condition  such  as  that  shown  in 
Fig.  59.  The  line  AB  represents  the  wave-front  advancing; 
CD  represents  the  surface  on  which  it  impinges.  Every  point 
in  the  wave-front  acts  as  a  centre  of  disturbance.  Thus  the 
wave-front  advances,  parallel  to  its  former  plane  forms.  After 
the  lapse  of  a  certain  time  the  whole  of  the  wave-front  has 


impinged  on  the  obstacle:  what  is  then  its  condition?  The 
part  of  the  wave  corresponding  to  the  particle  A  would  have 
travelled  as  far  as  A',  if  there  had  been  no  obstacle.  After 
reflexion  it  has  travelled  to  a  corresponding  extent,  in  some 
direction  tending  away  from  the  surface — that  is,  to  some  point 
on  the  circumference  of  a  circle,  the  centre  of  which  is  at  A, 
and  the  radius  of  which  is  A  A'.  So  the  part  of  the  wave 
indicated  by  b  would  have  reached  b' ;  the  line  W  crosses 
the  surface  at  E ;  that  'part  of  the  wave  is  reflected  to  a  dis- 
tance limited  by  a  circle  whose  centre  is  the  point  E  on  the 
surface,  and  whose  radius  is  the  distance,  E6',  between  the  sur- 
face and  the  position  at  which  the  wave-front  would  have 
arrived  if  there  had  been  no  obstacle.  By  drawing  a  sufficient 
number  of  circles  in  this  way,  we  see  that  the  aggregate  dis- 
turbance produces  a  plane  wave-front  A"B',  receding  from  the 
surface  CD.  If,  as  it  approached  the  plane  surface,  it  had  been 


120 


KINEMATICS. 


[CHAP. 


parallel  to  that  surface,  it  would  retrace  its  path.  If  it  had 
approached  the  surface  obliquely,  so  that  the  direction  of  the 
wave  makes  an  angle  i,  with  the  normal  to  the  surface,  the 

direction  of  the  reced- 

Fig.60. 

ing  wave  will  make  an 
equal  angle  i  with  the 
normal,  but  on  the  other 
side  of  it.  This  is  ex- 
pressed by  saying  that 
the  Angle  of  Inci- 
dence, j,  is  equal  to 
the  Angle  of  Reflex- 
ion, p\  these  being  un- 
derstood to  be  angles  made  between  the  direction  of  the  wave 
and  the  normal  to  the  surface,  or,  what  amounts  to  the  same 
thing,  between  the  plane  of  the  wave  and  the  plane  of  the 
surface. 

The  same  proposition  may  be  otherwise  demonstrated.  In  Fig.  60  let 
AB  represent  the  direction  of  a  wave,  and  CD  the  reflecting  plane  surface. 
Every  movement  of  the  vibrating  body  with  reference  to  the  direction  AB 
may  be  resolved  into  two,  these  being  referred  to  the  axes  By  and  Bar.  On 
reflexion,  the  component  in  yB  has  its  direction  reversed ;  that  in  the  direc- 
0  tion  xR  is  not  thus  interfered 

with.  After  reflexion,  on  re- 
compounding  the  components, 
the  resultant  is  found  to  be 
a  disturbance  similar  to  the 
original  one,  but  in  the  direc- 
tion B A',  while  the  angle  ABy 
is  equal  to  the  angle  yBA'. 

Reflexion  of  a  curved 
wave-front  at  a  plane 
surface.  —  If  a  wave-front 
be  curved,  it  may  be  con- 
sidered as  consisting  of  a 
very  large  number  of  very 

''  /-''"  ^.. ..,    \  x  small  plane  surfaces.    To 

,s''  ,. ^  "\  "  each  of   these   a   normal 

/'  .*- -^   T\  may  be  drawn  ;  each  such 

''  /- — -x%  N  normal  indicates   the   di- 

/'"'^\  rection  of  the  correspond- 

ed ing  part  of  the  wave-front ; 

the  angle  which  each  such  normal  makes  with  the  reflecting  sur- 
face, when  its  own  part  of  the  wave-front  strikes  the  obstacle, 


Fig.61. 


REFLEXION  OF   WAVES. 


121 


is  the  angle  of  incidence  for  that  part  of  the  wave-front ;  to  this 
angle  the  angle  of  reflexion  for  that  part  of  the  wave-front  must 
be  equal. 

Let  the  convex  spherical  wave-front  AB  strike  the  surface 
CD.  Each  part  of  the  wave-front  is  reflected  at  its  own  angle. 
The  result  is  the  reflexion  of  a  convex  wave,  which  is  of  the 
same  form  as  AB  would  have  assumed  in  the  time,  but  which 
travels  in  the  opposite  direction.  Such  a  wave  is,  in  effect, 
exactly  such  a  wave  as  would  have  travelled  from  the  point  O', 
as  far  behind  the  reflecting  surface  as  O  is  in  front  of  it. 

In  the  same  figure,  if  the  directions  be  reversed,  so  that  a 
concave  wave-front  travels  towards  the  reflecting  surface,  con- 
verging upon  O',  it  will,  when  reflected,  become  reversed,  and 
on  receding  from  the  reflector,  it  will  converge  upon  O,  which 
is  as  far  on  the  one  side  of  CD  as  the  point  O',  upon  which  the 
wave  had  originally  been  converging,  is  on  the  other. 

General  construction  of  a  reflected  wave.  —  Let  AB  (Fig. 
62)  be  a  wave-front,  and  CD  a  reflecting  surface,  both  of  any 


Fig.62. 


form.  Draw  normals  to  AB  of  such  lengths  that  they  may  all 
cut  CD  ;  from  these  normals  cut  off  equal  portions,  and  join  the 
extremities  of  these  portions;  the  line  EF  is  thus  obtained, 
which  represents  the  form  that  the  wave  would  have,  assumed 
but  for  the  reflecting  obstacle.  Draw  a  number  of  circles ;  the 
centre  of  each  of  these  is  a  point  at  which  one  of  the  normals 


122 


KINEMATICS. 


[CHAP. 


to  AB  cuts  CD ;  the  radius  is  the  distance  along  the  normal  in 
question  from  the  surface  CD  to  the  surface  EF.  These  circles 
have  a  common  tangent,  the  curved  line  GH,  which  indicates 
the  form  of  the  reflected  wave.  Normals  drawn  to  this,  of  equal 
length,  may  indicate  the  form  of  the  reflected  wave  at  any  sub- 
sequent instant ;  if  these  be  drawn  backwards,  all  the  previous 
positions,  real  or  apparent,  may  be  investigated. 

This  is  the  general  construction ;  but  it  very  frequently  leads  to  difficul- 
ties where  different  parts  of  the  wave  cross  one  another.  In  most  cases, 
however,  the  following  method  is  effective.  Consider  a  part  of  an  incident 
wave  and  the  point  at  which  it  impinges ;  from  the  point  of  incidence  draw 
a  line  indicating  the  direction  in  which  that  part  of  the  incident  wave  will 
be  reflected.  Find  to  what  distance  behind  the  reflecting  surface  the  inci- 
dent wave  would  have  travelled  in  a  given  time  if  there  had  been  no  obstacle ; 
measure  off,  along  the  direction  of  the  reflected  wave,  a  distance  equal  to  this ; 
repeat  this  operation  for  several  parts  of  the  wave-front ;  join  all  the  points 


Fig.63. 


thus  obtained.  This  gives 
the  form  of  the  reflected 
wave  at  the  time  chosen. 
Equal  distances,  measured 
forwards  or  backwards 
along  the  normals  to  this 
wave,  will  give  the  form 
of  the  wave  at  instants 
subsequent,  and  its  true 
or  hypothetical  form  at  in- 
stants previous.  The  fol- 
lowing are  examples  :  — 

Problems. 

1.  Let  the  reflecting  sur- 
face be  a  paraboloid :  let 
the  advancing  wave-front 
be  plane.  The  "  focus  " 
of  the  parabola  is  at  F 
(Fig.63).  We  may  choose 
three  instants  for  con- 
sideration. 

a.  That  at  which  the 
whole  wave-front  would 
have  arrived  at  O.  Each 
part  of  the  wave-front  is 
reflected  as  shown  in  the 
diagram.  The  part  which  would  have  taken  the  course  aa'  is  turned  into 
the  direction  aF  ;  bb'  into  &F  ;  cc'  to  cF ;  and  so  on.  aF  =  aa' ;  W  =  bY  ; 
cc'  =  cF ;  AO  =  AF.  The  wave-front  is  reduced  to  a  point ;  it  is  at  that 
instant  passing  through  the  focus  F. 

b.  Any  instant  at  which  the  wave-front,  having  passed  the  point  A,  would 
not  yet  have  reached  the  point  O ;  the  reflected  wave  is  spherical  and  con- 
cave, converging  on  F. 


KEFLEXION  OF   WAVES. 


123 


c.  Any  instant  at  which  the  wave  would  have  reached  a  plane  farther 
away  from  the  reflecting  surface  than  a'O  ;  the  wave  is  spherical,  divergent 
from  the  focus  F. 

2.  In  the  same  figure,  the  wave-front  is  one  which  starts  from  F  as  a 
centre ;  it  meets  the  paraboloid  reflector ;  it  is  reflected  with  a  plane  wave- 
front. 

3.  The  reflecting  surface  is  spherical  and  concave,  the  incident  wave- 
front  flat. 

In  Fig.  64  the  reflecting  surface  is  represented  by  the  line  RS.  The 
wave  travels  from  right  to  left  and  meets  RS.  Each  part  of  the  front 
of  the  wave  is  turned  back 

at  its  own  angle  of  reflexion :  '°  Fig.64. 

the  wave-front  becomes  con- 
vergent. It  does  not,  how- 
ever, converge  on  any  one 
point  ;  it  is  not  spherical. 
The  figure  shows  that  there 
is  a  curved  line,  a  "  caustic 
by  reflexion,1'  in  which  lie 
all  the  foci  of  all  the  sepa- 
rately considered  parts  or 
elements  of  the  wave-front. 
In  each  wave  the  element 
reflected  from  the  outer  part 
of  the  surface  RS  will  sooner 
come  to  focus  than  that  re- 
flected from  the  centre  of 
that  surface ;  hence  if  the 
wave  be  single,  a  spot  of  maxi- 
mum disturbance  will  appear 
to  run  along  each  limb  of  the 
caustic,  and  to  disappear  in 
a  diverging  wave  at  its  apex. 
A  succession  of  waves  will 
keep  the  whole  of  the  caustic 
in  a  state  of  maximum  dis- 
turbance.* 

4.  A  reflecting  mirror  a  segment  of  a  sphere :  a  centre  of  disturbance 
midway  between  the  centre  of  the  sphere  and  the  reflecting  surface.     With 
ruler   and  compass  draw  the  form  of   the  reflected  wave  as  in  Fig.  64. 
Approximately  plane  at  its  centre. 

5.  A  complete  spherical  surface  used  as  a  reflector;  a  centre  of  disturb- 
ance at  the  centre  of  the  sphere :    a  divergent  spherical  wave  produced. 
Prove  that  after  reflexion  this  becomes  a  convergent  spherical  wave,  converg- 
ing on  and  passing  through  the  same  centre,  and  then  repeatedly  reflected 
and  alternately  converging  on  and  diverging  from  the  original  point  of 
disturbance. 


*Take  a  long  strip  of  bright  tinplate;  bend  it  into  a  semicircle;  place  it  on  its 
side  on  a  sheet  of  paper  in  the  sunlight,  exposing  the  concavity  to  the  sun :  a  bril- 
liantly-illuminated Caustic  Curve  will  be  seen  on  the  paper.  The  form  of  this  curve 
may  be  varied  by  altering  that  given  to  the  tinplate. 


124  KINEMATICS.  [CHAP. 

6.  Prove  that  a  spherical  wave  starting  from  one  focus  of  an  ellipse  con- 
verges after  reflexion  on  the  other  focus.   Hence  prove  that  if  the  centre  of 
disturbance  be  at  one  focus,  and  if  it  be  surrounded  by  a  complete  ellipsoidal 
reflecting  surface,  a  wave  passes  back  and  fore  between  the  foci,  alternately 
converging  and  diverging,  first  at  one,  then  at  the  other  focus. 

7.  A  reflecting  mirror  a  small  segment  of  a  circle.     Draw  another  circle, 
whose  diameter  is  the  radius  of  curvature  of  the  mirror.     Waves  proceeding 
from  any  point  of  this  new  circle  will,  after  reflexion  from  the  mirror,  con- 
verge upon  some  other  point  of  the  same  circle. 

Transmission  of  a  linear  wave  into  a  denser  medium.  — 

If,  as  in  Fig.  65,  the  particles  be  more  closely  placed  in  B  than 

F1  65  in  A,  B  is  the  denser  medium. 

9      9     *     9     9      ••,..?••••    A  linear  wave-motion  travels 

to  the  right  in  A ;   it  arrives 

at  P.  It  meets  a  relative  obstruction.  P,  the  first  particle  of 
the  denser  substance,  is  more  resisted  than  the  preceding  parti- 
cles set  in  motion  by  the  wave.  The  wave  is  not  entirely 
obstructed,  and  goes  on  into  B  ;  but  there  is,  to  some  extent, 
the  production  of  a  reflected  wave  in  A.  This  reflected  wave, 
like  that  of  Fig.  58,  is  of  the  same  phase  and  period,  and  of  the 
same  wave-length  as  the  original  wave.  It  cannot  be  of  the 
same  amplitude,  for  some  of  the  energy  of  the  wave-motion  has 
been  spent  in  setting  up  a  wave  in  B.  The  wave-motion 
propagated  along  B  must  necessarily  be  of  the  same  period  as 
that  in  A,  for  the  particles  in  B  must  move  in  unison  with 
those  of  A,  which  impel  them ;  it  must  be  of  the  same  phase,  for 
it  is  the  direct  continuation  of  the  wave  in  A.  Since  the 
particles  are  more  crowded  together  in  B  (a  less  distance  cor- 
responding to  the  same  number  of  particles),  a  wave  cannot 
propagate  itself  in  B  so  far  in  a  given  time  as  it  can  in  A,  for 
its  doing  so,  still  retaining  the  same  wave-length,  would  imply 
its  setting  a  greater  mass  in  motion.  This  would,  however, 
require  a  greater  amount  of  energy.  If  the  latter  be  definite 
in  amount,  as  it  must  be,  the  wave-length  and  the  speed  of 
propagation  must  be  less  in  the  denser  medium.* 

As  to  the  relative  amplitudes  of  the  respective  vibrations,  the  original, 
the  reflected,  and  the  transmitted,  the  amplitudes  of  the  two  latter  taken 
together  are  not  necessarily  equal  to  that  of  the  first ;  biit  in  every  case  the 
energy  of  vibration  of  the  original  wave  is  equal  to 'the  sum  of  the  energies 
of  the  reflected  and  the  transmitted  waves. 

*  To  avoid  misconception  it  may  be  remarked  here  that  in  concrete  cases,  while 
its  density  is  a  powerful  factor  in  determining  the  velocity  of  propagation  of  vibra- 
tion in  a  given  body,  this  also  depends  greatly  on  the  peculiar  molecular  properties 
— the  elasticity — special  to  each  substance. 


v.]  KEFRACTION  OF   WAVES.  125 

If  in  Fig.  65  a  wave  beginning  with  compression  be  sup- 
posed to  run  through  B  towards  the  left,  when  it  comes  to  the 
particle  P  —  which  may  be  considered  as  the  last  of  B  or  the 
first  of  A  —  that  particle,  meeting  less  resistance  than  its 
predecessors  in  B  had  encountered,  plunges  into  the  rarer 
medium  and  sets  up  in  A  a  wave  depending  on  the  original 
wave  in  B  for  its  period  and  phase,  but  of  greater  amplitude; 
and  the  wave  in  A  will  also  have  a  greater  wave-length  than 
that  in  B,  for  a  reason  the  converse  of  that  stated  in  the  last 
paragraph.  The  effect  on  the  denser  body  is,  however,  singu- 
lar. The  particle  P,  plunging  away  from  the  rest  of  the  par- 
ticles of  B,  produces  in  that  part  of  B  a  dilatation  which  is 
propagated  backwards,  and  there  then  travels  in  B  a  reflected 
wave,  agreeing  with  the  incident  wave  in  period  and  in  wave- 
length, necessarily  not  in  amplitude,  and  opposed  in  phase. 
A  maximum  compression  arriving  at  P  causes  that  particle  to 
yield  to  the  greatest  extent,  and  to  produce  a  maximum  dilata- 
tion in  B  ;  hence,  when  the  incident  wave  produces  a  maximum 
compression  among  all  the  particles  of  A  in  the  neighbourhood 
of  P,  P  itself  starts  a  wave  in  B,  commencing  with  a  maximum 
dilatation,  and  the  incident  and  reflected  waves  are  not  con- 
tinuations of  one  another  as  in  Fig.  58,  but  there  is  loss  of  half 
a  wave-length.* 

A  comparison  of  the  diagrams  in  the  preceding  discussion 
shows  that,  in  every  case  where  the  medium  in  which  the  wave 
has  travelled  is  the  same,  the  space  traversed  by  every  part  of 
the  wave,  reflected  or  not  reflected,  or  sooner  or  later  so  affected, 
must  necessarily  be  the  same  in  a  given  time ;  and  hence,  count- 
ing from  any  initial  condition  to  any  final  wave-front  form, 
the  space  traversed  before  reflexion  +  that  traversed  after  it 
=  a  constant  quantity  for  a  given  time,  and  that  for  every 
element  of  the  wave-front;  or,  as  it  is  often  expressed,  the 

*  This  curious  result  has  an  interesting  bearing  on  the  Conservation  of  Energy. 
Both  the  amplitude  and  the  length  of  the  wave  in  the  rarer  medium  are  greater  than 
in  the  denser.  The  body  B,  If  the  metaphor  may  be  allowed,  finds  itself  to  have 
done  more  work  on  the  body  A,  and  therefore  to  have  transmitted  more  energy  to  it 
than  it  had  intended  :  the  particle  P  has  compromised  the  body  B  by  giving  a  greater 
dash  forward  than  was  expected.  Under  the  circumstances,  matters  are  adjusted  by 
the  propagation  of  a  wave  of  opposite  phase  in  the  body  B.  If  each  of  these  waves, 
the  reflected  and  the  transmitted,  be  regarded  separately  as  containing  so  much 
energy,  the  sum  of  their  energies  may  appear  to  exceed  that  of  the  original  wave. 
The  whole  vibrating  matter  must,  however,  be  regarded  as  forming  one  system.  In 
this  system,  a  compression  in  one  wave,  and  a  dilatation  in  another,  pfoduce  a  rela- 
tive motion  amounting  only  to  their  difference ;  and  this  is  the  true  motion  of  the 
system,  the  energy  corresponding  to  which  is  equal  to  the  energy  of  the  original 
vibration. 


126 


KINEMATICS. 


[CHAP. 


"  incident  ray  "  +  the  "  reflected  ray  "  =  constant  for  the  whole 
wave. 

Refraction  of  a  plane  wave  at  a  plane  surface. —If  the 
incident  wave  strike  the  plane  surface  simultaneously  at  all 
parts  of  its  own  front,  it  will  simply  pass  more  slowly  through 
the  denser  medium,  while  a  reflected  wave  is  sent  back ;  but  if 
it  strike  it  obliquely,  there  are  some  changes  in  the  wave,  which 
result  from  one  part  of  it  being  hampered  in  the  retarding  sub- 
stance while  the  rest  is  still  moving  with  comparative  rapidity 
in  the  rarer  medium. 

In  Fig.  66  let  AB  be  the  wave-front  in  the  rarer  medium ; 
CD  the  surface  separating  the  denser  from  the  rarer  medium; 
A'B'  a  position  at  which  the  wave-front  would  have  arrived 


if  it  had  not  encountered  the  denser  substance.  The  lines  W, 
cc1,  ddf,  etc.,  are  normals  to  the  incident  wave-front,  meeting  the 
line  CD  in  b'r,  c",  d",  etc.  The  angle  BAB'  between  the  wave- 
front  and  the  surface,  or  iOn  between  the  direction  of  the  inci- 
dent wave  arid  the  normal  to  the  surface,  is  called  the  angle  of 
incidence. 

Let  us  suppose  that  the  velocity  of  propagation  in  the 
denser  medium  is  f  of  that  in  the  rarer.  Then  with  centres 
A,  6",  (?",  d",  etc.,  and  radii  =  |  AA',  f  b"b',  f  c"c',  etc.,  draw 
circles.  The  line  A"B',  which  is  their  common  tangent,  indi- 
cates the  position  of  the  wave-front  at  the  end  of  the  time  dur- 
ing which  it  would  have  advanced  to  A'B'.  The  wave  has  been 
rendered  somewhat  broader,  and  has  changed  its  direction.  The 
angle  AB'A"  or  n'Or  is  called  the  "angle  of  refraction." 


REFRACTION   OF   WAVES. 


127 


In  the  figure,  A  A" :  A  A' :  :  2  : 3 ;  but  in  the  two  triangles  AB'A',  AB'A", 
we  see  that  A  A"  :  A  A' :  :  sin  AB'A"  :  sin  AB'A'. 

Sin  AB'A"_    sin  ang.  refr.    _  sing  _  2  _  velocity  in  denser  medium 
* '  Sin  AB'A'  ~~  sin  ang.  incid.  ~  sin  i  ~       "    velocity  in  rarer  medium  ' 

...  vel.  in  denser  med.        1 
Generally,  if  — ^—. ; -=—  =  — ,  a  fraction  ;  then  sin  ang.  incidence  = 

(3  x  sin  ang.  refraction ;    sin  i  =  ft  sin  g,  and  ft  is  called  the  "  index  of 
refraction"  of  the  denser  substance  as  compared  with  the  rarer  one. 

This  formula  shows  that  in  Fig.  66  if  iO  indicate  the  direc- 
tion of  the  incident,  Or  that  of  the  refracted  wave ;  nOn'  the 
normal  to  the  plane  refracting  surface  ;  if  a  circle  be  drawn  with 
centre  O  and  any  radius,  — the  lines  iO  and  Or  will  cut  it  in  I 
and  R ;  from  I  and  R  draw  lines  at  right  angles  to  nn'9  as  in 
the  figure.  These  lines  always  bear  to  one  another,  whatever 
the  angle  of  incidence,  the  same  ratio  as  the  Velocities  in  the 
respective  media,  and  this  law  defines  the  relation  of  the  angle 
of  refraction  to  the  angle  of  incidence.  An  equivalent  construc- 
tion is  given  in  Fig.  182. 

Refraction  of  a  wave  at  a  surface :  General  construction. 
—  Let  AB  be  an  advancing  wave-front,  CD  the  bounding  surface 

A  B 


of  a  denser  medium.  Let  the  assumption  be  made  that  each 
several  element  of  the  wave-front,  so  long  as  it  is  in  the  same 
medium,  travels  mainly  in  the  direction  of  the  normal  drawn  to 
it.  In  this  way  the  whole  wave-front  is  always  simply  related 
to  all  its  previous  forms,  all  the  parts  of  it  having  at  any  instant 
travelled  along  their  respective  normals  to  an  equal  extent  dur- 
ing any  given  interval  of  time ;  and  a  line  once  normal  to  the 
wave-front  is,  if  produced,  always  normal  to  it  so  long  as  it 
travels  in  the  same  isotropic  medium. 

Then  a  number  of  these  normals  to  the  incident  wave,  such 
as  aaf  in  the  figure,  are  drawn  equal  to  each  other,  ami  of  length 
sufficiently  great  to  enable  the  surface  to  cut  them  all.  The 
extremities  of  these  equal  normals  are  joined;  in  this  way  a 


128 


KINEMATICS. 


[CHAP. 


curve  EF  is  produced,  which  indicates  the  form  that  the  incident 
wave  would  have  assumed  had  it  travelled  thus  far  in  the 
original  medium.  Lines  normal  to  AB  are  also  normal  to  EF. 
We  see  segments  of  these  normals,  such  as  da',  cut  off  between 
CD  and  EF.  The  fraction  1//8  (=  the  velocity  of  propagation 
in  the  denser  medium  -?-  that  in  the  rarer)  must  now  be  known. 
From  da'  cut  off  da" ,  which  is  to  da'  as  1//3  :  1  ;  with  d  as  a 
centre,  and  da"  as  radius,  draw  an  arc  of  a  circle  through  a". 
Treat  similarly  all  dans  fellow-normals.  A  number  of  arcs  are 
thus  obtained,  to  which  the  common  tangential  curve  must  be 
drawn.  This  gives  the  form  of  the  refracted  wave-front. 

This  having  been  obtained,  normals  may  now  be  drawn  to 
it ;  these  will  not  in  general  coincide  with  aa'  and  its  fellows. 
By  measuring  off  equal  distances  along  these  normals  to  the 
refracted  wave-front,  all  the  future  forms  of  the  refracted  wave 
and  all  its  apparent  past  forms  may  be  ascertained. 

Refraction  of  a  spherical  wave  at  a  plane  surface.  —  Let 
a  spherical  wave  whose  centre  is  at  F  strike  the  plane  surface 
RS  and  enter  a  denser  medium.  The  wave  would  at  a  certain 


Fig.68. 


instant  have  arrived,  say  at  aWc'd*.  According  to  the  preced- 
ing construction,  lines  aa",  W,  etc.,  are  cut  from  aaf,  W,  etc., 
to  which  they  respectively  bear  the  constant  ratio  1  :  /3.  Arcs 


v.]  REFRACTION  OF  WAVES.  129 

are  drawn  with  centres  a,  5,  c,  d,  etc.,  and  radii  aa",  bb",  cc'f, 
ddn,  etc. :  the  common  tangent  BCD — a  curved  line  —  is  found: 
this  gives  the  form  of  the  refracted  wave  in  the  denser  medium  ; 
it  is  hyperbolic. 

Normals  may  now  be  drawn  to  this,  by  means  of  which  the 
future  and  the  apparent  past  forms  of  the  wave  may  be  traced 
out.  In  the  denser  substance,  as  it  travels  onward  it  retains  the 
hyperboloid  form;  and  if  the  normals  5B,  cC,  etc.,  be  traced  back- 
wards, and  such  equal  lines  as  BB',  CC',  etc.,  measured  off 
along  them,  all  those  hypothetical  wave-fronts  may  be  drawn 
from  which  the  wave-front,  as  it  travels  through  the  denser 
medium,  presents  the  appearance  of  having  been  developed.  On 
tracing  back  far  enough,  we  find  that  the  wave  appears  as  if 
developed  from  a  wave-front  convergent  not  upon  the  centre  F, 
but  through  a  caustic  the  apex  of  which  is  at  G,  where  OG  is  to 
OFas/3:l. 

Problem. 

A  spherical  convergent-wave  meets  the  plane  surface  of  a  refracting 
substance.  By  construction  show  that  the  wave  converges  through  a  caustic, 
the  distance  of  whose  apex  from  the  surface  is  less  than  that  of  the  original 
centre  in  the  ratio  of  1 :  (3. 

Passage  of  a  spherical  wave  through  a  parallel-sided 
sheet  of  a  denser  substance.  —  At  its  entrance  into  the  denser 
substance,  the  wave-front  becomes  hyperbolic ;  at  exit  every 
part  of  the  wave  resumes  the  direction  which  had  pertained  at 
the  instant  of  the  first  refraction  to  that  part  of  the  wave-front 
from  which  it  had  been  developed,  and  thus  the  wave-front  again 
approximately,  but  not  exactly,*  resumes  its  original  spherical 
form.  In  Fig.  68  the  part  of  the  wave-front  which  passes 
through  a  is,  when  the  wave  approximately  resumes  its  spheri- 
cal form,  again  refracted,  so  that  that  element  assumes  a  direc- 
tion parallel  to  its  original  direction  Fa. 

Approximate  Foci.  —  In  Fig.  68,  if  of  the  hyperbolic  wave 
in  the  denser  medium  only  a  limited  portion  in  the  centre  be 
considered,  it  will  be  found  to  approximate  very  closely  to  a 
small  arc  of  a  circle  whose  radius  is  Go.  We  have  already 
learned  to  express  this  by  saying  that  its  radius  of  curvature 
is  Go.  If  accordingly  the  incident  wave-front  be  narrow,  the 
refracted  wave-front  appears  to  have  diverged  from  a  group  of 
points  in  the  immediate  neighbourhood  of  G,  the,  apex  of  the 

*  An  object  seen  through  a  pane  of  glass  is  never  as  distinct  as  the  same  seen 
through  the  intervening  air  alone. 

K 


130  KINEMATICS.  [CHAP. 

caustic ;  and  the  narrower  the  incident  wave,  or  the  less  its 
divergence,  the  more  nearly  will  it  appear  to  come  from  a  single 
point,  the  very  apex  of  the  caustic  itself.  Similarly  in  Fig.  64, 
the  narrower  the  incident  wave  is  in  comparison  with  the  breadth 
of  the  curved  reflecting-surface,  the  more  nearly  will  the  reflected 
wave  converge  on  the  very  apex  of  the  caustic,  midway  between 
the  surface  and  the  centre  of  the  sphere.  Consequently  these 
points,  the  apices  of  the  caustics,  are  approximate  foci  for  com- 
paratively narrow-fronted  waves. 

Refraction  of  a  plane  wave  at  a  spherical  surface.  —  Con- 
structions somewhat  similar  to  that  of  Fig.  64  will  show  that  if 
a  plane-fronted  wave  be  made  to  strike  the  convex  spherical 
surface  of  a  denser  or  the  concave  surface  of  a  rarer  medium, 
it  will  be  made  to  converge :  if  it  strike  the  concave  spherical 
surface  of  a  denser,  or  the  convex  spherical  surface  of  a  rarer 
medium  it  will  diverge  :  in  all  these  cases  it  will  form  a  Caustic, 
actual  or  virtual. 

The  tip  of  the  Caustic  (the  approximate  focus  for  central  rays)  is  at  a 
distance  (measured  along  the  Axis,  a  line  passing  through  the  sphere-centre 
at  right  angles  to  the  wave-front)  from  the  refracting  surface  equal  to 
rf3l/((3,—(3(})1  where  r  is  the  radius  of  the  surface,  and  /30,  /?y  the  refractive 
indices  of  the  original  and  the  refracting  media  respectively. 

Refraction  of  a  spherical  wave  at  a  spherical  surface.  — 

Again  there  will  be  convergence  or  divergence  produced:  and 
in  all  cases  except  that  in  which  the  wave-front  is  made  to 
become  approximately  plane-fronted,  there  will  be  a  Caustic, 
actual  or  virtual. 

The  centre  of  the  incident  wave  and  the  tip  of  the  caustic  are  respec- 
tively at  distances  d'  and  d"  from  the  refracting  surface;  then  /30/d' 
—  fii/d"  =  (/?0  —  /3,)  /r ;  d',  d",  and  r  (the  radius  of  the  refracting  surface) 
being  all  reckoned  positively,  that  is  towards  the  right,  and  being  therefore 
negative  if  they  lie  towards  the  left  of  the  refracting  surface.  The  centre 
of  the  incident  wave  and  the  point  upon  which  the  wave  converges  after 
refraction  are,  in  respect  to  one  another,  called  conjugate  points. 

If  the  waves  travel  in  the  rarer  medium  from  right  to  left  and  are  plane- 
fronted,  so  that  d'=  +  oo,  then  d"=  +  ^,r/(^t  -  /?0)  =  "/'  "  :  if  they  travel 
in  the  denser  medium  from  left  to  right,  dt'=  -co,  but  "  /?0 "  is  now  the  den- 
sity of  the  more  refracting  medium  and  is  equal  to  the  former  ft,,  and  simi- 
larly "/?,"  is  the  former  "00":  therefore  d,"= "/"  "  =  &)r/(£0-&) 
=  —  (/?o/ /?/)/'  I  the  distances  between  the  tip  of  the  caustic  and  the  vertex 
of  the  refracting  surface  in  these  two  cases  are  called  the  Principal  Focal 
Distances,/'  and/",  for  the  refracting  surface  between  the  two  media  in 
question. 

If  the  wave,  after  entering  the  second  medium,  again  return  into  the 
first  through  a  second  spherical  bounding  surface,  the  point  upon  which  it 


v.]  RAYS.  131 

now  converges  may  be  found  in  precisely  the  same  way  by  finding  what 
point  is,  with  respect  to  the  second  refracting  surface,  the  point  conjugate 
to  the  centre,  real  or  virtual,  of  the  wave  approaching  the  second  surface. 
By  this  method,  keeping  in  view  the  requisite  exchange  of  numerical 
values  of  (30  and  /?,  at  the  second  surface,  the  ordinary  lens-formulae  (p.  540) 
are  arrived  at ;  and  the  process  may  be  repeated  for  any  number  of  media 
and  surfaces,  at  any  mutual  distances. 

Utility  of  the  idea  of  "  Rays  "  in  geometrical  construc- 
tion. —  All  the  preceding  discussions  have  been  grounded  on 
consideration  of  the  various  forms  assumed  by  the  wave-front : 
we  have  shown  that  any  form  may  be  developed  from  any  of  its 
predecessors  by  taking  each  point  in  that  predecessor  as  a  cen- 
tre of  disturbance,  and  drawing  equal  circles  of  appropriate  radii 
which  indicate  the  extent  to  which  the  disturbance  has  trav- 
elled ;  then  of  these  circles  the  common  tangential  line  denotes 
the  developed  form  of  the  wave-front :  we  found  that  in  simple 
cases  every  line  normal  to  any  wave  was  normal  to  all  those 
developed  from  it,  and  to  all  those  forms  through  which  it  had 
passed;  in  all  this  it  being  supposed  that  the  medium  was 
one  in  which  the  velocity  of  transmission  was  the  same  in  all 
directions. 

We  also  made  an  assumption  that  when  the  form  of  the  origi- 
nal wave  was  complex  and  the  medium  isotropic,  the  same  law 
applied;  that  the  maxim  once  a  normal  always  a  normal 
was  true ;  that  there  was  no  lateral  expansion  of  the  wave-front 
beyond  the  limits  indicated  in  a  diagram  by  lines  set  down  to 
represent  such  normals.  The  assumption  is  approximately  true 
only  in  special  cases  —  in  general  there  is  lateral  expansion  of 
the  wave-front  beyond  such  limits ;  but  on  reference  to  what  was 
said  in  connection  with  Figs.  56  and  57,  we  are  reminded  that 
it  is  possible  for  us  to  conceive  a  point  of  disturbance  011  a  wave- 
front  as  one  in  a  wide  aperture ;  and  hence  if  the  wave-length 
be  very  small,  the  wave-front  propagated  from  each  little  element 
of  the  wave-surface  travels  along  the  normal  to  that  element. 
In  any  case  this  is  never  an  absolute  statement,  and  there  is 
always  more  or  less  lateral  divergence  ;  but  as  a  first  approxima- 
tion of  sufficient  value  for  most  purposes,  it  may  be  said  that  in 
an  isotropic  medium  (where  the  velocity  of  propagation  is  equal 
in  all  directions)  the  wave-front  is  developed  from  all  its  prede- 
cessors along  their  common  normals ;  that  this  is  nearly  the  case 
when  the  wave-length  is  comparatively  short :  but  the  greater 
the  proportionate  length  of  the  wave,  the  more  lateral  expansion 
there  is,  and  the  less  able  would  we  be  to  find  the  form  of  the 


132  KINEMATICS.  [CHAP. 

wave-front  at  any  moment  by  exclusively  considering  the  nor- 
mals to  its  previous  forms.  If,  however,  the  wave-length  be 
comparatively  short,  we  may,  by  considering  the  normals  only, 
erect  in  a  diagram  the  scaffolding  on  which  the  form  of  the 
wave-front  may  be  constructed.  In  reflexion,  for  instance,  as  in 
Fig.  64,  normals  may  be  drawn  to  the  front  of  the  incident  wave  ; 
the  reflected  wave  is  of  such  a  form  that  each  normal  to  it 
makes,  with  the  corresponding  normal  to  the  reflecting  surface, 
an  angle  equal  to  that  made  by  a  normal  to  the  incident  wave. 
But  the  wave-front  itself  might  have  been  omitted  from  the 
diagram,  and  the  same  results,  as  regards  focus  and  caustic, 
would  have  been  obtained.  Then  attention  might  be  fixed  on 
the  normal  lines,  and  on  the  way  in  which  these  change  their 
direction  on  reflexion  or  refraction.  They  might  be  treated  as 
if  they  were  physical  entities,  and  might  receive  special  names. 
This  has  actually  happened.  The  imaginary  straight  line  drawn 
at  right  angles  to  the  wave-front  at  any  point  has  been  called  a 
"  ray "  passing  through  that  point.  Each  ray  is  straight,  for 
the  normals  preserve  always  the  same  direction  ;  and  of  all  wave- 
motion —  such  as  Light  —  which  does  not  expand  perceptibly 
beyond  the  limits  laid  down  for  it  by  its  normals,  as  in  Fig.  57, 
the  progress  is  described  by  saying  that  its  rays  travel  in 
straight  lines  so  long  as  it  is  in  the  same  medium.  This  mode 
of  expression  has  both  advantages  and  disadvantages.  It  leads 
to  the  assumption  that  a  divergent  wave-front  is  a  divergent 
"  pencil "  of  rays,  each  of  which  is  somehow  distinct  from  its 
fellows ;  it  leads  to  these  rays  being  conceived  as  themselves 
reflected,  refracted,  etc. ;  it  isolates  the  physics  of  those  phe- 
nomena— those  of  Light — in  which  waves  approximately  follow 
their  normals  only,  from  those  in  which  this  approximation  is 
much  less  complete,  as  in  the  case  of  Sound.  On  the  other 
hand,  it  presents  certain  advantages ;  it  simplifies  diagrams ;  it 
enables  any  problem  to  be  reduced  to  its  simplest  elements  by 
an  absolute  rejection  of  all  lateral  disturbances,  and  of  the  effects 
produced  by  any  parts  of  the  wave  other  than  those  at  the  points 
of  the  fronts  crossed  by  the  normals  ;  and  it  gives  results  which 
in  the  theory  of  Light  are,  up  to  a  certain  point,  of  sufficient 
accuracy.  This  advantage  persists,  however,  up  to  a  certain 
point  only,  and,  on  the  whole,  a  habit  of  referring  the  phenomena 
of  wave-motion  to  the  form  of  the  wave-front  is  to  be  preferred, 
though  hereafter  we  shall  make  free  use  of  the  device  of  refer- 
ence to  rays  whenever  it  is  found  convenient  to  do  so. 


RAYS. 


133 


The  law  can  easily  be  verified  that  the  path  traversed  by 
every  element  of  the  wave  in  the  rarer  medium,  together  with 
ft  x  that  traversed  in  the  denser,  is  a  constant  quantity. 

Ptolemy's  Law.  — If  a  ray  pass  from  A  to  B,  striking  some 
point  of  a  plane  reflecting  surface  in  its  course,  and  being  thence 
reflected  to  B,  there  is  no  path  from  A  to  B  vid  any  point  of 
the  mirror,  soshortas  that  actually  traversed  by  the  ray,  under 
the  law  of  reflexion. 

Fermat's  Law.  —  If  a  ray  pass  from  A  in  one  medium  to  B 
in  another,  there  is  between  these  points  no  path  which,  the 
relative  velocities  in  the  two  media  being  taken  into  account, 
could  be  traversed  insoshorta  time  as  that  actually  traversed 
under  the  law  ft  sin  ang.  refr.  =  sin  ang.  incid.  If  the  construc- 
tion be  attempted,  it  will  be  seen  that  the  actual  law  allows  the 
ray  the  greatest  possible  proportion  of  time  in  the  rarer  medium. 

Superposition  of  simultaneous  wave-motions  on  an  in- 
definite cord.  —  We  shall  here  simply  discuss  the  single  case  in 


which  two  equal  waves  travel  in  opposite  directions  on  the  same 
cord.  In  Fig.  69  are  seen  two  waves  approaching  one  another 
on  a  cord  of  indefinite  length ;  they  meet  at  A,  and  pass  through 
one  another.  Where  crest  meets  crest  and  trough  trough,  as  at 
A,  the  amplitude  is  doubled  ;  where  crest  meets  trough,  as  at  B 
and  C,  there  is  no  movement  so  long  as  the  waves  are  passing 
through  one  another:  B  and  C  are,  during  this  mutual  interfer- 
ence of  the  waves,  non- vibrating  or  "  nodal  "  points,  between 
which  the  cord  vibrates. 


[tFHI7BRSIT7] 


134  KINEMATICS.  [CHAP. 

If  the  waves  had  been  equal,  but  opposite  in  phase,  so  that 
crest  met  trough  at  A,  then  A,  the  point  of  meeting,  would  have 
been  a  nodal  point. 

The  two  nodal  points,  B  and  C,  are  seen  to  be  at  a  distance 
of  half  a  wave-length  from  each  other. 

If  the  waves  meeting  each  other  had  been  indefinitely 
numerous,  the  interference  would  occur  in  every  region  of  the 
indefinite  cord,  and  there  would  be  an  indefinite  number  of 

Pig.70, 


O  X 

nodal  points,  half  a  wave-length  distant  from  each  other.  In 
Fig.  70  such  waves  are  seen  running  on  an  indefinite  cord,  and 
the  nodal  points,  where  crest  meets  trough,  are  marked  with 
crosses. 

Cord  of  definite  length.  —  In  Fig.  70  let  us  limit  our  atten- 
tion to  a  part  of  the  string  comprised  between  two  nodal  points, 
say  between  O  and  X.  Within  this  limited  part  of  the  string 
we  observe  two  waves  running  in  opposite  directions,  the  crest 
of  one  meeting  the  trough  of  the  other  at  four  points:  there  are 
five  loops,  each  of  which  is  equal  to  half  a  wave-length  ;  the  cen- 
tres of  the  loops,  the  points  of  greatest  vibration,  are  the  points 
A,  B,  C,  D,  E ;  the  wave-length  is  here  equal  to  -|  of  OX. 

Nodes  and  Loops. — If  the  nodal  points  O  and  X  in  Fig. 
70  had  been  the  fixed  ends  of  the  string,  and  if  it  had  been 
possible  to  establish  in  OX  two  waves,  each  of  wave-length  =  ^ 
OX,  these  meeting  one  another  so  that  trough  always  coincided 
with  crest,  the  necessary  result  would  be  a  continuous  vibration 
of  the  cord  in  five  segments,  marked  off  by  four  non-vibrating 
points,  and  these  segments  would  always  be  in  opposite  phases 
of  vibration.  The  vibration  would  in  such  a  case  be  said  to  be 
Stationary  Vibration. 

This  is  precisely  the  case  where  a  wave  running  from  O  to 
X  meets  its  own  reflexion  returning  from  X  to  O,  the  point  X 
not  being  free  to  move,  but  being  held  fixed.  If  the  wave- 
length be  f,  f,  f,  |,  f,  etc.,  of  the  length  of  the  cord  OX,  the 
result  of  the  composition  of  the  wave  and  its  reflexion  will  be  a 
stationary  vibration,  in  which  the  string  will  vibrate  in  the  first 
case  as  a  whole,  or  in  the  others  in  two,  three,  four,  five,  etc., 
vibrating  segments  or  Loops,  separated  by  non- vibrating  points 
or  Nodes. 


NODES  AND  LOOPS. 


135 


Fig.71. 


The  nodes  are  points  where  there  is  no  displacement,  and  a 
maximum  variation  of  density :  at  the  centres  of  the  loops,  on 
the  other  hand,  there  are  maximal  displacements  and  velocities. 

If  the  end  at  which  reflexion  occurs  be  not  fixed  —  as  in  the  case  of 
water-waves  running  up  against  a  cliff  —  the  first  reflexional  node  is  at  a 
distance  =  A/4  from  the  reflecting  obstacle,  and  the  succeeding  nodes  at 
distances  =  A./ 2  from  the  first  node  and  from  each  other  successively. 

Vibrations  of  a  cord  whose  extremities  are  fixed.  —  A 

string  acted  on  by  any  force  tending  to  bring  the  particles  back 
to  their  mean  position,  and  varying  as  the  displacement  —  a 
criterion,  as  we  have  seen,  of  harmonic  motion  —  will  enter  into 
vibrations  of  a  type  obeying  Fourier's  law,  and  in  the  general 
sense  any  periodic  disturbance  of  such  a  cord  will  thus  be  con> 
pounded  of  vibrations  such  as  those  shown  in  Fig.  71.  These 
simultaneous  vibrations  will,  as  regards  amplitude,  be  inde- 
pendent of  one  another, 
and  will  also  from  mo- 
ment to  moment  neces- 
sarily differ  in  their 
relative  phases.  The 
whole  motion  is,  how- 
ever, periodic. 

If  a  point  situated 
in  the  loop  of  any  one 
of  these  harmonic  components  be  held  fixed,  the  corresponding 
oscillation  is  prevented.  If  the  centre  of  a  vibrating  string  be 
touched,  the  oscillations  corresponding  to  the  whole  string,  to 
one-third,  to  one-fifth,  etc. — all  the  odd  components — are 
suppressed,  and  only  the  even  components — those,  namely, 
which  already  have  a  node  at  the  point  fixed — are  allowed  to 
go  on.  If  the  string  be  touched  at  J  of  its  length  from  the  end, 

string  string  string 
all  vibrations  except  those  corresponding  to  — g — »  — g — »  — g — » 

etc.,  cease ;  those  still  continue,  for  the  effective  fixing  of  their 
nodes  does  not  affect  them.  Similarly,  if  the  string  be  held 
steady  at  a  point  £  of  the  string-length  from  the  end,  the  4th, 
8th,  12th,  16th,  etc.,  components  remain  unaffected,  while  all 
the  rest  are  stopped. 

Longitudinal  vibrations  of  a  string  or  rod— for  a  rod 
acts  in  this  case  like  a  bundle  of  parallel  strings  —  whose  ends 
are  held  fixed  obey  the  same  principles  as  transverse  vibrations. 
Fourier's  law  holds  good ;  and  if  any  point  be  held  steady, 


136  KINEMATICS.  [CHAP. 

those  component  vibrations  which  have  a  node  at  the  point  held 
steady,  and  those  components  only,  will  remain  unaffected.  In 
longitudinal  vibration,  where,  as  at  the  centres  of  the  loops, 
there  is  the  greatest  velocity  and  displacement,  there  is  least 
actual  change  of  density ;  and  at  the  extremities  of  a  rod  fixed 
at  both  ends,  and  at  the  nodes,  while  there  is  no  displacement, 
there  are  maximum  variations  of  density. 

If  longitudinal  vibrations  occur  in  a  string  or  rod,  or  in  a 
cylindrical  mass  of  gas  —  such  as  the  air  in  an  open  organ-pipe 
—  which  is  free  at  both  extremities,  it  is  plain  that  at  the  free 
ends  there  can  be  no  change  of  density,  but  that  there  is  free- 
dom of  movement ;  hence  each  extremity  must,  as  regards  dis- 
Fig  72  placement,  be  the  cen- 

tre of  a  loop.  The 
component  vibrations 
which  make  up  the 
Fourier-motion  in  such 
a  case  are  such  as 
those  shown  in  Fig.  72. 
In  this  case,  as  well 
as  in  the  preceding,  all 
the  components,  even  and  odd,  are  possible,  and  the  wave- 
length of  the  slowest  or  fundamental  vibration  is  equal  to 
twice  the  length  of  the  rod  or  string  vibrating  longitu- 
dinally. 

In  these  cases  we  see  that  the  wave-length  of  the  fundamental  as  well  as 
of  the  concomitant  vibrations  is  determined  by  the  length  of  the  vibrating 
string  or  rod  itself.  We  have  seen  that  v  =  A/T ;  X,  the  wave-length,  may 
easily  be  found  from  I  the  length  of  the  rod,  for  A.  =  21 ;  T,  the  period,  may 
be  measured  by  acoustical  or  graphic  methods ;  these  being  experimentally 
known,  we  may  find  the  quotient  X/T  =  i/,  the  velocity  of  propagation  of  an 
undulatory  disturbance  in  a  vibrating  string  or  rod. 

If  the  central  particle  of  the  system  of  Fig.  72,  vibrating 
longitudinally,  be  held  fixed,  those  vibrations  (2,  4,  6,  etc.)  are 
suppressed  which  have  not  their  nodes  at  the  centre  of  the  rod. 
Thus  only  the  odd  components  are  left ;  but  the  rate  of  these  is 
unaffected.  If  now  one  half  of  the  rod  were  removed  altogether, 
we  would  have  remaining  a  rod  fixed  at  the  one  end,  free  at 
the  other.  This  rod  would  have  component  vibrations,  as  shown 
in  Fig.  73.  A  rod  thus  vibrating  longitudinally  will  have  a 
fundamental  vibration  whose  wave-length  will  be  four 
times  the  length  of  the  rod;  the  concomitant  components  will 
have  wave-lengths  equal  to  ^,  £,  -f-,  etc.,  of  that  length. 


NODES   AND   LOOPS. 


137 


If  the  same  rod  be  supposed  to  be  set  in  longitudinal  vibra- 


Fig.73. 


tion,  first  with  both  ends  free,  and  next  with 
one  end  fixed,  the  fundamental  wave-length 
will  in  the  latter  case  be  doubled,  and  the 
period  of  vibration  will  also  be  doubled. 

Nodes  and  Loops  in  a  vibrating  mem- 
brane. —  A  membrane  may  vibrate  in  such 
a  way  that  certain  lines  may  be  at  rest. 
The  number  of  these  lines,  if  they  extend 
from  the  centre  to  the  circumference,  must 
be  even,  for  on  each  side  of  a  node  the 
directions  of  movement  are  opposite,  and  there  cannot  be  an 
uneven  number  of  nodes. 

The  forms  of  these  lines  vary  according  to  the  shape  of  the  membrane 
and  the  mode  of  disturbance.  In  a  square  membrane,  for  example,  the 
nodal  lines  may  be  one  diagonal  —  two  diagonals  —  lines  joining  the  centres 
of  opposite  sides  —  lines  more  numerous  parallel  to  these  —  curved  lines 
symmetrical  with  reference  to  the  centre  —  complex  lines  obtained  by  the 
superposition  of  these.  In  a  circular  membrane  we  may  have  concen- 
tric circles,  or  radial  lines  even  in  number. 

In  a  circular  membrane  of  which  the  centre  and  one  point 
of  the  circumference  are  held  fixed,  the  frequency  of  the  funda- 
mental vibration  varies  inversely  as  the  radius. 

The  frequency  of  vibration  of  a  circular  membrane  vibrat- 
ing as  in  Fig.  74  (a)  being  taken  as  1,  that  of  the  same  mem- 

Fig.74. 


brane  vibrating  as  in  (6)  is  f  nearly;*  as  in  (e)  f  nearly;  as  in 
(cT)  2  nearly. 

Waves  from  two  different  centres  —  Interference.  —  In 
Fig.  75  let  A  and  B  be  the  centres  of  disturbance ;  Al  the  wave- 
length: the  dotted  circles  indicate  troughs,  the  plain  circles 
crests  of  waves.  Where  crest  coincides  with  crest,  the  eleva- 
tion or  compression  produced  will  be  the  sum  of  those  produced 
by  the  two  waves ;  where  trough  meets  trough,  the  converse  will 
hold ;  but  where  the  trough  of  one  wave  coincides  with,the  crest 
of  another,  if  that  crest  be  equal,  the  resultant  motion  at  that 


*  Lord  Rayleigh,  Theory  of  Sound,  i.  275. 


138 


KINEMATICS. 


[CHAP. 


point  is  null.  This  is  the  result  of  the  mutual  interference  of 
waves.  Join  the  points  at  which  there  is  maximum  move- 
ment, whether  of  crest  or  trough ;  join  also  those  at  which  crest 
and  trough  coincide :  we  thus  obtain  a  series  of  hyperbolas 
indicated  in  the  figure.  Along  Oaf  there  is  motion  due  to 
the  concurrent  effects  of  the  disturbances  at  A  and  B ;  along 


bb',  or  a  line  very  closely  approximating  to  it,  there  is  rest ; 
along  ccf,  concurrence;  along  dd\  approximate  rest;  and  so  on. 

The  hyperbolic  lines  W  and  dd'  would  be  lines  of  perfect  rest  if  it  were 
not  that  the  one  wave  is  half  a  wave-length,  one-and-a-half  wave-length,  etc., 
behind  the  other,  and  hence  the  amplitudes  are  not  equal.  The  divergence 
of  the  true  lines  of  rest  from  the  true  hyperbola,  occasioned  by  this,  could 
not  be  indicated  in  the  diagram,  is  less  the  greater  the  distance  from  the 
centre,  and,  if  the  wave-length  be  very  small,  will  approximately  vanish. 

If  these  two  points  were  the  only  centres  of  disturbance,  and 
if  a  screen  MN  were  placed  in  the  field  of  the  wave,  there  would 
be  movement  at  a',  <?',  e',  rest  in  the  neighbourhood  of  6',  d',f'. 

Propagation  of  waves  along  normals  in  isotropic  media. 
—  The  principle  has  been  already  stated  (see  Fig.  54)  that  the 
propagation  of  any  wave-front  is  due  to  the  sum  of  the  effects 
produced  by  all  the  points  of  it  acting  as  centres  of  disturbance. 

Let  AB  in  Fig.  76  be  a  wave-front  whose  normal  at  the 
point  N  is  NP.  Trace  the  effect  of  the  wave-front  when  the 
wave-length  is  comparatively  small.  The  point  P  is  situated  on 
the  normal;  Pf  is  situated  laterally.  From  P  as  a  centre  draw 
circular  arcs,  whose  radii,  PN,  Pa,  P5,  Pe,  Pd,  etc.,  differ  from 
one  another  successively  by  half  a  wave-length.  The  disturb- 
ance caused  at  P  by  the  movement  of  Na  is  to  some  extent 
counteracted  by  that  derived  from  a&,  aided  by  that  from  be. 


v.]  INTERFERENCE   OF   WAVES.  139 

counteracted  by  that  from  cd,  and  so  on.    But  Na  is  greater  than 
ah,  ab  than  be,  be  than  cd,  and  so  on.  Fig  76 

On  the  other  side  of  N  the  circum- 
stances are  similar ;  and  thus,  on  the 
whole,  P  is  disturbed  by  the  wave- 
front  on  both  sides  of  N. 

If  on  the  other  hand  the  point  P' 
be  considered,  it  will  be  seen  that  if 
the  wave-front  be  wide  enough  in 
comparison  with  the  wave-length,  the 
disturbances  radiating  from  it  inter- 
fere with  one  another;  for  points  on 
the  wave-front  can  always  be  chosen  and  set  off  in  pairs,  differ- 
ing in  distance  from  P'  by  half  a  wave-length;  and  consequently 
there  is  no  disturbance  produced  at  any  such  lateral  point  as  P' 
by  the  wave-motion  at  JV,  and  the  wave-front  travels  along  the 
normals  without  any  lateral  expansion. 

The  narrower  the  wave-front  or  the  greater  the  wave-length, 
the  greater  will  be  the  difficulty  in  this  construction,  and  the 
greater  will  be  the  lateral  divergence  of  the  wave.  The  wave- 
front  must  be  at  least  one  wave-length  in  breadth  before  this 
construction  begins  to  become  possible- 

Effect  of  a  screen.  —  In  Fig.  76  we  may  neglect  the  influ- 
ence on  P  of  the  part  of  the  wave-front  lying  beyond  d,  for  the 
extent  to  which  it  disturbs  P  is  very  small.  If  a  screen  were 
thrust  between  the  wave-front  and  the  point  P,  so  as  to  cut  off 
the  influence  of  the  part  cd,  the  disturbance  of  P  would  be 
increased:  if  the  screen  come  to  b,  the  motion  of  P  will  be* less 
than  at  first:  if  it  come  to  a,  it  will  be  greater  than  if  there 
were  no  screen :  if  to  N,  it  will  be  less,  being  about  one  half  of 
the  original  amount.  If  the  screen  be  pushed  still  farther,  P 
will,  in  the  same  way,  be  in  more  or  less  active  motion  accord- 
ing to  the  position  of  the  screen.  The  waves  therefore  pass 
round  the  edge  of  the  screen,  producing  fringes  of  alternate 
maximum  and  minimum  motion.  The  possibility  of  this  result 
depends  on  the  smallness  of  the  wave-length. 

Effect  of  a  very  small  screen. — If  a  wave-front  be  inter- 
rupted by  a  very  small  screen  placed  so  as  to  allow  the  wave- 
motion  to  pass  it  all  round  its  edge,  Fig.  77  shows  that  dis- 
turbances passing  round  this  obstacle  produce,  even  behind 
the  screen,  hyperbolic  fringes  of  maximum  disturbance,  between 
which  there  may  be  traced  hyperbolic  fringes  of  minimum  dis- 


140  KINEMATICS.  [CHAP. 

turbance  where  crests  coincide  with  troughs ;  and  that,  further, 
even  beyond  the  shadow  of  the  screen,  there  are  hyperbolic 
fringes  of  maximum  and  minimum  disturbance.  The  centre  of 
the  shadow  of  the  screen  is  a  spot  of  maximum  disturbance. 

Fig.  77. 


Wave  traversing  an  aperture.  —  A  wave  whose  wave- 
length is  small,  passing  through  an  aperture,  gives  similar 
fringes  of  maximum  and  minimum  disturbance,  beyond  the 
edge  of  the  aperture.  A  very  curious  result  is,  that  a  wave- 
front  passing  through  an  aperture  may  produce  no  movement 
at  a  point  situated  immediately  opposite  its  centre,  for  there 
may  be  complete  interference  between  the  waves  proceeding 


INTERFERENCE   OF    WAVES. 


141 


from  the  edges  and  those  from  the  central  regions  of  the 
aperture. 

Relation  of  the  wave-lengths  to  Fringes.— The  position  of 
the  fringes  depends  on  the  wave-length.  The  smaller  the  wave- 
length, the  nearer  to  one  another  will  the  fringes  be.  If  the 
incident  wave,  diverging  from  a  point,  be  compound,  each 
component  will  form  its  own  set  of  fringes  without  reference  to 
the  others ;  and  any  particle  within  the  field  of  fringes  may  be 
stationary  as  regards  one  of  the  component  vibrations,  while  at 
the  same  time  it  is  affected  by  the  others. 

Broken  wave-front.  —  In  Fig.  77  a,  let  a  plane  wave-front  be  repre- 
sented only  by  equal  and  equidistant  portions,  O,  O'.  Then  in  the  first 
place,  there  is  propagation  in  the  original  direction  along  the  normal.  Next, 
there  are  directions  laterally  situated,  along  which  there  are  maxima  of  dis- 
turbance. To  find  the  first  of  these,  construct  the  right-angled  triangle 
OO'L,  in  which  OL  is  one  wave-length.  Disturbances  at  O'  and  those  at  L 
from  O  are  in  the  same 


Fig.  11  a. 


are 

phase.  Plane  fronts  are 
thus  developed,  parallel  to 
O'L;  the  wave  travels  to- 
wards an  infinitely  distant 
point,  or  may  by  refraction 
be  made  to  converge  upon 
a  point  P.  The  deflection 
is  the  angle  8' ;  let  the  dis- 
tance OO'  be  I/N  cm. ;  then 
since  OL=OO'  •  sin  8',  sin  8' 
=  NX.  Hence  the  greater 
the  wave-length  the  greater 
the  deflection ;  and  the  more  finely  the  wave-front  is  broken  up,  again  the 
greater  the  deflection  8'.  On  the  other  side  of  the  normal  there  is,  symmet- 
rically situated,  another  such  deflection  8'.  If  the  construction  be  repeated, 
making  OL  equal  to  two  wave-lengths,  we  find  another  system  of  deflected 
wave-fronts,  whose  deflection  8"  is  such  that  sin  8"=  2NA  ;  and  with  consecu- 
tive similar  constructions  we  find  sin8'"=3NX,  sin8""=4NA,  and  so  on. 
This  principle  is  utilised  in  the  use  of  Diffraction-gratings  in  Optics. 

If  the  incident  wave-front  be  not  parallel  to  the  screen  OO',  the  con- 
struction is  similar ;  but  it  will  be  found  that  the  deviation  is  a  minimum 
when  the  angle  of  incidence  and  the  angle  of  diffraction  (the  angle  between 
OP  and  the  normal  at  O)  ar/3  equal.  When  the  angle  of  incidence  is  such 
that  the  angle  of  diffraction  is  nearly  zero,  the  ratio  between  a  difference  of 
wave-length  and  the  corresponding  difference  in  the  diffraction-angle  is 
approximately  constant  (Normal  Spectrum,  p.  550). 

Energy  of  S.H.M.  —  The  Energy  of  a  S.H.M.  is  proportional  to  the 
S  q  u  a  r  e  of  its  Amplitude.  The  angular  velocity  is  constant,  for  S.H.M.'s 
are  isochronous ;  the  velocity  in  the  circle  of  reference  varies  as  the  radius, 
i.e.  as  the  amplitude ;  the  Energy  varies  as  the  square  of  the  velocity  with 
which  the  body  executing  the  S.H.M.  passes  the  midpoint ;  this  velocity  is 
the  same  as  the  velocity  in  the  circle  of  reference;  therefore  the  Energy 
varies  as  the  square  of  the  amplitude. 


142  KINEMATICS.  [CHAP,  v.] 

If  a  S.H.M.  be  wholly  in  one  line,  its  energy  is  wholly  kinetic  as  the 
moving  body  passes  the  midpoint  with  velocity  v ;  it  is  equal  to  ^mv2. 

If  another  S.H.M.  of  equal  amplitude  and  period,  and  in  the  same  line 
with  the  former,  be  compounded  with  it,  the  amplitude  is  doubled,  and  the 
energy  therefore  quadrupled ;  there  must  therefore  be  a  draft  of  energy  from 
elsewhere  before  this  superposition  can  actually  occur. 

Energy  of  Conical  Pendulum.  —  If  with  one  S.H.M.  there  be  com- 
pounded another  S.H.M.  of  equal  amplitude  and  period,  but  in  a  line  at 
right  angles  to  the  former,  the  energy  of  the  compounded  movement  is  simply 
double  that  of  either  of  the  components.  In  circular  motion  these  two 
S.H.M.'s  differ  by  £  period ;  when  the  energy  of  the  one  is  wholly  kinetic 
that  of  the  other  is  wholly  potential,  and  vice  versa ;  while  in  intermediate 
positions  the  one  has  gained  as  much  potential  or  kinetic  energy  as  the  other 
has  lost ;  so  that  the  amount  of  kinetic  energy  is  continuously  equal  to  the 
amount  of  potential  energy,  each  of  these  being  %mv2.  The  whole  energy 
of  a  circular  movement  of  velocity  v,  when  the  moving  body  is  attracted 
towards  the  centre  by  a  force  which  varies  directly  as  the  distance  from  the 
centre,  is  thus  wiu2;  of  which  half  is  kinetic,  half  potential. 

Energy  of  wave-motion.  —  The  energy  of  a  wave-motion  is  equally 
divided  between  the  potential  and  the  kinetic  forms.  Let  us  first  consider 
a  linear  wave  :  at  the  crest  and  at  the  trough  the  whole  energy  is  potential ; 
midway  between  crest  and  trough  the  energy  of  the  particles,  as  they  pass 
through  their  mean  position,  is  wholly  kinetic ;  elsewhere  all  the  particles 
are  symmetrically  and  continuously  losing  potential  and  gaining  kinetic 
energy,  or  gaining  potential  while  losing  kinetic  energy;  the  gains  must  be 
equal  to  the  losses,  for  there  is  no  change  in  the  type  of  vibration,  and  no 
change  in  the  amount  either  of  potential  or  of  kinetic  energy  (friction  being- 
imagined  absent).  Hence  the  whole  energy  of  the  wave  is  divided  into  two 
equal  moieties,  kinetic  and  potential  in  their  respective  forms.  In  a  circular 
wave  the  kinetic  energy  is,  under  the  same  supposition,  invariable  in  its 
absolute  amount,  and  the  potential  energy  bears  to  it  the  same  symmetrical 
fixed  ratio  of  equality.  So  for  tridimensional  waves. 

The  Energy  of  tridimensional  waves  per  cubic  cm.  is  numerically  equal 
to  the  pressure  per  square  cm.,  which  is  exerted  on  the  bounding  surface  in 
consequence  of  the  continued  propagation  of  wave-motion.  This  is  equal,  if 
p  be  the  density  and  v  the  maximum  velocity  of  vibration,  to  £py2  dynes 
per  square  cm.,  or  ergs  per  cubic  cm. ;  or,  if  a  be  the  amplitude,  and  n  the 
frequency,  to  2  7r2a2pn2  dynes  or  ergs,  as  the  case  may  be.  It  is  therefore 
proportional,  for  waves  of  a  given  frequency,  to  a?p. 

In  interference-bands,  due  to  two  equal  sources,  the  amplitudes 
are  alternately  double  and  zero;  the  energies  are  accordingly  alternately 
quadruple  and  zero;  the  average  energy  is  double  of  the  energy  coming 
from  one  source.  There  is  thus  a  redistribution  of  the  energy. 

Rate  of  propagation  of  groups  of  waves.  —  In  every  case  where  the 
rate  of  propagation  of  an  individual  wave  does  not  depend  on  the  wave- 
length,—  e.g.,  sound-waves  in  air,  and,  so  far  as  yet  k,nown,  ether-waves  —  a 
group  of  waves  advances  with  the  same  speed  as  the  individual  waves  do. 
The  case  is,  however,  different  where  the  velocity  of  an  individual  wave 
depends  on  the  wave-length.  For  example,  in  waves  in  deep  water,  where 
the  restitution  of  form  is  due  to  gravity,  the  group  only  travels  half  as  fast 
as  the  individual  waves,  and  the  individual  waves  appear  to  travel  through 
the  group,  dying  away  towards  its  anterior  margin. 


CHAPTER  VI. 

KINETICS. 

GENERAL  PROPOSITIONS  relating  to  the  possible  forms  of  Motion 
find  their  parallel  in  those  relating  to  Forces.  The  formula 
F  =  ma,  already  established,  shows  that  every  force  is  measured 
by  the  Quantity  of  Motion  produced  in  unit  of  time.  In  this 
way  we  see  that  the  truth  of  the  propositions  entitled  the  Par- 
allelogram of  Velocities  and  that  of  Accelerations  involves  that 
of  a  similar  proposition  in  regard  to  Forces.  Two  forces  acting 
on  a  single  particle  produce  the  same  result  as  would  be  pro- 
duced if  a  single  force  were  acting  on  it,  represented  in  magni- 
tude and  direction  by  the  diagonal  of  a  parallelogram,  the 
adjacent  sides  of  which  represent,  in  the  same  respects,  the  two 
simultaneous  forces.  This  is  the  proposition  of  the  "  Parallelo- 
gram of  Forces." 

This  proposition  shows  us  that  if  we  have  two  component 
forces  at  right  angles  to  one  another,  the  square  of  the  resultant 
force  will  be  equal  to  the  sum  of  their  squares.  Refer  to  Fig. 
11  (a)  ;  let  it  be  desired  to  apply  force  to  a  particle  tying  at  A,  so 
as  to  make  it  move  in  the  direction  AC:  the  available  force  is  rep- 
resented in  magnitude  and  direction  by  the  line  AD  :  then  obvi- 
ously this  force  cannot  exert  its  full  effect  in  the  direction  AC  ; 
it  is  only  its  effective  component  in  that  direction  that  can  pro- 
duce any  such  effect :  this  component  is  represented  by  the  line 
AC.  The  other  component  AB,  at  right  angles  to  AC,  can  pro- 
duce no  such  effect.  This  is  an  example  of  the  Resolution  of 
Forces.  Plainly,  we  may  resolve  any  Force  into  two  compo- 
nents at  any  angle  to  one  another,  just  as  we  may  so  resolve 
a  Velocity.  In  tridimensional  space  we  may  resolve  a  force  into 
three  components.  Suppose,  then,  such  a  question  as  the  follow- 
ing:—  A  pull  is  made  in  the  direction  AD  ;  this  pull  is  designed 
to  draw  an  object  through  a  tube  whose  direction  is  AC :  what 
proportions  do  the  component  effective  in  pulling  down  the 

143 


144 


KINETICS. 


[CHAP. 


object,  and  the  lateral  pressure  on  the  walls  of  the  tube,  respec- 
tively bear  to  the  force  applied?  If  Fig.  11  (#)  were  drawn 
with  the  proper  angle  CAD  =  £  between  the  direction  of  applica- 
tion of  the  force  and  the  line  along  which  the  object  is  to  be 
drawn  :  then  the  effective  component  would  be  represented  by 
AC,  and  the  component  producing  lateral  pressure  by  AB. 
But  AB  =  AD  sin  f ;  AC  =  AD  cos  f :  or  — 

AB  :  AD  :  :  sin  f  :  1 

AC  :  AD  :  :  cos  f  :  1. 

Hence,  if  AD  be  taken  as  unity,  AB  and  AC  may  easily  be 
found,  if  the  angle  f  be  known,  by  finding  the  value  of  sin  f 
and  cos  f  in  a  table  of  trigonometrical  ratios.  If  AD  have  any 
other  value  than  unity,  the  values  of  sin  f  or  cos  £  derived 
from  the  tables  must  be  multiplied  proportionately. 

Problem.  —  A  force  of  50  Ibs.  is  applied  to  a  solid  body  drawn  down  a 
fixed  canal  which  the  solid  body  exactly  fits  ;  the  angle  £  which  the  direction 
of  traction  makes  with  the  axis  of  the,  canal  is  29°.  What  is  the  effective 
component  available  in  pulling  the  solid  body?  and  what  is  the  pressure 
produced  by  the  traction-force  on  the  walls  of  the  canal?  In  Fig.  11  (a), 
if  £  be  29°,  AD  represents  the  force  applied,  equivalent  to  the  weight  of  a 
50-lb.  mass ;  AC  represents  the  effective  component  =  AD  x  cos  £ ;  AB 
represents  the  component  at  right  angles  to  AC  —  that  is,  the  detrimental 
pressure  =  AD  x  sin  £  But  cos  29°  =  -8746197  ;  sin  29°  =  -4848096.  The 
effective  component  is  thus  -8746197  of  the  force  applied  —  i.e.  it  is  equal 
to  the  weight  of  43-731  Ibs. ;  the  detrimental  pressure  is  24-2405  Ibs. 

It  seems  rather  surprising  that  the  effective  and  lateral  components 
together  should  appear  to  be  so  much  greater  than  the  original  force  applied. 
But  geometrically  this  is  the  same  thing  as  to  say  that  two  sides  of  a  triangle 
are  greater  than  the  third  side  ;  and  further,  we  have  already  seen  that  there 
is  no  law  of  the  Conservation  of  Force,  though  there  is  a  law  of  the  Con- 
servation of  Energy.  The  principle  of  the  Conservation  of  Energy  is 

maintained,  for  the 
work  done  in  enforc- 
ing the  detrimental 
press  ure,together  with 
the  work  done  in  pull- 
ing the  body  down  the 
canal,  is  equal  to  the 
energy  imparted. 

Experimental 
proof  of  the  Paral- 
lelogram of  Forces. 
—  A  cord,  two  masses 
of  12  and  13  Ibs.  re- 
spectively, a  pulley, 
and  a  dynamometer  are  arranged  on  a  beam  as  shown  in  Fig.  78.  The  string 
can  be  adjusted  so  that  the  angle  BAG  may  have  a  wide  range  of  values : 
for  every  position  there  is  a  corresponding  stress  on  the  dynamometer. 


Fig.78. 


vi.]  EQUILIBRIUM   OF  FORCES.  145 

When  AB  and  AC  are  fixed  at  right  angles  to  one  another,  the  spring 
of  the  dynamometer  is  pulled  out  to  the  same  extent  as  it  would  have  been 
by  the  weight  of  a  5-lb.  mass.  The  dynamometer  may  be  replaced  by  a  5-lb. 
mass  suspended  over  a  pulley;  in  that  case  the  cord  and  the  three  sus- 
pended masses  would  so  adjust  themselves  that  the  angle  A  would  be  a 
right  angle.  Here  52+122=132,  or  25  +  144  =  109  j  and  the  law  is  confirmed. 

In  a  similar  way,  the  propositions  known  as  the  Triangle  of 
Velocities,  the  Polygon,  etc.,  are  replaced  in  Kinetics  by  the 
Triangle,  the  Polygon,  etc.,  of  Forces ;  and  the  resultant  force 
is  the  missing  side  of  the  triangle  or  polygon,  of  which  all  the 
sides  except  the  missing  one  represent  the  various  forces  acting, 
in  precisely  the  fashion  already  studied  under  Kinematics. 

In  tridimensional  space  we  have  the  propositions  of  the  parallelepipedon, 
the  skew-polygon,  etc.,  of  forces,  parallel  to  similar  propositions  under 
Velocity  and  Acceleration. 

Equilibrium  of  Forces.  —  There  here  emerges  an  important 
and  difficult  question  of  nomenclature.  It  may  appear  not 
strictly  consistent  with  the  definition  of  Force  to  speak  of  two 
forces,  equal  and  opposite,  balancing  one  another  and  produc- 
ing no  effect,  as  being  still  two  distinct  Forces ;  for  the  essence 
of  Force  is  acceleration  observed.  But  it  is  not  more  incon- 
sistent than  it  is  to  speak  of  two  equal  and  opposite  simul- 
taneous velocities  being  equivalent  to  rest,  or  of  two  equal  and 
opposite  simultaneous  accelerations  resulting  in  no  change  of 
velocity.  There  is  no  doubt  that  we  are  entitled  so  to  speak ; 
and  if  so,  then  we  are  entitled  to  speak  of  equal  and  opposite 
simultaneous  forces  balancing  one  another,  for  propositions 
concerning  forces  are  strictly  parallel  to  those  concerning  veloci- 
ties or  accelerations. 

The  inconsistency  finally  vanishes  when  we  observe  that 
the  Forces  which  apparently  destroy  one  another  are  not  physi- 
cal entities,  but  mental  artifices,  "paper  bullets  of  the  brain;" 
and  that  we  do  not  really  think  of  them  as  simultaneous:  we 
look  first  to  the  one  and  then  to  the  other,  and  see  that  what 
the  one  does  the  other  undoes;  in  the  end  there  is  thus  no 
result.  The  imaginary  Forces  are  in  equilibrium;  the  actual 
body  is  in  a  condition  of  Stress  of  some  kind.  But  "  Force  " 
is  a  compendious  phrase ;  its  use  saves  many  words ;  and  bear- 
ing this  in  mind,  we  may  admit  that  it  is  often  more  convenient 
to  study  the  balanced  forces,  each  in  its  turn,  than  it^is  to  con- 
sider the  actual  stressed  condition  of  the  body :  and  thus  while 
we  cannot  speak  of  a  single  force  unless  there  be  actual  motion 


146  KINETICS.  [CHAP. 

produced  or  checked,  we  may  allow  ourselves  to  speak  of  the 
equilibrium  of  the  several  forces  acting  upon  a  body  at  rest. 

If  we  suppose  a  stone  perched  upon  a  height,  we  say  that  the  earth 
attracts  it  with  a  definite  and  measurable  Force.  Yet  there  is  no  move- 
ment. Here,  however,  we  have  really  a  downward  Action  of  the  earth,  an 
upward  Reaction  of  the  support ;  and  the  phenomenon  is  an  equilibrium  of 
two  "  forces,"  of  which  we  may  confine  our  attention  to  one  only,  the  down- 
ward Force  of  Gravitation. 

So  we  may  say  that  an  electrified  body,  freely  suspended  at  a  certain 
distance  from  another  electrified  body,  is  attracted  or  repelled  with  a  definite 
Force ;  and  so  it  is,  as  we  may  see  if  it  be  free  to  move ;  but  if  it  be  not  free 
to  move,  the  attraction  or  repulsion  is  still  the  same,  but  is  now  balanced  by 
an  equal  and  opposite  Reaction  of  the  support ;  and  this  Force  and  Reaction 
together  correspond  to  a  Stress,  a  mutual  Pressure  or  Tension,  between  the 
body  attracted  or  repelled  and  its  support.  Hence  we  may  say  that  the 
body  is  in  every  such  case  impelled  to  move  (repelled  or  attracted)  with  or 
by  a  certain  definite  Force. 

Centre  of  Figure,  Centre  of  Mass,  or  Centre  of  Inertia.  — 

We  have  already  seen  that  a  rigid  body,  of  which  the  several 
particles  are  subject  to  accelerations  which  are  equal  and  par- 
allel to  one  another,  moves  as  if  concentrated  at  its  centre  of 
figure,  and  as  if  this  were  subject  to  a  single  acceleration.  A 
body  may  thus  be  acted  upon  by  parallel  forces  affecting  its 
particles,  the  result  being  the  same  as  if  a  single  force  had  acted 
at  the  centre  of  figure ;  while  conversely,  if  a  single  force  act 
at  the  centre  of  figure,  the  result  is  to  impart  parallel  and 
equal  accelerations  to  all  the  particles,  and  thereby  to  effect 
a  Translation  of  the  body.  Hence  most  of  the  propositions  of 
Kinematics,  which  describe  the  motion  of  a  single  point,  may 
be  transferred  to  Kinetics,  not  only  as  relating  to  the  move- 
ments of  single  particles,  but  also  as  relating  to  translation  of 
material  bodies. 

If,  however,  the  body  be  not  uniform  in  density,  then,  as  in  the  case  of 
the  moon,  the  centre  of  mass  may  not  coincide  with  the  centre  of  figure. 

Inertia  of  Matter.  — If  in  any  system  of  bodies  there  be  no 
force  acting,  the  formula  ~F  =  ma  =  Q  shows  that  a  =  0,  that 
there  is  no  acceleration :  hence  if  there  be  no  force  acting,  there 
is  no  change  in  the  speed  with  which  a  body  is  moving,  or  in 
its  state  of  rest,  as  the  case  may  be;  in  other  words,  Matter 
has  Inertia.  This  is  Newton's  first  law  of  motion,  and  it 
appears  to  be  here  derived  from  the  formula ;  but  it  will  be 
remembered  that  the  formula  was  itself  derived  by  implication 
from  that  law. 


vi.]  INERTIA.  147 

Coefficient  of  Inertia.  —  If  a  body  have  translational  momentum  m'  v', 
then,  in  order  to  bring  it  to  rest  in  time  t',  the  mean  negative  accelera- 
tion required  is  —  (v'/*')>  and  the  mean  retarding  force  required  is 
F=(m'v'/t').  The  body  offers  a  mean  Resistance  to  Stoppage 
equal  to  (m'v'/*').  If  the  required  acceleration  (v"/'"=v'/O  be  the 
same  in  another  body,  but  the  momentum  m"v"  different,  the  resistance  to 
stoppage  is  now  m"v"/t"=m"v'/t'.  Generally,  the  resistance  to  change  of 
momentum  is  thus  equal  to  the  product  of  m  into  the  necessary  acceleration ; 
and  m  is  the  Coefficient  of  Translational  Inertia  for  a  body 
whose  Mass  is  m. 

Examples  of  Inertia.  —  Examples  of  this  abound.  Col- 
lisions between  ships  and  between  trains,  which  do  not  stop  if 
there  be  not  sufficient  retarding-force  at  command ;  trains  pass- 
ing stations  when  their  speed  is  great  and  the  rails  are  slippery ; 
a  person  falling  off  the  stern  of  a  boat  or  the  back  of  a  car, 
when  the  vehicle  makes  a  sudden  movement  forwards  in  which 
his  body  does  not  participate ;  the  onward  motion  retained  by  a 
rider  when  his  horse  stops  under  him;  the  jerk  received  by 
a  horse  suddenly  starting  in  order  to  set  in  motion  a  heavy 
waggon ;  when  the  waggon  is  running,  if  the  horse  suddenly 
stop,  he  is  bruised,  for  the  massive  waggon  does  not  stop  at 
once ;  a  greyhound  chasing  a  hare  is  carried  forward  and  cannot 
stop  or  turn  his  path  instantly  at  the  spot  where  the  hare 
doubled  or  turned  abruptly  from  her  course ;  the  inertia  of  the 
dust  of  a  carpet,  when  the  carpet  is  beaten — the  carpet  moves 
forwards  at  each  blow,  but  the  dust  remains,  and  is  thus  sepa- 
rated from  the  carpet  and  blown  away  by  the  wind ;  the  inertia 
of  dust  when  it  is  shaken  off  a  book  —  the  book  and  the  dust 
are  made  to  describe  together  a  rapid  movement  in  the  air — 
the  book  is  suddenly  arrested  by  a  smart  blow,  while  the  dust 
does  not  stop  but  moves  onwards ;  the  inertia  of  the  snow 
which  in  the  same  way  is  kicked  off  one's  boots  —  the  boot  is 
suddenly  stopped,  but  the  snow  goes  on,  and  is  thus  shaken  off; 
the  inertia  of  loose  grain  cargo  in  a  ship — it  acquires  a  certain 
velocity  when  the  ship  rolls,  and  does  not  stop  when  the 
ship  arrives  at  its  normal  limit,  but  pours  on  so  as  sometimes 
to  make  the  ship  roll  beyond  the  limits  of  safety ;  the  oscilla- 
tions of  mercury  in  an  ordinary  barometer  at  sea,  the  mercury 
being  jerked  up  by  each  roll  of  the  ship ;  the  jerking  of 
the  blood  against  the  valves  of  the  blood-vessels  of  a  bad  rider ; 
the  inertia  of  the  mercury  in  a  mercury  manometer  used  to 
investigate  fluid-pressure  —  the  variations  in  the  height  of  the 
mercurial  column  being  greater  than  the  real  variations  in 


148  KINETICS.  [CHAP. 

the  pressure,  for  the  mercury  does  not  stop  moving  when 
the  fluid-pressure  ceases  to  rise  or  fall;  the  inertia  of  a 
mass  suspended  on  a  spring-balance,  by  reason  of  which  the 
weight  is  apparently  increased  when  the  balance  is  suddenly 
raised,  and  lessened  when  it  is  suddenly  lowered;  the  inertia 
of  water  in  house  water-pipes  if  it  be  set  to  run  and  then  sud- 
denly stopped  —  the  water  is  compressed  against  itself  and  a 
violent  jerk  is  produced,  which  is  utilised  in  the  hydraulic  rain ; 
the  inertia  of  water  in  the  case  of  the  water-supply  of  the 
locomotive  engines  of  passenger  express  trains  on  the  L.  and 
N.W.  Railway  system — the  engine  puts  down  a  tube,  the  lower 
end  of  which  acts  as  a  scoop  for  the  water,  which  tends  to 
remain  in  its  trough  on  the  ground  between  the  rails  and  at  rest 
relatively  to  the  ground ;  but  this  being  equivalent  to  a  back- 
ward movement  relatively  to  the  engine,  the  water  slips  up  the 
inclined  tube  into  the  tender  if  the  train  be  moving  at  sufficient 
speed. 

There  are  some  further  remarkable  consequences  of  the 
inertia  of  matter.  A  body  may  be  struck  or  pressed  so  suddenly 
that  it  stands  practically  at  rest  during  the  time  that  the  blow 
is  being  spent  on  it,  and  it  may  be  crushed  or  broken  by  such  a 
blow.  A  candle  can  thus  be  fired  through  a  board;  a  bullet 
may  be  fired  through  glass  without  cracking  it ;  and  a  cannon- 
ball  through  a  half-open  door  without  opening  it.  Water  will 
reflect  a  cannon-ball  or  flatten  a  bullet.  A  dynamite-charge, 
exploded  upon  a  stone,  developes  pressure  between  the  stone 
and  the  air,  so  suddenly  that  the  stone  is  shattered  before  the 
air  has  time  to  move  away.  A  grain  of  corn  or  a  granule  of 
gold-quartz,  if  thrown  up  into  the  air  and  struck  a  blow  by  an 
iron  bar  moving  at  the  rate  of  about  180  feet  a  second,  will 
be  crushed  by  compression,  and  will  be,  by  a  succession  of  such 
blows,  very  effectively  pulverised.  Milling  machinery  has  been 
constructed  on  this  principle.  A  bullet  in  a  gun,  though  free 
to  move  onwards,  is  crushed  against  itself  before  it  fairly  starts, 
so  that  the  soft  lead  is  moulded  into  the  grooves  of  the  rifle- 
barrel  by  the  rapidly-applied  pressure,  due  to  the  explosion  of 
quick-burning  gunpowder.  V 

Another  example  is  afforded  by  that  instrument  with  the  aid  of  which 
M.  Rosapelly*  investigated  the  movements  of  the  larynx  during  the  emis- 
sion of  sounds.  A  heavy  mass  of  metal  is  suspended  in  a  light  frame-work 
which  is  tied  over  the  larynx :  as  this  mass  cannot  at  once  participate  in 

*  Trav.  du  Laboratoire  de  M.  Marey,  1876. 


vi.]  INERTIA.  149 

the  rapid  movements  which  the  vibration  of  the  larynx  communicates  to 
the  light  framework,  it  forms  a  kind  of  fixed  point,  and  the  light  frame- 
work, as  it  vibrates  in  contact  with  the  skin  over  the  larynx,  may  strike  the 
heavy  mass  a  series  of  blows ;  these  may  cause  an  electric  current  to  be 
alternately  made  and  broken  ;  the  number  and  frequency  of  these  interrup- 
tions may  be  registered  on  an  appropriate  recording-instrument. 

Further,  the  inertia  of  matter  is  a  property  of  retaining 
whatever  motion  an  object  has,  and  that  in  a  plane  fixed  in 
space,  without  reference  to  the  movements  of  surrounding  ob- 
jects, unless  these  are  so  connected  with  it  as  to  be  able  to 
affect  its  motion.  A  hammock  retains  its  position  in  space 
independently,  in  the  main,  of  the  pitching  and  rolling  of  the 
ship.  The  statement  would  be  approximately  accurate  that  the 
hammock  does  not  swing  in  the  ship,  but  that  the  ship  swings 
enclosing  the  hammock,  which  may  for  any  short  period  of  time 
be  regarded  as  moving  onward  in  space  with  the  average 
velocity  of  the  ship,  but  independently  of  it.  A  long  and  heavy 
pendulum  set  to  swing  in  one  plane,  and  connected  by  a  very 
slender  attachment  to  the  roof  of  the  building  in  which  it  is 
suspended,  will  swing  in  the  same  plane  in  absolute  space 
though  the  earth  rotate  under  it :  the  apparent  result  is,  that 
the  plane  in  which  the  pendulum  swings  gradually  alters  its 
aspect,  so  that  the  pendulum  swings  successively  in  every  pos- 
sible direction.  The  real  state  of  the  case  is  not  that  the 
heavy  pendulum  alters  its  direction  of  oscillation,  but  that  the 
earth  rotates  or  has  a  component  of  rotation  under  the  pendu- 
lum, except  at  the  equator.  If  a  heavy  wheel  be  set  in  motion, 
it  will  in  the  same  way,  if  it  can  rotate  for  a  sufficiently  long 
time,  show  the  same  phenomenon,  for  it  tends  to  continue  to 
rotate  in  the  same  plane  in  space. 

Momentum.  —  The  product  my  of  m,  the  mass  of  a  moving 
body,  into  v,  its  velocity  in  any  given  direction,  is  called  the 
Momentum  of  the  body  in  that  direction. 

If  a  shell  explode,  its  fragments  form  a  system  of  bodies 
moving  at  different  velocities.  The  average  velocity  of  the 
centre  of  mass  of  the  whole  system  is,  however,  unchanged: 
some  fragments  travel  with  a  greater,  some  at  a  less  velocity 
than  that  with  which  the  shell  had  travelled  before  the  explo- 
sion ;  but  the  mass  m  is  unchanged  though  differently  arranged, 
the  mean  velocity  of  the  system  is  the  same  as  that  of  the 
original  shell,  and  thus  the  momentum  of  the  whole  system,  in 
the  direction  of  movement  before  the  explosion,  is  the  same  after 
explosion  as  before  it. 


150  KINETICS.  [CHAP. 

Impact.  —  If  there  be  two  inelastic  bodies,  of  masses  ml  and 
mn  respectively,  of  which  the  first  moves  with  velocity  vy,  while 
the  second  is  at  rest :  if  the  moving  one,  whose  momentum  is 
wyvy,  strike  the  other,  it  will  divide  its  momentum  with  that 
other ;  it  itself  will  travel  more  slowly,  while  the  other  is  set 
in  motion  ;  but  the  two  will  travel  together  with  a  common 
velocity  v,  in  the  original  direction  of  the  mass  mt. 

The  whole  mass  moving  with  this  new  velocity  v  is  (m,  +  m/y)  ;  its 
momentum  is  equal  to  the  original  w/v/ ;  hence  the  velocity  v  may  be 
found  by  stating  this  equality  of  momenta  in  the  form  of  the  equation  — 

(m,  +  wi;/)v 


.'.  V    =• 


If  the  mass  mn  be  large  in  comparison  with  m^  the  new 
velocity  v  is  much  less  than  vy.  If  a  man  lie  with  an  anvil  on 
his  chest,  and  if  the  anvil  be  struck  a  blow  with  a  hammer  rela- 
tively not  too  heavy,  the  person  lying  down,  if  he  can  support 
the  anvil,  will  not  be  much  affected  by  the  blow,  for  the  move- 
ment imparted  to  the  anvil  will  be  slow  as  compared  with  that 
of  the  hammer. 

Let  the  two  inelastic  masses  be  ml  and  mni  moving  with  the  respective 
velocities  vy  and  v/y,  and  together  moving  after  impact  with  the  velocity 
v;  the  respective  momenta  of  the  masses  before  impact  were  wi-yvy  and 
m/yvy,  ;  that  of  the  conjoined  mass  after  impact  is  (m,  -f  mlt)  v.  Hence 

mv  +  m    v=m  +  m      v. 


It  was  found  experimentally  by  Newton  that,  in  such  a  case,  the  momentum 
lost  by  one  body  was  equal  to  that  gained  by  the  other.  To  express  this 
algebraically,  if  M  represent  the  momentum  gained  by  one  and  lost  by  the 
other, 

Tfyv—  7WyVy     =  M.     (2.) 

mu  v//  -  w?/yv=M.     (3.) 
From  either  of  these,  with  the  aid  of  equation  (1)  we  find 


Apparent  loss  of  Energy.  —  In  this  case  the  kinetic   energy   after 
impact 


1  mass  X  v*  =  i  (m,  +  »,,)(m'T'  +  ">"*"}*=  ^^  +  "'"V")') 

"\     mt  +  mn      ]  m,  +  mlt        J 

is  less  than  the  sum  of  the  kinetic  energies  before  impact  (which  were 
^m/v/2  and  Jwi^v,,2  respectively).  This  is  not  true  if  vy  be  equal  to  v//5  but 
in  that  case  the  two  bodies  would  be  travelling  in  the  same  direction  with 


vi.]  IMPACT.  151 

equal   speed,  and  the  one  could  not  overtake  and  strike  the  other.     The 
energy  which  has  apparently  disappeared  has  assumed  the  form  of  Heat. 

Impact  of  Elastic  Bodies.  —  We  may  here  anticipate  a 
statement  of  the  nature  of  Elasticity  so  far  as  to  say  that  a 
perfectly  elastic  body,  possessed  of  a  certain  amount  of  kinetic 
energy,  and  striking  a  perfectly  rigid  body,  will  rebound,  and 
will  possess  as  much  kinetic  energy  after  the  impact  as  before  it  ; 
for  it  leaves  the  rigid  body  with  a  velocity  equal  to  that  with 
which  it  had  approached  it.*  The  mass  and  the  velocity  being 
numerically  unchanged,  the  momentum  is  numerically  equal 
after  impact  to  that  before  it;  but  as  it  is  no  longer  my  but 
m  x  (  —  v)  =  —  my,  it  has  become  negative,  and  has  therefore 
altered  by  an  amount  equal  to  2my.  If  the  body  be  imperfectly 
elastic,  so  that  the  velocity  is  not  completely  regained,  it  is 
found  experimentally  that  it  returns  with  a  certain  fraction, 
X,  of  its  original  velocity  (this  fraction,  X,  being  called  the 
coefficient  of  restitution),  and  the  change  of  velocity  is  not  2v 
but  (1  +  X)  v  ;  and  its  momentum  has  become  negative  and 
=  —  X  •  mv,  so  that  it  has  changed  by  the  amount  (1  +  X)  my. 

If  two  masses  m,  and  mlt,  moving  with  velocities  vy  and  v/;  in  the  same 
direction,  and  formed  of  such  material  that  the  coefficient  of  restitution 
between  them  is  X,  strike  one  another,  they  will,  after  impact,  travel  with 
velocities,  say,  v/  and  vy/.  The  momentum  gained  by  the  one  is  equal  to 
that  lost  by  the  other  ;  but  it  is  not  equal,  as  it  is  in  the  case  of  inelastic 

bodies  where  X  =  0,  simply  to  m'm"^"  ~  V/%   but  to  (1  +  X)  x  that  quan- 

m/  +  mn 

tity.     This  equality  of  momenta  is  expressed  by  the  equations  — 
(1)    Gained  by  m,  ;  n»,v/  -  ro,v,  =  (1  +  X)  m'm^''  ~  v'>. 

/Aiy       -f-       //<yy 

(2)   Lost  bym,,;  m/yvy/-  m,,v,/  =  (l  +  X)  TO'm"(v"  "  ^ 

'«/      ~T      "«•// 

Whence 


Take,  as  a  particular  instance,  the  case  in  which  the  elasticity  is  perfect 
or  the  restitution  complete  (i.e.  X  =  1),  and  the  balls  which  strike  one 
another  are  of  equal  weight,  so  that  m,  =  mn  ;  then  v/  =  vy/  and  vy/  =  v,  ; 
i.e.  :  — 

*  Such  is  the  elementary  theory.  There  is,  however,  no  perfectly  etastic  body, 
and  even  if  there  were,  a  part  of  its  energy  would  necessarily  be  spent,  upon  impact, 
in  setting  up  vibrations  in  it,  and  the  speed  of  rebound  could  never  come  up  to  the 
theoretical  limit. 


152  KINETICS.  [CHAP. 

Two  equal  and  perfectly  elastic  balls,  striking  one  another 
directly  in  the  line  joining  their  centres,  exchange  their  velocities, 
and  that  whether  they  meet  or  overtake  one  another. 

Oblique  Impact.  —  If  a  ball  strike  a  rigid  surf  ace  obliquely, 
its  motion  relative  to  that  surface  may  be  resolved  into  two  com- 
ponents :  one  parallel  to  it,  which  is  not  affected  by  the  impact  ; 
the  other  at  right  angles  to  it,  which,  after  impact,  will  be 
wholly  or  in  part  restored  in  the  reverse  direction.  Reference 
to  Fig.  60  will  show  that  if  X  =  1,  the  angle  of  reflexion  will  be 
equal  to  the  angle  of  incidence  ;  while  if  X  be  less  than  unity, 
the  angle  of  reflexion  will  be  proportionately  less  acute,  to  an 
extent  easily  determined  by  construction. 

If  the  oblique  impact  be  between  two  balls,  the  investigation  is  based  on 
similar  principles.  Take  a  line  joining  their  centres  of  figure  ;  in  a  direc- 
tion at  right  angles  to  this  line  the  motion  is  unaffected  by  the  impact,  and 
the  component  in  this  direction  will  be  the  same  after  impact  as  before  it  ; 
in  the  line  joining  the  centres  of  figure,  the  velocities  v/  and  v//  may  be 
found  as  in  the  preceding  discussion.  By  compounding  the  component 
velocities  after  impact,  the  resultant  velocities  and  their  directions  may  be 
found.  In  practice,  as  the  two  balls  are  passing  one  another  in  contact, 
friction  between  their  surfaces  causes  a  relative  delay  of  one  aspect  of  each, 
and  causes  rotation  of  the  balls  ;  the  energy  necessary  for  this  is  taken  from 
that  theoretically  available  for  the  direct  translational  movements  of  the 
balls  as  wholes. 

Energy  in  impact  of  Elastic  Bodies.  —  In  the  case  of  per- 
fectly elastic  bodies,  the  energy  after  impact  is  equal  to  that 
before  it;  ^w/v/2  4-  i^y/v,/2  =  ^/vy2  +  ^mltyn2.  If  X  be  less 
than  1,  there  is  apparent  loss  of  Energy,  which  has  assumed 
some  other  form  than  that  of  motion  of  the  mass.  If  a  horse 
with  loose  traces  rush  forward  and  jolt  a  car,  the  energy  which 
disappears  is  wasted  in  the  form  of  heat,  or  deleteriously  spent 
in  disintegration  of  the  materials  of  the  car,  or  in  bruising  the 
animal. 

Accelerated  Motion.  —  The  discussion  of  the  accelerated 
motion  of  a  particle  moving  with  constant  acceleration  —  i.e. 
under  the  continuous  influence  of  a  constant  force  in  the  line 
of  the  existing  motion  —  has  already  (pp.  69-70)  led  us  to  the 
formulse  — 


=  J  (v0 


s  = 


',  =  V0  ±  at, 

00 

t  =  V0«  ±  |a«2, 

(ii.) 

•?  =  V02  ±  2as, 

(iii.) 

=  (v,2-V)  + 

2a,  (iv.) 

vi.]  ACCELERATED   MOTION.  153 

where  V0  represents  the  velocity  of  a  particle  at  the  beginning, 
and  vt  that  at  the  end  of  time  £,  s  the  space  traversed  during 
that  time,  and  a  the  acceleration  (positive  or  negative,  as  the 
case  may  be)  per  unit  of  time.  The  most  familiar  examples  of 
this  kind  of  movement  are  those  in  which  bodies  exposed  to  the 
constant  influence  of  gravity  fall  with  constantly-increasing 
speed. 

Problems. 

1.  A  train  travelling  at  50  miles  an  hour  comes  into  collision  with  a  fixed 
obstacle  and  is   abruptly  stopped  :  the  passengers  receive  a  blow.     What 
height  must  one  fall  in  order  to  receive  a  similar  blow?  —  Ans.   50  miles  an 
hour  =  73-3  feet  per  second.     A  body  falling  from  rest  (v0  =  0)  acquires  a 
speed  of  73-3  feet  per  sec.  in  falling  83-51  feet,  for 

v,2  =  v02  ±  2as  =  v02  +  2gh.   (iii.)  ; 
(73-3)2  =  0  +  (2  x  32-2  x  h)  ;  .-.  h  =  83-51. 

A  blow  in  a  collision  at  50  miles  an  hour  is  equivalent  to  a  blow  received  in 
consequence  of  a  fall  of  83-51  feet  ;  for  a  body  which  has  fallen  83-51  feet 
is  in  consequence  travelling  at  50  miles  an  hour  at  the  instant. 

2.  A  ball  weighing  5  ounces  is  hurled  upwards.     It  is  supposed  that 
while  it  is  in  the  hand  it  is  swung  through  4  feet  ;  the  thrower  during  this 
swing  continuously  exerts  an  accelerating  pressure  on  it.     This  pressure 
must  be  equivalent  to  the  effort  which  would  be  put  forth  in  raising  some 
definite  mass  in  the  same  position  of  the  body.     What  is  this  mass  if  the 
ball  rise  100  feet? 

The  ball  leaves  the  hand  with  an  unknown  upward  velocity  v0;  it 
rises  through  a  vertical  height  h  =  100  feet  against  a  force  (gravity),  the 
negative  or  downward  acceleration  produced  by  which  is  32-2  ft.-per-sec. 
per  second  ;  it  comes  for  an  instant  to  rest  (vt  =  0)  at  the  top  of  its  course. 

v,2  =  v02  ±  2as  =  v02  -  2gh.   (iii.) 

0  =  v02  -  (2  x  32-2  x  100). 
v02  =  6440.       v0  =  80-25. 

The  question  thus  becomes  —  Under  the  influence  of  a  force  F  acting 
upwards  through  4  feet,  an  upward  velocity  80-25  feet  per  second  is  imparted 
to  a  mass  m  =  T%  lb.  Find  P  ;  or,  since  P  =  ma.  and  m  =  T\,  find  a. 

v,2  =  v02  ±  2as  (iii.)  ;  v0  =  0  ;  a  is  positive. 

.-.  vf2  =  2as  ;  but  v,  =  80-25,  the  velocity  with  which  the  ball  leaves  the  hand. 
6440  =  2a  x  4  =80.       a  =  805  ft.-per-sec.  per  second. 
F  =  mo.  =  226  T9^  British  units  of  force. 
But  wt.  of  1  Ib.-mass  =  32-2  Brit,  units  of  force. 


P  =  wt.  of  Ib.-masses  =  wt.  of  7|f  Ibs. 

tKS*2 

The  effort  then  is  the  same  as  that  put  forth  in  lifting  a  mass  of  7  Ibs. 
13  oz.  against  gravity  ;  the  time  is  0-0997  sec.,  for  t  =  2s  -4-  (v()  +  vt)  ;  and 
the  activity  =  Ts/t  =  7-8125  Ibs.  x  4  ft.  -s-  0-0997  sec.  =  313-475  ft.-lbs.  per 
sec.  =  0-57  horse-power. 


154  KINETICS.  [CHAP. 

3.  A  shot  is  fired  vertically  from  a  gun  whose  barrel  is  30  inches  long  : 
it  rises  half  a  mile.     Compare  the  acceleration  of  a  body  falling  under  the 
influence  of  gravity  with  that  under  which  the  bullet  acquires  such  velocity 
in  the  space  of  30  inches. 

First  find  the  velocity  of  the  shot  as  it  leaves  the  gun.  In  its  course  in 
the  air  (friction  being  entirely  'neglected)  it  commences  with  the  unknown 
velocity  v0,  traverses  height  h  =  2640  feet  against  gravity  which  produces 
a  vertically  downward  acceleration  a  =  g  =  —  32-2,  and  comes  to  rest 
(v,  =  0)  for  an  instant. 

v,2  =  v02  ±  2as  =  v02  -  2gh.   (iii.)  _ 

0  =  v02  -  (2  x  32-2  x  2640).    v0  =  V170016  =  412-3. 

In  the  gun-barrel  a  velocity  of  412-3  feet  per  second  is  acquired  within  a 
space  s  =  2|-  feet.  We  want  to  know  the  acceleration.  Now  412-3  ft.  per 
sec.  =  vt  ;  v0  =  0  ;  B  =  2-5  ;  a  is  unknown,  but  is  positive. 

v,2  =  v02  ±  2as.       (iii.) 
(412-3)2  =  0  +  (2a  x  2*). 
170016    =  5a.     a  =  34003-2  ft.-per-sec.  per  second. 

If  the  barrel  were  long  enough  to  expose  the  projectile  to  the  influence  of 
such  an  accelerating  force  for  a  whole  second,  this  velocity  would  be  acquired, 
and  the  barrel  would  be  17001-6  feet  long  ;  but  the  time  in  the  barrel  is 
t  =  v,/a  =  0-012  second  only. 

But  gravity  produces  an  acceleration  of  32-2  ft.-per-sec.  per  second.  Hence 
the  acceleration  due  to  the  gunpowder  is  greater  than  that  of  gravity  in  the 

OJAAO.Q 

ratio  of       Q     *",  or   1056  :  1.     The  force  exerted  by  the  powder  =  ma  = 

34003-2m,  and  in  the  case  of  an  ounce-bullet  would  be  equal  to  2125-2 
British  units  of  force. 

4.  With  what  "  force  "  will  a  10-lb.  mass  falling  100  feet  strike  at  the 
end  of  its  course  ?     As  it  stands,  this  question  is  devoid  of  sense,  for  it  does 
not  specify  the  time  during  which  the  momentum  is  changed  on  impact. 
If  the  body  struck  were  rigid,  and  the  falling   mass  perfectly  elastic,  it 
would,  apart  from  vibrations,  rise  on  rebounding  to  an  equal  height  of  100 
feet.     During  the  impact  it  must  have  come  to  rest.     Let  us  arbitrarily 
assume  it  to  have  taken  ^W  sec>  ^°  come  to  rest,  and  an  equal  period, 
y^  sec.,  to  gain  its  upward  initial  velocity  :  this  upward  initial  velocity  is 
80-25  feet  per  second,  for 

v,2  -  v02  ±  2as  =  v02  -  2gh. 
0  =  v02  -  (2  x  32-2  x  100). 

v0  =  80-25. 

The  question  thus  becomes  —  What  is  the  mean  pressure  between  the  body 
which  has  fallen  and  that  on  which  it  falls,  if  a  speed  of  80-25  feet  per 
second  can  be  arrested  or  developed  by  it  in  ^¥  sec.?  The  answer  is  — 
Since  vt  =  at  :  v,  =  80-25  ;  t  =  ^V  <y  J  •'•  a  =  160,500  ;  and  F  =  ma  =  1,605,000 


British  units  of  force  =  the  wt.  of    '    „  '        =  49844-7  Ibs. 

c>2'2 

5.  A  ball  weighing  10  Ibs.  falls  from  a  height  of  100  feet  on  a  rigid 
floor.  It  is  flattened  to  the  extent  of  -£$  inch,  measured  in  the  direction  of 
its  motion  :  it  recovers  its  form  and  rebounds.  What  is  the  time  taken  to 
bring  the  ball  to  rest,  and  what  is  the  mean  total  pressure  between  the  ball 


vi.]  ACCELERATED   MOTION.  155 

and  the  floor  on  which  it  falls?    Here  a  velocity  of  80-24954  ft.  per  sec.  is 
arrested  in  the  space  of  ^  inch :  what  is  the  retarding  acceleration  a?  what 
is  the  corresponding  pressure  P  ?  what  is  the  time  t  f    It  is  again  assumed 
that  the  ball  is  perfectly  elastic,  and  that  there  are  no  vibrations. 
V  -  v02  -  2as.     (iii.) 

0  =  (80-24954)2  -  (2a  x  &  inch)  =  6440  -  (2a  x  ^  foot.) 
a  =  £  (6440  x  360)  =  1,159,200  ft.-per-sec.  per  second. 
Again,  P  =P  =  ma  =  1,159,200  x  10  =  11,592,000  British  units  of  force, 

=  the  weight  of  (11,592,000  -5-  32-2)  =  360,000  Ibs.  =  mean  pressure. 
Lastly,  v,  =  at  (i.);  80-24954  =  1,159,200*;  ...  t  =  T¥       sec. 


The  Principle  of  Moments.— In  Fig.  79  a  linear  body  is 
poised  at  the  point  F ;  at  A  suppose  a  force  F  equal  to  10  units, 
at  B  a  parallel  force  equal  to  1  unit.  The  former,  acting  alone, 
would  turn  the  bar  round  F  through  an  angle  0,  and  the  work 
done  at  A  by  the  force  F  is  equal  to  Fs  =  10  units  x  Aa  = 

id 

i 


0 


i 


10  AFsin  6.  The  latter  force  applied  at  B  would  turn  the  bar 
round  F  through  an  angle,  say  </>,  and  would  in  producing  rota- 
tion do  work  =  1  x  FB  sin  <£.  The  work  done  by  the  force 
applied  at  A  during  any  small  displacement  is  opposite  in  sense 
to  that  done  by  the  force  applied  at  B.  Together  they  may 
balance  one  another,  and  produce  equilibrium.  If  the  bar  be 
rigid,  <f>  —  6.  If  the  one  force  raise  as  much  as  the  other 
depresses  every  point  of  the  bar,  there  is  equilibrium.  As 
regards  rotation  round  F,  the  forces  are  of  opposite  sign ;  they 
are  accordingly  4-  10  at  A,  tending  to  produce  positive  rotation 
in  the  direction  opposed  to  that  of  the  hands  of  a  watch,  and  —  1 
at  B,  tending  to  produce  a  negative  rotation.  Hence,  if  there 
be  equilibrium,  the  work  done  by  force  =  +  10  acting  at  A 
and  that  done  by  force  —  1  acting  at  B  are  together  =  0. 

(10  x  AF  sin  0)  +  (  -  1  x  BF  sin  6)  =  0 ;  (1.) 
(10  x  AF)  +  (-l  x  BF)  =  0; 

or,  generally,  if  the  parallel  forces  be  P  and  Q,  — 

(P  x  AF)  +  (Q  x  BF)  =  0; 
or  P  :  Q  :  :  BF  :  AF. 


156  KINETICS.  [CHAP, 

The  parallel  forces  which  balance  one  another  are  inversely  pro- 
portional to  their  distances  from  the  fixed  point  F. 

Thus  a  smaller  force  acting  at  a  greater  distance  can 
balance  a  greater  force  acting  at  a  less  distance.  The  Impor- 
tance of  the  greater  force  with  reference  to  the  point  F  is  exactly 
the  same  as  that  of  the  smaller  force,  which  has  the  advantage 
of  greater  distance,  or  greater  "  leverage  "  or  "  purchase." 

This  Importance  of  a  force  not  passing  through  a  point  is 
called  the  Moment  of  that  force  round  that  point.  It  is  equal 

to  the  amount  of  the  force  x  the 
shortest  distance  from  the  point  to 
the  line  of  application  of  the  force. 
The  shortest  distance  from  a  point 
to  a  line  is  well  known  to  be  a  line 
drawn  from  the  point  to  the  line  in 
"*  question,  at  right  angles  to  the  lat- 

/& B\    ter.     In  Fig.  79  the  moments  of  the 

forces  round  the  point  F  are  respec- 
tively 10  x  AF  and  -  1  x  BF.  In  Fig.  80  the  forces  acting  at 
A  and  B  are  not  parallel ;  their  lines  of  application  are  AE  and 
BE ;  the  distances  of  these  lines  from  F  are  FC  and  FD  at 
right  angles  to  AE  and  BE :  the  moments  of  the  respective 
forces  round  F  are  Force  A  x  distance  CF,  and  Force  B  x  dis- 
tance FD ;  and  if  the  forces  are  to  produce  equal  and  contrary 
rotational  effects  round  F,  so  that  there  may  be  rest  and  statical 
equilibrium,  their  moments  must  be  equal  and  of  opposite  sign, 
so  that  their  sum  =  0.  This  is  the  Principle  of  Moments. 

Moments  should  be  specified  in  terms  of  dyne-centimetres,  or  of  poundal- 
f eet ;  or,  it  may  be,  in  pound-feet,  or  in  ton-feet,  if  the  engineers'  gravita- 
tional units  be  employed. 

If  in  Fig.  79  the  forces  at  A  and  B  acted  at  the  ends  of  an 
immovable  rod,  there  would  be  a  reaction  =11  units  spread 
over  the  whole  extent  of  the  rod,  but  more  intense  near  the 
point  A :  if  the  rod  be  held  fast  only  at  one  fixed  point  F,  all 
the  reaction  (  =11  units)  is  concentrated  at  that  point,  if  it  be 
such  a  point  that  there  is  no  tendency  to  rotation  round  it  —  i.e. 
if  the  moments  round  the  point  of  resistance  =  0  ;  if  these  be 
not  =0,  the  pressure  on  it  is  still  11  Ibs.,  but  the  energy  is 
partly  spent  in  producing  rotation  round  that  point.  If  the 
reaction  pass  through  the  point  round  which  the  moments  =0, 
there  is  neither  translation  nor  rotation,  and  hence  the  three 
forces  are  in  equilibrium :  these  are,  10  units  at  A,  1  unit  at  B, 


vi.  J  THE    PRINCIPLE    OF   MOMENTS.  157 

parallel  to  the  former,  aiid  11   units  at  F,  parallel  but  in  the 
opposite  direction. 

Thus  Fig.  81  is  established  as  indicating  the  conditions  of 
equilibrium  of  two  parallel  forces,  P  and  Q ;  a  third,  R,  equal  to 
their  sum,  must  act  in  the  opposite  direc-  6 

tion    at  F,  a  point  round  which  their    3  Fig.si. 

moments  vanish  or  are  together  equal  to 

zero.     If  the  two  conditions  be  satisfied    p JT 

(1)  that  P  +  Q  =  -  R,  and  (2)  that  the 

moments  of  P  and  Q  round  F  be  equal 

and    opposite,    there    will   be    statical  |R 

equilibrium  :  if  the  former  be  not  satis-  8 

fied  there  will  be  translation  ;  if  the  latter,  there  will  be  rotation ; 

if  both  be  violated  there  will  be  both  translation  and  rotation. 

Now  let  the  point  of  application  of  the  force  Q  be  shifted  to 
the  right :  the  force  P  must  increase  in  order  that  its  moment 
may  remain  equal  to  that  of  Q.  If  Q  be  transferred  to  an 
indefinite  distance  the  force  P  would  have  to  become  indefinitely 
great  in  order  to  balance  it.  Two  unequal  forces,  tending  to 
produce  rotation,  may  be  balanced  by  a  single  force  :  P  and  R 
are  balanced  by  Q.  In  this  case  P  and  Q  have  opposite  and 
equal  moments  round  F ;  R  has  no  moment  round  F,  its  own 
point  of  action.  There  is  equilibrium  here  between  P,  Q,  and 
R;  their  moments  round  F  are  together  =0.  So  are  their 
moments  round  any  other  point,  as  may  be  easily  proved.  In 
general,  whatever  point  is  considered,  if  there  is  to  be  no  rotation 
round  that  point,  the  sum  of  all  the  moments  of  all  the  forces 
acting  round  that  point,  each  taken  with  its  proper  sign,  must 
be  equal  to  0. 

Example.  —  Suppose  a  slab  AB,  where  AB  =  100  cm.,  and  the  mass  of 
the  slab  is  100  kilogrammes,  to  be  supported  upon  two  feet,  D  and  E,  each  at 
a  distance  of  10  cm.  from  A  and  B  respectively :  and  let  a  mass  of  40  kg.  be 
hung  over  the  end  B  by  means  of  a  cord  :  what  will  be  the  pressures  between 
the  slab  and  its  two  supports  D  and  E  ?  The  Weight  of  the  slab  acts  as  if 
at  its  midpoint  C,  40  cm.  from  D  and  from  E ;  the  forces  involved  are  100 
kg.  wt.  at  C,  and  40  at  B ;  the  upward  reactions,  RD  and  RE,  at  D  and  E  are 
required.  There  is  no  rotation  round  E :  therefore  the  sum  of  the  moments 
round  E  =  0;  i.e.  (RD .  ED)  +  (100  kg.  x  EC) +  (40 kg.  x  EB)=0;  or  (RDx 
(_  80)  +  (100  x  (-  40))  +  (40  x  10)  =  0  ;  whence  RD  =  -  45  kg.,  a  negative 
or  upward  Reaction  of  the  support  D  against  the  slab.  Similarly,  round  D, 
(100  x  DC)  +  (RE  x  DE)  +  (40  x  DB)  =  0 ;  and  DC  =  40,  DE  =  80,  DB  =  90; 
whence  RE  =  —  95  kg.  These  upward  reactions  are  equal  and  opposite  to 
the  downward  pressures  of  the  slab  on  its  supports  D  and  E.  Ai  the  mass 
hung  over  B  be  raised  to  800  kg.,  RD  becomes  equal  to  zero,  and  the  slab  is 
about  to  tilt  over 


158  KINETICS.  [CHAP. 

Torque.  —  The  Torque  or  Turning  Power  of  a  Force  round 
a  Point  is  measured  by  its  Moment  round  that  point.  In  rota- 
tory movements,  Torques  are  analogous  to  Forces  in  translatory 
movements.  For  example,  a  body  tends  to  remain  at  rest  or  in 
a  state  of  uniform  rotation  round  its  centre  of  mass  except 
in  so  far  as  it  may  be  acted  upon  by  an  external  Torque  ;  and 
the  Angular  Acceleration  is  proportional  to  the  applied  Torque. 
Force  causing  rotation  constant  in  direction.  —  If  a  body 
be  caused  to  rotate  by  force  whose  direction  is  the  same  or 
nearly  so  throughout  the  movement,  the  effect  of  the  force  varies 

greatly.  In  Fig.  82  let  AC,  AD,  AE, 
AF,  be  successive  positions  of  a  rod 
rotating  round  A,  and  acted  upon  by  a 
force  applied  at  the  extremity  remote 
from  A,  and  always  parallel  to  the  lines 
I  Cc,  DC?,  E#.  In  the  position  AF  the 
effect  produced  by  the  force  is  a  maxi- 
mum, because  the  force  is  there  applied 
with  the  greatest  "leverage,"  or  so  as 
to  have  the  greatest  possible  Moment 
or  Torque,  in  this  way  the  forearm  moves  with  the  greatest 
swiftness  at  the  middle  of  flexion. 

Couples.  —  Two  forces  not  directly  opposed,  and  concurring 
in  producing  rotation,  may  sometimes,  as  in  some  of  the  Exam- 
ples of  Couples  below,  be  called  a  Couple,  whether  these  forces 
be  equal  and  parallel  or  not.  In  another  sense  the  word  Couple 
is  restricted  to  two  equal  and  parallel  forces  causing  rotation  of 
a  symmetrical  body  round  its  centre  of  mass,  when  that  centre 
is  situated  midway  between  these  forces  and  in  the  straight  line 
joining  their  points  of  application.  The  standard  definition  of  a 
Couple  is,  however,  a  generalised  one,  is  independent  of  any 
fixed  point  of  rotation,  and  is  based  upon  the  following  consid- 
erations. Two  forces  always  have  a  resultant  and  may  be  bal- 
anced by  a  third  except  in  one  case,  viz.  that  in  which  the  iwo 
forces  are  equal  and  opposed  in  their  direction,  but  not  opposed 
in  the  same  straight  line.  This  pair  of  equal  forces  constitutes 
a  Couple,  strictly  so-called ;  and  the  Standard  Definition  of  a 
Couple  is  —  two  equal  and  parallel  forces  opposed  in 
direction,  but  not  in  the  same  straight  line. 

A  Couple,  as  thus  defined,  has  the  following  properties :  — 
1.    It  cannot  be  balanced  by  any  one  force  at  any  finite 
distance.     If  P  become  equal  to  R  in  Fig.  83,  Q  vanishes. 


vi.]  COUPLES.  159 

2.  It  can  be  balanced  by  an  opposed  couple. 

3.  It  produces  rotation  round  any  point  which  may  happen 
to  be  fixed,  whether  within  the  same  plane  or  not. 

If,  on  the  other  hand,  two  unequal  forces  (P  and  R  of  Fig.  81,  in  the 
absence  of  Q)  tend  to  produce  rotation,  there  is  always  at  least  one  point 
such  that,  if  this  point  be  held  fixed,  there  can  be  no  rotation. 

4.  It  produces  no  pressure  upon  the  point  fixed,  wherever 
that  point  may  be  situated. 

If,  on  the  other  hand,  two  unequal  forces,  opposed  in  direction  but  not 
opposed  in  the  same  straight  line,  act  upon  a  body,  there  is  a  tendency  to 
translation.  If  the  handles  of  a  copying-press  be  equally  acted  upon  by  the 
two  hands,  there  will  be  rotation  simply ;  if  the  hands  act  unequally,  the 
copying-press,  with  the  table  on  which  it  is  fixed,  may  be  pulled  or  pushed 
over. 

5.  The  algebraic  sum  of  the  Moments  or  Torques  of  the 
components  of  the  couple  is  the  same  round  all  points  in  space. 

6.  Applied  to  a  freely  moving  mass,  a  Couple  produces  no 
translation  of  that  mass,  wherever  it  maybe  applied;  the  centre 
of  mass  remains  unmoved  ;  the  mass  is  set  in  rotation  round  the 
centre  of  mass.     The  consequent  angular  acceleration  is  deter- 
mined by  the  algebraic  sum  of  the  two  torques  of  the  couple ; 
but,  by  (5),  this  is  the  same  for  all  points ;  hence,  the  angular 
acceleration  round  the  centre  of  mass  is  the  same,  wherever  the 
couple  may  be  applied. 

These  special  properties  have  earned  for  this  pair  of  equal 
forces  the  specific  name  of  Couple  ;  and  an  example  of  a  Couple, 
as  thus  defined,  is  furnished  by  every  case  in  which  Reaction, 
though  equal  and  opposite  to  Action,  is  not  in  the  same  straight 
line  with  it;  and  that  whether  the  reaction  be  due  to  a  support, 
to  friction,  or  to  the  inertia  of  the  body  acted  upon. 

If  a  couple  be  applied  to  a  body  of  which  no  particle  is  held 
fixed,  there  will  accordingly  be  rotation  round  the  centre  of  mass ; 
but  the  direction  of  the  axis  of  rotation  round  the  centre  of  mass 
will  depend  upon  the  relative  unwieldiness  of  the  body  in  respect 
of  the  various  possible  axes  passing  through  the  centre  of  mass. 

A  single  force,  or  Resultant  of  forces,  on  the  other  hand, 
applied  in  a  line  which  does  not  pass  through  the  centre  of 
mass,  produces  rotation  round  that  centre  plus  translation  of  that 
centre  parallel  to  the  line  of  application  of  the  force.  In  Fig. 
83  f?,  the  concurrence  of  these  two  movements  causes  the  point  A 
to  be  at  rest  when  C  is  struck  a  sudden  blow. 

Moment  of  a  Couple.  —  The  sum  of  the  moments  round 
all  points  in  space  is  the  same.  Take  the  midpoint ;  the  whole 


160  KINETICS.  [CHAP. 

distance  between  the  forces  is  I;  the  moment  of  each  force 
round  the  midpoint  is  F  •  1 7,  where  F  is  either  force :  both  forces 
concur  in  producing  rotation  ;  the  joint  moment  is  Fl  =  M,  the 
Moment  of  the  Couple,  the  product  of  either  of  the  equal 
forces  into  the  distance  between  them.  The  turning  power  or 
Torque  of  a  Couple,  like  that  of  a  Force,  is  equal  to  its  Moment. 

Examples  of  Couples.  —  The  action  of  the  two  hands  on  the  handles 
of  a  copying-press  is  that  of  a  couple :  one  pulls,  the  other  pushes. 

Examples  abound  in  the  muscular  and  osseous  system.*  Such  are — the 
elbow  joint,  where  the  triceps  pulls  the  olecranon  process  backwards,  and 
the  reaction  of  the  articular  surface  of  the  humerus  against  the  sigmoid  cav- 
ity of  the  ulna  constitutes  the  other  member  of  the  couple ;  the  jaw  in  lateral 
chewing,  where  the  external  pterygoid  muscle  may  pull  one  side  of  the  jaw 
forward  while  the  result  of  the  action  of  the  hinder  fibres  of  the  opposite 
temporal  muscle,  together  with  the  corresponding  muscles  below  the  jaw,  is 
to  pull  the  opposite  side  of  the  jaw  backwards ;  the  weight  of  the  head  when 
a  person  stands  in  a  very  erect  position  is  equivalent  to  a  force  acting  along 
a  line  passing  through  a  point  a  little  behind  the  occipital  condyles,  and  this, 
together  with  the  reaction  between  the  atlas  and  the  occipital  condyles,  forms 
a  couple  which  is  equilibrated  by  an  opposing  couple  due  to  the  contraction 
of  the  muscles  of  the  front  of  the  neck,  together  with  the  additional  reac- 
tion between  the  atlas  and  the  occipital  condyles  which  is  produced  thereby ; 
the  same  weight  of  the  head,  when  this  bends  forward  a  little,  passes  along 
a  line  a  little  in  front  of  the  condyles,  and  it  forms  with  the  reaction  of  the 
atlas  a  couple,  which  is  balanced  in  the  same  way  by  the  contraction  of  the 
muscles  of  the  back  of  the  neck :  when  these  contractions  slacken,  as  when  a 
person  is  falling  asleep,  the  head  is  rotated  by  the  couple  on  a  transverse  axis, 
and  it  drops  forwards  or  backwards  according  to  the  position  in  which  it  hap- 
pens to  be  at  the  time  when  muscular  contraction  ceases  to  balance  its  weight. 

Equilibrium  of  Couples.  —  Let  a  couple,  consisting  of  two  equal  forces, 
act  always  in  one  and  the  same  direction,  pulling  the  particle  A  (Fig.  83 a) 

,  and  pushing  the  particle 
Fig.  83 a.  ,.-B  j^  an(j  iet  A  an(j  ]3  be  so 

connected  as  to  form  a 
system  capable  of  rotation 
round  the  point  O  midway 
between  them.  When  AB 
is  at  right  angles  to  the 
couple,  the  Moment  of  the 
Couple  is  equal  to  twice 
the  product  of  either  force 
into  the  arm  OA  or  OB ; 

A''  it  is   therefore   equal   to 

either  force  F  x  the  length 
AB.  Let  the  system  rotate 
into  the  position  A'B' 
making  an  angle  0  with 
its  previous  direction ;  the  couple  acts  upon  a  rod  whose  virtual  length  is 

*  Numerous  examples  may  be  found  discussed  in  Hermann  Meyer :  Die  Statik 
u.  Mechanik  des  menschl.  Knochengerustes ;  Leipzig,  Engelmann. 


VI.] 


COUPLES. 


161 


Fig.  83  b. 


reduced,  by  way  of  projection,  to  cd  or  AB  cos  0.  The  moment  of  the  couple 
is  now  F  •  AB  cos  6,  and  when  0  is  90°  the  moment  of  the  couple  is  reduced 
to  zero,  and  there  is  110  further  effect. 

Now  let  two  similar  couples  act  upon  the  same  system  AB,  and  let  their 
directions  be  at  right  angles  to  one  another  and  their  actions  opposed.  There 
will  be  equilibrium  when 
F  (Fig.  83  ft)pushes  B  so  as 
to  diminish  6  just  as  much 
as  F'  pushes  it  so  as  to  in- 
crease 9.  At  that  moment, 
and  in  that  position  of 
AB,  the  effective  moments 
of  the  two  couples  are 
equal.  The  one  is  F'  •  AB 
cos  0 ;  the  other  is  F  •  AB 
cos  6'.  Expressing  this 
equality  by  means  of  an 
equation,  we  have  F'«  AB  cos  0  =  F  •  AB  cos  0'  =  F  •  AB  sin  0.  Hence  F'  cos  0 
=  F  sin  0  or  F' :  F : :  tan  6  : 1,  where  0  is  the  deflection  from  a  position  par- 
allel to  FF.  This  proposition  is  applied  in  the  construction  of  the  Tangent 
Galvanometer. 

Again,  let  the  one  couple  F'F'  have  a  direction  always  at  right  angles 
to  the  direction  of  AB,  while  the  other,  FF,  has  any  direction  whatsoever  not 
at  right  angles  to 
AB.  AB  is  deflected 
through  an  angle 
0  from  a  position 
parallel  to  FF.  The 
moment  of  the  F' 
couple  is  F'-AB;F' 
that  of  the  F  couple 
is  F  .  AB  sin  0  as 
before.  These  cou- 
ples being  in  equi- 
librium, we  have 

F'.  AB=F-  AB  sin 0  or  F' :  F  : :  sin0  : 1. 
Sine  Galvanometer. 


Fig.  83c. 


This  proposition  is  applied  in  the 


Rotation.  —  Propositions  concerning  rotational  movement 
run  parallel  to  those  concerning  translational  movement. 

Rotations  are  produced  by  accelerations  which  are  radial,  directed  towards 
a  point  or  line,  which  is  or  becomes  the  instantaneous  or  the  permanent 
centre  or  axis  of  rotation.  In  the  most  general  case,  a  force  applied  in  any 
direction  to  a  moving  body  may  have  a  component  in  the  direction  of  the 
existing  motion,  and  a  component  radial  towards  some  point,  and  producing 
rotation  round  that  point. 

If  a  particle  move  along  a  circular  path,  of  radius  r,  with  uniform  angu- 
lar velocity  <o  radians  per  second  (see  p.  75),  the  angle  swept  round  in  time 
t  will  be  0  =  a>t.  If  the  motion  be  accelerated,  the  angular  acceleration 
being  such  as  to  increase  or  diminish  the  angular  velocity  <o  by'd>  radians- 
per-sec.  each  second,  we  have,  corresponding  to  the  equations  of  p.  152,  the  fol- 
lowing equations :  iot  =  <DO  ±  u>£  (i.)  ;  6  =  |(o)0  +  w*)  t  =  ay  ±  £  <irf2  (ii.)  ; 

M 


162  KINETICS.  [CHAP. 

and  o>t2  =  o>02  ±  2d>0  (iii.)-  When  the  angular  velocity  is  <o  radians  per  second, 
the  actual  linear  velocity  along  the  circumference  is  v  =  r<a  cm.  per  second ; 
the  Energy  of  movement  is  \mv*-  =  ^mr2o>2.  The  Force  required  to  impart 
this  velocity  v  in  one  second,  starting  from  rest,  is  mv  or  mr<a  ;  that  required 
to  do  so  in  t  seconds  is  F  =  ma  =  mv/t  =  mro)/t  =  mrtu.  The  Torque  required 
is  Fr  =  mr2o).  When  torque  is  considered  as  the  cause  of  rotation,  we  have, 
parallel  to  the  expression  F  =  ma  in  linear  translation,  the  expression  Torque 
=  (mr2)  o>  =  (mr2)  x  Angular  Acceleration.  Hence  wr2  is,  in  rotational 
kinetics,  the  analogue  of  m,  the  coefficient  of  inertia,  in  translational ;  the 
inertia  opposed  to  setting  a  mass  m  in  rotational  movement  under  a  given 
torque  depends  not  only  on  the  quantity  m  of  the  mass  to  be  moved,  but  also 
on  its  position  with  regard  to  the  axis  of  rotation. 

Moment  of  Inertia.  —  This  product,  wr2,  is  called  the  Moment  of  Inertia. 
In  a  mass  m  whose  several  particles  are  respectively  at  different  distances 
from  the  proposed  centre  or  axis  of  rotation  (which  may  be  either  internal 
or  external  to  the  mass  itself),  the  Moment  of  Inertia  is  found  by  summing 
up  in  appropriate  units  the  products  of  the  mass  m  of  each  particle  of  the 
mass  into  the  square  of  its  corresponding  distance  f  from  the  axis  of  rotation. 
This  operation  generally  requires  the  aid  of  the  Integral  Calculus,  but  the 
resultant  sum,  %mr2  =  N,  is  a  numerical  quantity,  and  is  always  positive. 
Then,  Torque  =  No>. 

Radius  of  Gyration,  or  Radius  of  Inertia.  —  Suppose  a  uniform 
disc  of  radius  r  to  rotate  round  its  centre,  with  a  given  quantity  of  rota- 
tional energy ;  it  rotates  with  less  angular  velocity  than  it  would  have 
assumed  if  the  same  matter  had  been  gathered  nearer  the  centre ;  for  the 
energy  of  rotation  of  each  particle  is  ^mr2o>2,  and  if  the  mean  value  of  ~r 
be  greater,  the  value  of  <o  must  be  less,  if  the  energy  of  rotation  be  constant. 
Such  a  disc  rotates,  on  the  other  hand,  with  greater  angular  velocity  than 
that  which  it  would  have  assumed  if  the  matter  had  been  gathered  near 
the  circumference.  Between  these  two  limits  there  must  be  a  mean  dis- 
tance from  the  centre,  such  that  if  the  whole  mass  had  been  concentrated 
there,  the  angular  velocity  would  have  been  the  same  as  that  actually 
assumed  by  the  disc.  This  mean  distance  is  the  Radius  of  Gyration 
or  the  Radius  of  Inertia,  with  reference  to  that  point.  If  i  be  the 
radius  of  inertia,  t2  =  $mr'2/m  =  N/ra ;  and  if  the  whole  mass  m  were 
placed  at  the  distance  t  from  the  point  of  suspension,  it  would  have,  with 
reference  to  that  point,  the  same  Moment  of  Inertia  as  that  actually  pos- 
sessed by  the  physical  mass  in  question.  Hence  wit2  =  N  =  Swr2. 

If  the  body  do  not  continuously  rotate  round  the  same  point,  the  radius 
of  inertia  may  continuously  change  in  value. 

Radii  of  Inertia  and  Moments  of  Inertia  in  particular  cases. — 
(1.)  A  uniform  rod  of  length  /,  suspended  at  its  extremity,  rotates  with  the 
same  angular  velocity  as  if  its  mass  were  accumulated  at  a  distance  //  V3 
from  the  point  of  suspension.  The  rad.  of  inertia  t  =  I/  V3 ;  the  mom.  of 
inertia,  N  =  m/2/3. 

(2.)  A  uniform  rod  of  length  Z,  poised  on  its  centre;  i=Z/2\/3; 
N  =  m£2/12. 

(3.)  A  rectangular  lamina  of  sides  a  and  6,  poised  at  its  centre ;  rotation 

round  an  axis  at  right  angles  to  the  lamina;  i  =  V(a2  +  b*)/ 2  V3 ; 
N  =  wi(a2-f  &2)/12. 


sr^ 

[UII7SRSIT7] 

vi.]  ROTATION? 


(4.)  A  circular  disc  of  radius  r,  and  of  uniform  thickness  ;  rotation  round 
an  axis  at  right  angles  to  the  disc  and  passing  through  its  centre  ;  i  =  r/  V2  ; 
N  =  £mr2. 

(5.)  The  same,  round  a  diameter  ;  t  =  r/2  ;  N"  =  \mr\ 

(6.)  A  solid  cylinder  rotating  round  its  axis;  same  as  (4). 

(7.)  A  solid  ring  cut  out  of  a  uniform  disc  of  any  thickness  ;  inner  and 
outer  radii  r,  and  rn  ;  rotation  round  an  axis  passing  through  the  centre  of 
the  ring  and  at  right  angles  to  the  plane  of  the  ring  ;  i  =  V^r,2  +  r,f)/2  ; 
X  =  m(r?  +  rl*)/2. 

(8.)  Solid  ring,  whose  cross-section  is  a  circle  of  radius  a;  distance 
between  the  centre  of  the  ring  and  the  centre  of  this  circle  =  b  ;  rotation 
round  an  axis  passing  through  the  centre  of  the  ring  and  at  right  angles  to 
its  plane  ;  t  =  \/62  +  fa2  ;  X  =  m  (b2  +  fa2). 

(9.)  Spherical  shell  of  radius  r;  rotation  round  any  diameter;  i=r  V2/3  ; 
N  =  |  ?nr2. 

(10.)  Solid  sphere  of  radius  r  ;  rotation  round  any  diameter  ;  t  =  r  V2/5  ; 


(11.)  Let  mi02  be  the  moment  of  inertia  round  an  axis  passing  through 
the  centre  of  gravity  of  a  mass  of  any  form  ;  round  any  other  axis,  parallel 
to  the  former  and  at  a  distance  h  from  it,  the  moment  of  inertia  mif  is 
m  (t02  +  A2)  *  where  m  is  the  whole  mass.  For  example,  if  the  disc  of  (4)  be 
rotated  round  a  point  in  its  edge,  t,  =  r  •  Vf  .  At  what  point  is  Lt  =  rl 

Angular  Momentum.  —  The  analogue  of  linear  momentum  mv  is  angu- 
lar momentum  Nw.  This,  like  ordinary  momentum,  cannot  be  increased  in 
one  body  without  an  equal  negative  momentum  being  developed  in  another. 
Further,  in  respect  of  one  and  the  same  body,  any  change  in  the  value  of  N" 
or  of  co  causes  a  corresponding  change  in  the  value  of  w  or  of  N",  for  N<o  is 
constant.  Thus  the  sun  or  the  earth,  shrinking  as  it  cools,  acquires  a 
smaller  moment  of  inertia,  and  o>,  the  angular  velocity,  tends  to  increase. 
Similarly,  the  water  in  a  basin,  when  the  central  portions  are  withdrawn, 
begins  to  swirl  ;  the  circumferential  portions,  coming  towards  the  centre, 
acquire  smaller  moments  of  inertia,  and  any  existing  rotation,  however 
small  at  first,  becomes  increasingly  more  rapid. 

Energy  of  a  rotating  body.  —  The  energy  of  a  particle  in  rotational 
movement  is  2wr2o>2;  that  of  a  system  of  particles,  each  at  its  own  dis- 
tance r  and  with  its  own  mass  TW,  must  be  2Qmr2eo2)  =  2to2S(mr2)  ;  o>,  the 
angular  velocity,  being  the  same  in  all  particles  of  a  rotating  body.  But 
2wr2  =  N",  the  moment  of  inertia  :  therefore  the  energy  of  a  rotating  body 
is  ^o>2N",  or,  where  t  is  the  radius  of  inertia,  =  ico2mt2.  The  energy  of  the 
ring  of  example  (7)  above  is  therefore  m  (rf  +  r//2)o>2/4. 

*  Draw  a  triangle  ABC  ;  A  'represents  the  centre  of  gravity  of  the  object  spun, 
through  which  the  central  axis  of  rotation  passes,  perpendicular  to  the  paper  ;  B  is 
the  position  of  the  other  axis  parallel  to  the  former  ;  C  any  point  whatsoever  in  the 
mass  rotated.  Draw  a  line  CD  from  C  at  right  angles  to  AB  or  to  AB  produced.  Then 
BC2  =  AC2  +  AB2  ±  2  AB  •  AD.  AB  is  the  distance  h  between  the  two  axes  ;  BC  the 
distance  of  the  particle  C  from  the  new  axis,  AC  its  distance  from  the  centre-of- 
gravity  axis.  If  the  particle  at  C  have  mass  m,  m-BC2  =  m  •  AC2  +  w-  h2±  2m-h-AD. 
Now  sum  up  for  all  such  particles  as  C,  and  we  have  2  (m-BC2)  =  2(m-  AC2) 
-f  2(m  •  ft2)  ±  2h  •  2(m  •  AD)  .  The  last  term  disappears,  for  all  round^the  centre  of 
gravity  AD  has  as  many  positive  as  negative  values  ;  2(m  •  BC2)  is  mi,2,  the  moment 
of  inertia  round  B;  2(m-AC2)  is  mt02,  the  moment  of  inertia  round  A;  2(m-A2) 
Whence  wt,2  =  m  (t02  +  h  2). 


164  KINETICS.  [CHAP. 

A  flywheel  in  motion  possesses  a  large  amount  of  kinetic 
energy;  and  if  an  obstacle  be  placed  in  the  way  of  the  engine, 
the  engine  cannot  be  stopped  by  it  unless  the  flywheel  can  be 
arrested  also :  this  would  involve  the  sudden  exercise  of  a  very 
great  force ;  hence  an  engine  with  a  heavy  flywheel  rapidly 
rotating  can  overcome  a  very  great  resistance,  and  in  this  way, 
for  ordinary  resistances,  it  is  prevented  from  manifesting  any 
very  great  irregularity  of  motion. 

If  a  flywheel  whose  energy  is  |o>2mi2  were  called  upon  to  expend  W 
units  of  energy  in  overcoming  a  certain  resistance,  the  energy  in  it  after 
doing  so  would  be  ( Ja»2nua  —  W),  and  a  new  angular  velocity  w,  would  be 
assumed,  such  that  Ja*/*»ita  =  (|o)2wi2  —  W).  The  amount  of  kinetic  energy 
in  a  flywheel  thus  fluctuates.  If  a  very  large  flywheel  have  a  heavy  rim, 
and  if  the  spokes  be  relatively  thin,  the  radius  of  inertia  is  practically  the 
distance  between  the  centre  of  the  wheel  and  the  middle  of  the  thickness  of 
the  rim;  and  the  energy  is,  approximately,  {\w2m  •  (mean  radius)2}. 

Minimum  Angular  Velocity.  —  In  examples  (4)  and  (5)  above,  the 
respective  energies  of  rotation  for  a  given  angle  of  velocity  to  are  :  axial 
rotation,  <l<D2-wr2/2;  diametral,  ^w2-  7w>2/4.  The  former  is  twice  the  latter. 
For  a  given  amount  of  energy  W,  the  respective  angular  velocities  are : 
axial,  <i)a  =  2/r  •  V  \V /in ;  diametral,  <Dd  =  2/r  •  V2  W/m.  Rotation  in  a  freely 
suspended  body,  if  effected  round  its  shortest  axis  of  symmetry,  involves  the 
smallest  angular  velocity  for  a  given  supply  of  kinetic  energy ;  and  this  is 
the  most  stable  form  of  rotation.  The  Earth  rotates  round  its  shortest 
axis.  A  hard-boiled  egg  or  an  egg-shell,  freely  suspended,  does  the  same ; 
though  if  spun  on  a  table  it  rises,  as  explained  on  p.  76.  A  mass  of  liquid 
set  in  rotation  always  tends  to  spread  out  or  to  set  itself  so  as  to  have  a 
maximum  moment  of  inertia  and  a  minimum  angular  velocity ;  and  in  the 
spinning  of  an  unboiled  egg  on  a  table  this  tendency  overcomes  that  of  the 
egg-shell  itself  to  rise  on  the  table :  an  unboiled  egg  does  not  rise. 

Suspended  Body.  —  Suppose  a  heavy  body  of  mass  m  to  be  suspended 
at  a  point,  and  then,  that  point  of  suspension  retaining  its  fixed  position, 
to  swing  down  so  far  that  its  centre  of  figure  sinks  through  a  vertical  height 
k,  acquiring  an  angular  velocity  <o.  The  kinetic  energy  acquired  by  the 
body  considered  as  rotating  round  the  point  of  suspension  is  ^<o2i"J;  the 
potential  energy  lost  by  descent  of  the  mass  m  through  height  h  is  mgh. 
These  are  equal.  Hence  ^o>2X  =  mgh ;  or  w2  —  2mgh/N.  But  N  =  m2,  where 
i  is  the  radius  of  inertia;  whence  w2  =  2//A/12. 

Centre  of  Oscillation.  —  In  Fig.  80  d  let  A  be  the  point  of  suspension 
of  a  body,  B  its  centre  of  figure  or  of  mass  (centre  of  gravity),  t  the  length 
of  the  radius  of  inertia  of  the  mass  with  reference  to  the  point  of  suspension 
A :  then  there  is  in  the  same  straight  line  with  A  and  B,  and  on  the  oppo- 
site side  of  B  from  A,  a  point  C,  called  the  Centre  of  Oscillation,  which  has 
the  following  properties  :  — 

(1.)  The  body  may  be  swung  upon  A  or  upon  C  indifferently,  and  in 
either  case  it  will  oscillate  pendulum-wise  with  equal  rapidity. 

(2.)  The  body  thus  suspended  at  A  or  at  C  will  oscillate  at  the  same  rate 
as  an  ideal  simple  pendulum  of  the  length  AC.  (Proved  at  p.  214.) 

(3.)  This  body  will,  if  struck  at  C,  oscillate  round  A  without  producing 


vi.]  ROTATION.  165 

any  pressure  on  the  supporting  axis  at  A.  In  batting  at  cricket,  A  repre- 
sents the  shoulder-joint  and  C  the  proper  point  of  impact  on  the  bat. 

(4.)  Though  the  support  at  A  were  withdrawn  —  as,  for  instance,  if  the 
body  float  submerged  in  water  —  yet  if  the  point 
C  were  struck  by  a  properly  directed  blow,  the 
point  A  would  remain  at  rest,  and  all  the  part  of 
the  body  lying  above  A  would  move  in  a  direction 
opposite  to  that  in  which  C  is  struck.  For  every 
point  C  at  which  a  body  may  be  struck,  or  for 
every  centre  of  percussion,  there  is  a  correspond- 
ing point  A  on  the  other  side  of  the  centre  of  figure, 
through  which  passes  an  instantaneous  axis  of 
spontaneous  rotation  round  which  the  body  im- 
pulsively rotates.  If  the  lower  part  of  any  object 
be  suddenly  pulled  forwards,  the  upper  part  will 
move  backwards.  This  property  is  found  applied 
in  the  jaw  of  echinoidei ;  the  upper  end  of  each  of 
the  five  jaws  is  suddenly  tilted  outwards,  and  the 
lower,  the  tooth -bearing  ends,  are  tilted  together. 

(5.)  The  distance  AC  is  equal  to  i2/AB  when 

the  body  is  suspended  at  A,  t  being  the  radius  of  inertia  in  this  case ;  or  to 
i^/CB  when  suspended  at  C,  i,  being  the  radius  of  inertia  in  this  case.  These 
radii  of  inertia  are  so  related  that  i2/AB  =  t/2/CB.  (See  p.  214;  there 
written  i2/h  =  i,2/fly.) 

(6.)  AB  :  t0 : :  t0 :  BC,  where  i0  is  the  radius  of  inertia  round  B. 

The  table  on  p.  166  gives  a  comparative  conspectus  of  certain  quanti- 
ties in  Translational  and  llotational  Kinetics. 

"  Centrifugal  Force,"  so-called.  —  It  has  been  already 
shown  (p.  79)  that  when  a  body  describes  a  curved  path,  there 
is  an  acceleration  towards  the  centre  =  v2/r,  where  v  is  the  tan- 
gential velocity  in  the  curve,  and  r  the  instantaneous  radius  of 
curvature.  This  centripetal  acceleration  changes  the  direction 
of  motion  without  changing  the  velocity.  If  the  path  be  a 
circle  of  radius  r,  this  acceleration  is  constantly  =v2/r.  The 
component  force  drawing  the  body  from  the  tangential  path  is 
therefore  one  which  produces  an  acceleration  towards  the 
centre  =  v2/r;  and  it  is  itself  F  =  mv*/r. 

Since   v/r  =  w,   the   centripetal   acceleration    vz/r  =  <oV,  and  the  cen- 
tripetal Force  =  w<o2r. 

Suppose  a  stone  of  mass  m  to  be  whirled  round  like  a  sling- 
stone  by  a  string,  but  in  a  perfectly  circular  path.  This  circular 
path  may  be  supposed  to  be  made  up  of  very  numerous  short 
straight  lines  or  elements  (p.  58),  each  of  which  is  tangential. 
The  actual  velocity  along  any  one  of  these  tangential  elements 
during  any  one  instant  may  be  hypothetically  resolved  during 
the  next  instant  into  two  components ;  one  along  the  circle  =  v  ; 


166 


KINETICS. 


[CHAP. 


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Quantities. 

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measured  in  radian 
Angular  Velocity  .  . 
Angular  Acceleration 

[Torque  .  .  .  .  , 
i  Moment  of  Force  .  , 
[Moment  of  Couple  , 

Moment  of  Inertia  , 

Radius  of  Inertia  .  . 

[Angular  Momentum  . 
•j  Moment  of  Momentu 
[Moment  of  Impulse 

[  Work  =  Torque  x  A: 
j  Energy,  Kinetic  .  , 
[Energy,  Potential  .  . 

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* 

VL]  "CENTRIFUGAL  FORCE."  167 

one  away  from  it,  in  the  line  of  the  radius,  and  corresponding  to 
an  outward  acceleration  a  =  —  v2/r.  Of  these  two  components 
the  former  freely  manifests  itself  as  velocity  v  along  every  suc- 
cessive element  or,  practically,  as  a  continuous  velocity  v  in  the 
circular  path;  the  latter,  the  outward  component,  never  mani- 
fests itself,  for  it  is,  at  every  instant,  counteracted  by  tension  in 
the  string.  This  tension  is  a  stress  set  up  in  the  string  by  the 
action  of  its  molecular  forces,  when  the  whirling  ball  tends  to 
pull  the  outer  end  of  the  string  outwards ;  and  numerically  it 
is,  across  every  complete  cross-section  of  the  string,  a  Total 
Tension  equal  to  that  which  would  have  been  established  by  the 
application  of  mv2/r  units  of  force.  If  the  string  snapped  or  were 
suddenly  cut,  this  tension  would  cease  ;  there  would  then  be 
nothing  to  hinder  the  actual  tangential  velocity  at  the  instant  of 
snapping  from  persisting  during  the  next  and  succeeding  instants ; 
the  motion  of  the  stone  would  therefore  be  continuous  motion  in 
a  straight  line,  the  tangent  to  the  curved  path  at  the  point  where 
the  stone  had  happened  to  be  at  the  instant  of  snapping,  and 
the  stone,  thus  liberated  and  flying  off  at  a  tangent,  would  then 
obey  Newton's  First  Law  of  Motion. 

As  the  ball  flies  off  in  its  tangential  path,  it  will  spin  :  for  in  its  circular 
path,  its  outer  particles  had  travelled  with  greater  velocity  than  its  inner. 

The  stone  flies  off  at  a  tangent,  and  not  straight  from  the 
centre ;  there  is  therefore  no  counteraction,  on  the  part  of  the 
string,  of  any  tendency  on  the  part  of  the  stone  to  fly  off  in 
some  direction  straight  away  from  the  centre  ;  there  is  there- 
fore no  so-called  "  centrifugal  force,"  in  the  old  sense  of  the 
term,  counteracted  by  the  tension  of  the  string.  The  tension  of 
the  string  is,  however,  equivalent  to  a  force  F  =  mv2/r,  acting 
upon  the  stone,  directed  inwards  along  the  string,  and  the  inward 
acceleration  of  which,  a  =  v2/r,  balances  at  every  instant  the 
opposite  tendency  on  the  part  of  the  whirling  stone  to  increase 
its  actual  distance  from  the  centre  by  pursuing  a  tangential  path. 

Any  string  will  snap  if  force  be  applied  to  it  beyond  a  cer- 
tain limit.  If  a  string  be  just  so  strong  that  x  grammes  of 
matter  may  be  suspended  on  it  without  its  snapping,  it  can 
survive  the  application  of  force  equal  to  981#  dynes.  If  this 
string  be  used  to  whirl  a  slingstone  of  mass  m,  it  will  snap 
unless  the  velocity  v  be  such  that  mvz/r  is  less  than,  981#  — 
that  is,  v  must  be  less  than  V981#r/ra.  If  the  velocity  exceed 
this  limit,  the  string  will  snap.  As  the  velocity  increases,  its 


168  KINETICS.  [CHAP. 

centrifugal  component  increases,  and  requires  a  greater  force  or 
reaction  to  be  exerted  in  a  direction  towards  the  centre  in  order  to 
bend  the  path  into  the  same  curve  in  a  shorter  time.  In  the  same 
way,  if  a  fly  stand  on  the  rim  of  a  rotating  wheel,  the  adhesion 
between  the  foot  of  the  fly  and  the  rim  of  the  wheel  necessary 
in  order  to  enable  the  fly  to  retain  its  footing  may  become  so 
great  that  the  fly  cannot  hold  on,  and  is  hurled  off  at  a  tangent. 

When  a  grindstone  or  flywheel  is  rotated  too  rapidly,  the 
molecular  forces  of  cohesion  cannot  keep  the  particles  together 
against  their  tendency  to  fly  off  at  a  tangent. 

If  the  earth  rotated  on  its  axis  seventeen  times  as  fast  as  it 
does,  the  attraction  of  gravitation,  the  effect  of  which  is  even 
now  masked  to  some  extent  by  the  rotation  of  the  earth,  would 
only  just  be  able,  at  the  equator,  to  keep  bodies  from  flying  off 
its  surface  at  a  tangent. 

The  greater  the  velocity  of  a  railway  train  the  greater  is  its 
tendency  to  fly  off  the  track  as  it  is  rounding  curves. 

If  a  drop  of  oil  be  suspended  in  a  mixture  of  spirit  and 
water,  so  that  it  is  free  to  assume  any  form,  and  if  a  motion  of 
rotation  be  communicated  to  it,  the  globular  drop  assumes  the 
form  of  an  oblate  spheroid,  and  bulges  at  its  equator ;  for  par- 
ticles at  its  original  equator  have,  when  set  in  motion,  a  greater 
velocity  than  those  nearer  its  poles.  For  the  same  reason  the 
earth  itself  has  assumed  the  form  of  an  oblate  spheroid. 

In  the  trundling  of  a  wet  mop,  when  the  drops  fly  off  because 
they  do  not  adhere  firmly  enough  to  enable  them  to  retain  their 
position  —  in  the  rotation  of  a  steam  governor,  the  balls  of  which 
fly  asunder  as  the  speed  of  the  engine  increases,  thereby  actuat- 
ing an  appropriate  train  of  mechanism  which  to  a  greater  or 
less  extent  shuts  off  the  steam — we  find  examples  of  this  phe- 
nomenon. If  a  man  were  placed  on  a  revolving  table,  with  his 
feet  towards  the  centre,  the  blood  in  his  body  would  be  urged 
towards  his  head ;  and  this  has  actually  been  proposed  as  treat- 
ment in  bloodlessness  of  the  brain. 

When  a  circular  cylindrical  vessel  containing  water  is 
rotated  on  its  axis,  the  water  is  heaped  up  towards  the  sides  of 
the  vessel.  If  the  speed  exceed  a  certain  limit,  the  water  will 
be  hurled  over  the  sides  of  the  vessel,  and  if  the  supply  of 
water  and  the  rotation  be  continuous,  an  engine  may  expend  its 
energy  in  thus  continuously  lifting  water  against  gravity.  This 
principle  is  applied  in  Siemens's  governor  for  machinery  ;  when 
the  engine  goes  too  fast  it  begins  to  spend  energy  in  producing 


vi.]  "CENTRIFUGAL  FORCE."  169 

this  current  of  water.     The  form  of  the  surface  of  water  thus 
produced  is  always  parabolic. 

When  light  and  heavy  particles  in  mixture  are  whirled,  the 
heavier  fly  outwards ;  thus  milk,  if  rotated,  separates  into 
heavier  milk  externally  and  lighter  cream  internally. 

A  badly-balanced  flywheel  exercises  a  fluctuating  pressure  on  its  bear- 
ings, which  varies  as  o>2. 


THE  MECHANICAL  POWEES. 

The  principle  of  moments  or  —  what  is  essentially  the  same 
thing  —  the  principle  that  the  work  done  by  or  on  a  machine 
=  0,  or  that,  on  the  whole,  there  is  no  accumulation  of  work  in  a 
machine,  is  the  key  to  the  explanation  of  the  action  of  many  of 
the  Simple  Machines  or  so-called  Mechanical  Powers.  The  work 
done  by  a  simple  machine  is  equal  to  that  done  upon  it,  and 
upon  the  machine  itself  there  is  no  work  done.  This  is,  of 
course,  not  strictly  accurate  ;  but  simple  machines  are  supposed, 
in  the  first  instance  and  for  the  sake  of  theory,  to  be  them- 
selves without  weight,  and  to  work  without  friction. 

The  Lever.  —  This  is  a  bar  of  any  substance,  rigid  enough 
to  retain  its  form  under  the  forces  applied  to  it.  We  consider  it 
at  the  moment  when  the  forces  or  pressures  applied  to  it  are  all 
in  equilibrium,  so  that  there  is  no  movement.  If  the  point  A  be 
pressed  down  with  force  P,  if  the  fixed 
point  or  Fulcrum  be  at  F,  and  if  the 
point  B  be  pressed  down  with  a  parallel  Fig.84. 

force  Q,  then  round  F  the  moments 
(P-AF)  +  (-Q-BF)  =  0.  If  AF be  shorter 
than  BF,  Qis  numerically  less  than  P;  then 
the  smaller  Q  can  balance  the  greater  P ; 
a  practical  mechanical  advantage.  As  an 
example  of  this  take  a  crowbar  :  the  man's 
strength  is  exerted  at  B ;  the  fixed  point 
is  at  F,  and  the  weight  of  the  body  to  be  lifted  acts  downwards 
at  A.  Suppose  the  lever  to  be  42  inches  long,  the  point  F  to  be 
2  inches  from  A,  and  the  man's  strength,  which  is  competent  to 
raise  56  Ibs.,  to  be  exerted  at  B  :  then  AF  =  2,  FB  =  40,  and 
Q  :  P  : :  2 :  40 ;  whence  the  man  can,  by  exerting  at  B  a  force  of 
56  Ibs.,  keep  a  mass,  weighing  1120  Ibs.  and  resting  on,A,  from 
moving  downwards  ;  an  effort  a  little  greater  will  lift  it. 

On  the  other  hand,  a  force  equal  to  the  weight  of  1120  Ibs. 


170  KINETICS.  [CHAP. 

applied  at  A  can  only  balance  a  weight  of  56  Ibs.  at  B :  here 
there  is  great  mechanical  disadvantage. 

An  example  of  this  occurs  in  the  case  of  forceps,  with  blades  relatively 
long :  the  pressure  which  they  can  exert  at  the  tip  is  relatively  small,  for 
each  blade  of  the  forceps  is  a  lever  supported  at  the  hinge.  Extremely  long 
scissors  do  not  cut  so  well  at  the  point  as  near  the  joint. 

But  what  is  gained  or  lost  in  force  is  lost  or  gained  in  range 
of  movement;  for  the  work  done  (  =  Fs)  by  the  one  arm  is 
always  the  same  as  that  done  upon  the  other. 

Whatever  the  special  arrangement  of  the  two  forces  and  the 
reaction,  the  principle  is  always  the  same,  that  the  lever  is 
studied  at  the  instant  of  equilibrium,  when  round  the  fulcrum 
the  sum  of  the  moments  =  0 ;  then  an  excess  either  of  the  Force 
applied  or  of  the  Resistance,  beyond  their  proportions  at  that 
instant,  will  cause  rotation  round  the  fulcrum. 

There  is  a  popular  division  of  levers  into  three  classes,  which  it  is  well 
to  explain  ;  Fig.  84  illustrates  them  all. 

Class  I. —  The  fixed  point  is  at  F  in  Fig.  84.  A  crowbar,  a  handspike, 
a  pair  of  forceps,  scissors,  or  shears,  a  poker,  a  common  balance,  —  all  these 
have  the  fulcrum  or  fixed  point  between  the  point  of  application  of  the  force 
and  that  of  the  resistance. 

Class  IT.  —  If  the  fulcrum  be  at  A  in  Fig.  84,  and  the  force  be  applied  at 
B,  the  resistance  overcome  at  F  is  numerically  greater  than  the  force  applied 
at  B,  as  is  found  by  taking  moments  round  A;  (AF  x  R)  =  (AB  x  Q). 
Examples  of  this  are  furnished  by  nut-crackers,  where  the  resisting  nut  is 
nearer  the  hinge  than  the  hand  is  ;  by  the  oar  of  a  boat,  in  which  the  force 
is  applied  at  the  handle,  while  the  tip  of  the  oar  is  approximately  at  rest,  and 
the  resistance  of  the  boat  is  overcome  between  these ;  by  a  claw  hammer 
used  for  extracting  nails,  where  the  fulcrum  is  at  the  end  of  the  claw,  the 
force  is  applied  through  the  handle,  and  the  resisting  head  of  the  nail  is 
between  these  points ;  by  a  wheelbarrow,  in  which  the  fulcrum  is  at  the 
axle  of  the  wheel,  the  raising  force  is  applied  at  the  handle,  and  the  resist- 
ance to  be  overcome  is  the  weight  of  the  substance  in  the  barrow  between 
the  handle  and  the  wheel. 

Class  III.  —  This  is  the  same  as  the  second  class,  except  that  the  Force 
and  the  Resistance  have  changed  places.  As  an  example  of  this  we  find 
that  in  a  pair  of  tongs  for  sugar  or  for  coal,  in  which  the  fixed  point 
is  at  the  hinge  or  the  flexible  end,  the  resistance  is  encountered  near  the 
other  end,  and  the  force  is  applied  between  these  points.  The  pressure 
that  can  be  applied  by  such  an  arrangement  is  comparatively  feeble,  while 
to  overcome  any  given  resistance  the  force  applied  must  be  proportionately 
very  great.  This  is  seen  in  opening  a  gate  by  pressing  on  it  near  the  hinges ; 
a  considerable  force  has  to  be  exerted.  Such  an  arrangement,  in  which  force 
is  sacrificed  in  order  to  gain  amplitude  of  movement,  is  of  ordinary  occurrence 
in  the  muscular  system.  The  biceps  is  inserted  into  the  radius  at  a  point 
about  one-sixth  of  the  distance  between  the  axis  of  rotation  of  the  elbow  joint 
and  the  centre  of  the  palm  of  the  hand.  In  order  to  raise  a  pound-mass  in 
the  hand,  that  muscle,  if  it  acted  alone,  would  have  to  exert  a  force  which 


vi.]  THE   MECHANICAL  POWERS.  171 

would  directly  lift  6  Ibs. ;  but,  on  the  other  hand,  the  forearm  has  a  range 
and  rapidity  of  movement  which  it  would  not  have  had  had  the  muscles  been 
inserted  in  the  position  of  greatest  mechanical  advantage,  not  to  mention 
the  inconvenience  of  having  muscles  extending  from  prominence  to  promi- 
nence of  the  skeleton  like  the  rigging  of  a  ship.  The  pectoral  muscle  of  a 
bird,  the  deltoid  muscle  of  man,  his  glutei  muscles,  actuate  conspicuous 
examples  of  osseous  levers  of  the  third  order. 

Problems.  —  1.  Two  porters  bear  a  burden,  56  Ibs.  in  weight,  by  means 
of  a  bar  of  such  length  that  the  distance  between  shoulder  and  shoulder  is 
70  inches.  The  weight  is  suspended  from  a  point  40  inches  from  the 
shoulder  of  one  of  the  porters.  What  share  of  the  burden  is  borne  by  the 
shoulder  of  each  respectively  ?  —  A  ns.  This  is  a  case  of  Fig.  84,  in  which 
the  weight  of  the  burden  corresponds  to  R,  and  the  upward  shoulder-reactions 
to  P  and  Q.  There  is  no  tendency  to  rotation  round  F,  which  is  relatively 
fixed ;  hence  the  reactions  at  A  and  B  must  be  such  that  their  moments 
round  F  are  equal.  Hence  the  two  equations,  P  -f  Q  =  56  and  40  P  =  30  Q, 
give  P  =  24,  Q  =  32.  The  porter  nearer  to  the  burden  carries  32  Ibs.,  the 
one  farther  from  it  carries  24  Ibs. 

2.  A  nut-cracker  6  inches  long  has  a  nut  in  it  an  inch  from  the  hinge. 
The  hand  exerts  a  force  equal  to  the  weight  of  4  Ibs. :  what  is  the  total  stress 
on  the  hinge  ?  —  Am.  The  nut,  so  long  as  it  does  not  yield,  affords  a  fixed 
point :  the  total  stress  on  the  hinge  =  the  weight  of  20  Ibs. 

The  Wheel  and  Axle.  —  The  lever,  when  it  has  done  work 
and  raised  a  burden  against  resistance,  moves  into  a  position 
where  the  leverage  and  the  corresponding  Moment  or  Torque  are 
diminished  (see  Fig.  82),  if  the  force  retain,  or  nearly  retain,  its 
original  direction.  If  by  any  means  matter  could  be  so  arranged 
that  a  lever  would,  when  it  had  moved  out  of  its  position  of 
greatest  advantage,  be  replaced  in  the  most  favourable  position 
by  another  lever,  to  which  the  burden  and  the  force  applied 
were  shifted,  the  apparatus  thus  constructed  would  in  some 
respects  be  more  useful  than  a  simple  lever. 

This  criterion  is  satisfied  as  regards  levers  of  the  first  order 
by  the  Wheel  and  Axle.  This  consists  of  a  large  wheel  or  cyl- 
inder and  a  small  one,  both  on  the  same  axis,  and  capable  of 
rotating  together  on  that  axis. 

Each  wheel  may,  if  solid,  be  regarded  as  consisting  of  an 
infinite  number  of  spokes.  One  of  these  spokes  in  the  larger 
wheel,  and  one  running  in  the  opposite  direction  from  the  centre 
in  the  smaller  wheel,  together  make  up,  when  they  are  for  an 
instant  at  right  angles  to  the  lines  of  application  of  the  force 
and  the  resistance,  a  lever  in  the  most  favourable  position.  As 
soon  as  this  has  left  the  position  of  greatest  advantage,  by  reason 
of  rotation  of  the  system,  its  place  is  at  once  taken  by  another. 

The  weight  of  a  large  mass  hung  on  the  smaller  wheel  and 
that  of  a  smaller  mass  hung  on  the  larger  wheel  will  balance  one 


172  KINETICS.  [CHAP. 

another,  if  their  moments  round  the  axis  of  rotation  be  equal. 
The  weights  may  be  replaced  by  a  force  applied  at  the  circum- 
ference of  the  larger  wheel,  and  a  larger  resistance  balancing 
this  at  the  margin  of  the  smaller  wheel.  This  is  the  principle 
of  the  capstan  and  the  winch — the  former  used  on  ships  for 
raising  the  anchor,  the  latter  in  use  for  drawing  buckets  up 
wells.  In  the  former  the  spokes  of  the  larger  wheel  are  few, 
while  the  smaller  takes  the  form  of  a  cylinder  or  drum ;  in  the 
latter  the  smaller  takes  the  same  cylindrical  form,  while  the 
larger  consists  virtually  of  only  one  spoke,  the  handle,  which  is 
turned  through  all  successive  positions  in  a  circle. 

The  wheel  and  axle  is  a  statical  instrument  so  long  as  its 
moments  round  the  axis  are  together  =  0 ;  but  when  one  of  the 
moments  is  numerically  greater  than  the  other,  there  is  rotation. 

Wheelwork.  —  If  a  force  be  applied  to  the  first  wheel  of  a 
chain  of  wheelwork,  so  that  it  acquires  an  angular  velocity  &>, 
and  if  the  last  wheel  of  the  chain  have,  in  consequence  of  this, 
an  angular  velocity  a>;,  the  force  which  the  last  wheel  can 
exert  is,  as  compared  with  that  which  the  first  wheel  alone 
might  exert  when  running  with  angular  velocity  o>,  as  o> :  «.. 
The  principle  holds  good,  whatever  the  nature  or  complication 
of  the  mechanism  which  intervenes  between  the  first  and  the 
last  wheel.  In  a  crane  or  in  a  lathe  arranged  for  metal-cutting, 
we  see  the  wheelwork  so  devised  that  the  last  axis  moves  very 
slowly,  and  with  a  correspondingly  great  power  of  overcoming 
resistance. 

The  Inclined  Plane.  —  The  mechanical  advantage  of  this 
machine  depends  on  the  principle  of  the  resolution  of  a  force 
into  its  components. 

When  a  body  is  pushed  up  an  inclined  plane  by  a  force  or 
push  just  sufficient,  and  no  more,  to  prevent  it  from  moving 
down  the  plane  in  obedience  to  gravity,  there  is  equilibrium 
between  three  forces  —  viz.  this  Force,  acting  along  the  slope 
of  the  incline,  the  Weight  of  the  body  acting  vertically,  and 
the  Reaction  between  the  body  and  the  surface  of  the  plane, 
acting  at  right  angles  to  the  latter.  These  three  forces  can  be 
represented  by  the  sides  of  a  right-angled  triangle,  in  which  the 

Hypotenuse        :  Height  :  Base 

as  Weight  of  body :  Push  up  the  plane  :  Reaction. 

If  the  push  be  applied  horizontally,  the  three  forces  in 
equilibrium  —  which  are  the  Weight  of  the  body  downwards, 


VI.] 


THE   MECHANICAL  POWERS. 


173 


the  Reaction  at  right  angles  to  the  surface  of  the  plane,  the 
Push  up  the  plane  applied  horizontally  —  will  have  the  relation 
of  the  sides  of  a  right-angled  triangle,  in  which 

Hypotenuse  :  Height  :  Base 

as  Reaction       :  Horizontal  Force  :  Weight  of  body. 

Velocity  of  fall  down  a  frictioiiless  inclined  plane.  —  If  a  body 
slip  down  an  inclined  plane,  the  potential  energy  lost  by  it  in  virtue  of  its 
vertical  descent  h  is  mgh  ;  there  is  none  gained  or  lost  in  virtue  of  horizon- 
tal motion,  in  which  there  is  no  work  done  by  or  against  gravity.  The 
kinetic  energy  acquired  is  ^my2.  These  must  be  equal;  hence  v  =  \/'2gh, 
the  same  speed  as  would  have  been  acquired  by  a  vertical  fall.  In  the 
latter  case,  however,  the  direction  of  motion  would  have  been  directly  down- 
wards ;  in  the  former  it  is  in  the  direction  of  the  plane.  The  reaction  of 
the  plane  has  not  modified  the  speed  of  the  fall  ;  it  has  modified  its  direc- 
tion. The  speed  at  which  the  body  is  moving  down  the  plane  after  effect- 
ing a  vertical  descent  h  is  thus  the  same  velocity  as  that  which  it  would 
have  acquired  if  it  had  fallen  vertically  through  that  height  h.  But  it  has 
travelled  through  a  greater  space  in  order  to  attain  this  speed.  The  accelera- 
tion down  the  plane  is  therefore  smaller  ;  and  a  body  slipping  down  a  smooth 
slope  of  1  in  20  would  take  a,  greater  time  to  reach  the  bottom  than  it  would 
take  to  fall  vertically  through  an  equivalent  height,  in  the  ratio  of  20  :  1. 

If  the  body  travelled  down  a  succession  of  inclined  planes,  or  down  a 
curve,  the  same  ultimate  velocity  would  be  acquired  :  the  reaction  of  the 
curve  alters  the  direction  but  not  the  speed.  If  it  rolled  down  the  plane  or 
curve,  a  part  of  its  energy  would  be  rotational  ;  it  would  acquire  a  corre- 
spondingly smaller  velocity  of  fall. 

The  Screw.  —  In  Fig.  85,  across  the  rectangular  parallelo- 


gram Aaa'B  are  drawn  equidistant  lines 
at  right  angles  to  Aa  and  Ba'.  The  lines 
abf,  be',  cdf,  etc.,  are  drawn  as  there  shown. 
If  the  surface  Aaa'B  be  wrapped  round  a 
cylinder  whose  circumference  is  equal  to 
AB,  the  line  Aa  will  coincide  with  Baf, 
and  the  lines  ab',  be',  cd',  etc.,  will  form 
a  continuous  spiral  line  abed  round  the 
cylinder,  and  will  trace  out  the  form  of 
the  thread  of  a  screw  whose  pitch  is  ab, 
the  distance  between  the  equidistant  lines 
ab',  be',  etc.  Hence  the  thread  of  a  screw 
is  seen  to  correspond  to  a  narrow  inclined 
plane  wrapped  round  a  cylinder. 

A  screw  is  usually  used  as  a  mechani- 
cal power  for  the  sake  of  moving  a  body 
through  a  small  space  with  great  force,  as 
in  the  copying-press. 


bbf,  cc1,  dd\  etc., 


Fig.85. 


174  t  KINETICS.  [CHAP. 

The  less  the  pitch  of  the  screw  —  the  greater  the  number 
of  turns  to  the  inch — as  well  as  the  greater  the  leverage  of 
the  handles,  the  greater  the  mechanical  advantage  that  can  be 
derived  from  its  use. 

Problem. 

What  is  the  mechanical  advantage  which  can  be  obtained  in  a  copying- 
press  of  the  following  construction  :  —  Effective  radius  of  the  arms  12  inches 
— screw  1}  inches  thick — pitch  $  inch?  —  Ans.  The  hands  move  through  1 
inch,  while  the  point  of  the  screw  descends  ^  inch.  .  • .  F1  =  603  F.  The 
thickness  of  the  screw  is  of  no  consequence,  except  as  a  means  of  securing 
structural  strength. 

The  Wedge.  —  A  wedge,  as  seen  in  Fig.  86,  is  practically 
a  double  inclined  plane  movable  between  resistances.  During 
Fig.86.  a  blow  there  are  at  work  (1)  the  driving  force 
B  acting  downwards  through  the  centre  of  AB, 
(2)  a  reaction  at  right  angles  to  AC,  and  (3) 
one  at  right  angles  to  BC ;  these  latter  being 
through  the  point  of  contact,  or,  if  there  be 
contact  over  the  whole  of  AC  and  BC,  through 
the  centre  of  these  lines.  These  must  cross 
in  a  point  if  the  equilibrium,  which  subsists 
the  instant  before  the  wedge  commences  to 
move,  be  considered ;  and  they  must  be  repre- 
sented by  the  sides  of  a  triangle.  Round  the  point  at  which 
they  meet,  the  moments  =  0. 

Pulleys.  —  These  well-known  objects  are  wheels,  solid  or 
spoked,  mounted  in  a  framework  or  block,  which  is  either 
movable  or  fixed  to  a  beam  or  other  solid  attachment.  The 
simplest  use  of  a  pulley  is  to  change  the  direction  of  applica- 
tion of  a  force  applied  to  a  cord.  The  total  tension  of  the  cord 
on  one  side  of  a  pulley  would,  if  there  were  no  friction,  be 
equal  to  that  on  the  other  side  of  it,  while  the  motion  of  the 
cord  on  the  one  side  of  the  pulley  is  in  any  case  equal  to  that 
on  the  other  side,  whatever  be  the  size  of  the  pulley,  and  what- 
ever be  the  amount  of  the  flexure  to  which  the  cord  is  sub- 
jected. A  single  pulley  thus  produces  no  mechanical  advantage 
if  it  simply  serve  this  purpose,  except  in  so  far  as  the  change 
of  direction  of  the  cord,  produced  by  the  intervention  of  the 
pulley,  may  itself  be  of  advantage ;  but  if  this  pulley  be  itself 
movable  against  a  resistance — if,  for  instance,  a  heavy  mass  be 
suspended  from  it,  while  the  other  end  of  the  cord  is  attached, 
say,  to  the  roof — a  movement  of  the  suspended  mass  through 


TI.]  THE   MECHANICAL  POWERS.  175 

one  inch  would  correspond  to  the  pulling  in  of  two  inches  of 
cord,  and  the  hand  exerting  the  force  would  move  through  a 
space  twice  as  great  as  that  traversed  by  the  pulley.  Thus,  by 
the  intervention  of  a  cord,  one  end  of  which  is  fixed,  and  of  a 
single  pulley  round  which  the  cord  is  bent  through  180°,  so  as 
to  make  it  turn  back  parallel  to  itself,  a  resistance  2  may 
be  overcome  by  a  force  just  greater  than  1.  This  principle  of 
reduplication  of  a  string  round  a  pulley  is  taken  advantage  of 
and  practically  turned  to  use  in  combinations  of  pulleys,  in  any 
of  which  the  mechanical  advantage  is  the  numerical  ratio  of  the 
amount  of  string  pulled  out  to  the  corresponding  movement  of 
the  body  pulled  upon. 

The  Bell-crank.  — If  in  Fig.  87  the  rigid  body  ABC,  which 
can  rotate  round  A,  have  a  force 
applied  to  it  at  C,  its  tendency  to 
rotate  round  A  may  cause  motion  at  B 
against  a  resistance  R.  The  principle 
of  moments  shows  us  that,  whatever 
the  ornamental  shape  of  the  crank,  the  A(O) 

relation  of  the  resistance  R  overcome 

to  the  force  P  exerted  depends  on  the  |p 

relative  lengths  of  the  effective  arms 
AP  and  AR. 

The  Knee.— In  Fig.  88,  if  two  bars  be  jointed  at  O,  and 
their  ends  A  and  B  be  confined  to  a  given  straight  line  CD,  a 
movement  of  the  hinge  O  athwart  the  direction  of  the  line  CD 
corresponds,  especially  when  AO  and  OB  are  nearly  in  the  same 
straight  line,  to  a  relative  motion  of  A  and  B,  which  is  propor- 
tionately very  small.  Hence  A  and  B  are  thrust  asunder  with 
a  force  greater  than  that  which  acts  upon  the  hinge  and  presses 
it  into  its  central  position.  This  contrivance  is  used  in  some 
copying-presses,  hand  printing- 
presses,  and  railway-ticket  en- 
dorsing machines.  It  is  seen 
in  the  human  knee :  when  the 

leg  is  straightened  out  a  vig-    * . — ^      D 

orous  thrust  upwards  and  for- 
wards is  given  to  the  body,  and  a  corresponding  one  downwards 
and  backwards  to  the  earth  on  which  the  foot  presses. 

A  wire  stretched  between  two  points,  and  loaded  by  a  weight  or  by  the 
pressure  of  the  wind,  is  a  knee  whose  action  is  reversed.  It  tends  to  pull 
together  the  two  supports  to  which  it  is  fixed  ;  and  if  there  were  any  move- 


176 


KINETICS. 


[CHAP. 


ment  of  these  supports,  it  would  be  small  in  comparison  with  the  correspond- 
ing movement  of  the  centre  of  the  wire.  Thus  the  force  acting  upon  the 
supports  and  resisted  by  them  is  greater  than  that  acting  upon  the  wire 
itself. 

FRICTION. 

Statical  Friction  between  Solids.  —  Let  a  body  M,  of  mass 
w&,  be  supported  on  a  table  T ;  and  let  the  total  pressure  between 
M  and  T  be  P,  which  may  either  be  the  weight  mg  of  the  body 
M  or  have  any  other  value  or  source ;  and  let  a  force  F,  say  the 
weight  of  a  mass  m'  suspended  over  a  pulley,  be  employed  to 
pull  it  towards  the  edge;  then  the  body  M  will  not  begin  to  slide 


Fig.89. 


M 


along  T  unless  the  force  F  bear  a  certain  proportion  to  the  total 
pressure  P.  This  proportion,  a  fraction  less  than  unity,  is  the 
Coefficient  of  Statical  Friction,  is  usually  represented 
by  the  symbol  //,,  and  has  to  be  experimentally  found. 

The  force  P  encounters  a  Frictional  Resistance,  F,  which  has  a  maxi- 
mum numerical  value  //P.  When  F  is  less  than  /xP,  there  can  be  no  sliding ; 
when  F  =  /xP,  sliding  is  just  about  to  commence  ;  and  when  F  is  greater 
than  /xP,  there  will  be  sliding.  This  Frictional  Resistance  F  is  brought  into 
being  by  the  application  of  the  force  F ;  and  it  enters  into  calculations  as  if 
it  were  aft  oppositely  directed  Force,  equal  to  F,  and  preventing 
movement,  until  F  comes  to  be  equal  to  /xP,  but  unable  itself  to  exceed  that 
value. 

Experiment  has  shown  that  /x,  the  coefficient  of  statical 
friction,  depends  upon  (1)  the  nature  of  the  substances  of  which 
M  and  T  consist;  (2)  the  smoothness  or  roughness  of  their 
surfaces ;  (3)  the  presence  or  absence  of  thin  films  of  lubricat- 
ing material  —  oil,  soap,  blacklead  —  between  them. 

The  coefficient  of  friction  is  the  same  between  the  same 
two  substances,  whatever  the  value  of  F  or  that  of  the  press- 
ure P  may  be ;  and  hence  F,  the  statical  resistance  to  sliding, 


vi.]  FRICTION.  177 

being  numerically  equal  to  yaP,  varies  directly  with  the 
total  pressure  P  between  the  given  surfaces.  Again,  the  mass 
of  M  may  be  distributed  in  any  way,  and  the  contact  between 
M  and  T  may  be  by  a  surface  large  or  small.  If  the  area  of  contact 
be  diminished,  the  pressure  on  each  unit  of  area  will  be  increased, 
arid  therefore  the  friction  on  each  unit  of  area  will  also  be  pro- 
portionately greater ;  but  the  number  of  units  of  area  over  which 
the  resistance  is  exerted  is  correspondingly  smaller,  so  that  the 
force  which  is  just  competent  to  pull  the  body  M  towards  the 
edge  remains  the  same.  Hence  the  Total  Frictional  Resist- 
ance F  is  independent  of  the  area  of  contact  between 
two  given  masses  ;  but  the  Frictional  Resistance  per  Unit  Area 
of  contact  varies  directly  as  the  Pressure  per  Unit- Area. 

Limiting  Angle.  —  In  Fig.  89  we  may  consider  the  equi- 
librium subsisting  at  the  instant  before  the  body  M  begins  to 
slide  on  T.  The  body  M  is  at  rest  under  (1)  the  force  F  acting 
horizontally,  (2)  its  own  weight,  mg,  acting  vertically,  and  (3) 
the  reaction  R  between  M  and  T.  This  reaction  R  is  inclined  at 
an  angle  ^  to  the  vertical.  The  horizontal  component  of  R  is  to 
the  vertical  as  tan  ^  is  to  1.  But  (horiz.-compon./vert.-compon.) 
=  F/w#  =  F/P  =//,;  .'./*  =  tan%  =  the  Coefficient  of  Statical 
Friction. 

Any  force  applied  to  M,  or  any  set  of  forces  whose  result- 
ant acts  on  it,  in  a  line  making  with  the  normal  to  the  surface 
between  M  and  T  an  angle  less  than  ^,  will  not  produce 
sliding.  If  there  were  no  friction,  any  force  applied  to  M  in  a 
direction  differing  in  the  least  degree  from  the  normal  would 
have  a  component  which  would  produce  sliding ;  but  Friction 
makes  it  necessary  that  a  Force  should  be  wide  of  the  normal 
to  the  surfaces  of  contact,  by  something  more  than  the  Limit- 
ing Angle  %,  before  sliding  can  occur. 

If  a  flat  piece  of  wood  be  placed  on  a  table,  and  pressed  against  the 
table  by  a  stick  held  at  right  angles  to  it,  it  wDl  not  slip ;  the  stick  may 
be  inclined  somewhat,  and  still  it  will  not  slip ;  when  the  stick  is  inclined 
more  than  a  certain  amount,  the  piece  of  wood  begins  to  slip  on  the  table. 

If  M  be  pressed  against  T  by  a  force  F0,  acting  in  a  line  which  makes 
any  angle  <£  with  the  normal,  we  may  resolve  this  force  into  two  components  : 
one  at  right  angles  to  the  surface,  (F0-  cos^>}  =  P,  produces  pressure 
between  these  surfaces ;  the  other,  (F0  •  sin  <£}  =P,  is  the  component  which 
tends  to  produce  sliding.  If  the  sliding-component  -=-  the  pressure-com- 
ponent (i.e.  tan  <£)  be  less  than  or  merely  equal  to  /A,  there  will  be  no  sliding ; 
if  it  be  greater  than  p  (i.e.  if  <f>  be  greater  than  x),  sliding  wiH  occur.  If 
there  be  no  sliding,  the  component  which  tends  to  produce  it  sets  up  a  con- 
dition of  stress  between  the  particles  of  the  two  bodies,  and  thus  a  reaction 

N 


178  KINETICS.  [CHAP. 

is  set  up  by  molecular  forces,  equal  and  opposite  in  the  direction  of  the 
sliding-component.  The  amount  of  this  reaction  depends  on  the  molecular 
conditions  of  the  substances,  and  is  only  to  be  determined  by  experiment. 

Angle  of  Repose.  — Suppose  a  body  to  be  placed  on  a  table, 
and  the  table  to  be  tilted  up  until  the  body  is  just  about  to 
begin  to  slide.  At  that  moment  there  is  equilibrium.  To  what 
angle  can  the  table  be  tilted  up?  Let  ^  be  the  coefficient  of 
statical  friction  between  the  body  and  the  table.  The  one  will 
then  slip  on  the  other  if  a  force  be  communicated  between  them 
in  the  direction  of  a  line  making,  with  the  normal  to  the  sur- 
face of  contact,  an  angle  greater  than  %.  In  Fig.  90  are  shown 
Fig  90  two  positions  of  the  table  T  bear- 

ing the  body  M.  In  both  the 
dotted  lines  indicate  the  limiting 
angle  % ;  in  both  the  weight  G  of 
the  body  M  acts  vertically  down- 
wards. The  equilibrium,  then, 
is  between  (1)  the  Pressure  P  pro- 
/  x  \  L---''\  duced  between  the  body  and  the 

table,  at  right  angles  to  the  latter, 

=  G  cos  <f> ;  (2)  the  Sliding-Component,  F  =  G  sin  </> ;  and  (3) 
the  Reaction  of  the  table,  equal  and  opposite  to  G,  the  weight 
of  the  body.  In  the  first  case  of  Fig.  90  the  reaction  between 
the  body  and  the  table  falls  within  the  limiting  angle,  and  there 
is  no  sliding :  the  sliding  component  is  less  than  //,  x  the  press- 
ure. In  the  second  case  the  limit  is  reached ;  the  sliding  com- 
ponent is  just  equal  to  //,  x  the  pressure,  and  is  just  able  to 
balance  the  friction.  If  the  table  were  tilted  up  any  farther, 
sliding  would  occur.  But  in  the  second  case  it  is  easy  to  show 
that  the  angle  <£,  to  which  the  table  has  been  tilted,  is  itself 
equal  to  ^  the  limiting  angle.  Hence  the  angle  ^  is  also  called 
the  Angle  of  Repose.  Upon  the  coefficient  of  mutual  fric- 
tion depend  in  this  way  the  angles  at  which  heaps  of  sand,  of 
grain,  and  the  like,  will  adjust  themselves  when  poured  out  and 
allowed  to  find  their  own  position. 

This  angle  %  has,  then,  three  properties  :  (1)  the  Coeffi- 
cient of  Statical  Friction,  /m  =  tan  ^ ;  (2)  ^  is  the  Limiting 
Angle ;  (3)  %  is  the  Angle  of  Repose. 

Friction  of  a  rope  round  a  post.  —  This  is  familiar  in  the  example  of 
a  rope  passed  round  a  post  on  a  quay  in  order  to  hold  fast  a  ship.  If  any 
little  part  or  element  of  the  rope  be  considered,  it  will  be  seen  that  the  fric- 
tion is  proportional  to  the  pressure  of  that  part  of  the  rope  on  the  post,  and 


vi.]  FRICTION.  179 

that  to  a  certain  extent  it  tends  to  prevent  slipping ;  in  this  it  partly  coun- 
teracts the  tension  of  the  rope  ;  the  total  tension  communicated  to  the  end 
of  the  element  of  the  rope  farther  from  the  applied  force  is  less  in  conse- 
quence of  this  than  it  would  have  been  if  there  had  been  no  friction.  If  we 
trace  out  in  this  way,  along  the  rope,  the  gradual  diminution  of  tension,  we 
find  that  the  tension,  after  a  complete  turn  of  the  rope  round  the  post, 
dwindles  down  to  a  constant  fraction  of  the  original  tension.  Between  a 
flexible  rope  and  wood  this  constant  fraction  is  about  ^ ;  hence  a  lorce  of 
1  Ib.  could  prevent  a  force  of  9  Ibs.  from  pulling  a  flexible  rope  round  a  post 
round  which  it  had  been  passed  so  as  to  form  a  complete  turn.  After  two 
turns  the  tension  falls  to  -fa ;  after  ten  turns  it  becomes  1/910  =  1/3486,784401. 
Hence  a  man  exerting  a  pull  of  1  Ib.  at  the  end  of  a  rope  wound  ten  times 
round  a  post  would  be  able  to  resist  a  pull  of  about  one-and-a-half  million 
tons.  Of  course  this  is  not  attained  in  practice,  because  no  ropes  are  thor- 
oughly flexible,  and  none  are  strong  enough  to  stand  such  stresses ;  but  a 
perfectly  flexible  rope  would  diminish  tension  in  this  manner  without  refer- 
ence to  the  diameter  of  the  post  round  which  it  is  wrapt. 

Kinetical  Friction  between  Solids.  —  After  slipping  has 
begun,  the  motion  is  retarded  by  a  Frictional  Resistance,  which 
is  still,  at  low  speeds  and  with  moderate  values  of  P,  proportional 
to  the  pressure  P  between  M  and  T.  This  is  equivalent  to  a 
virtual  negatively-directed  pulling  force  —R-—  b  -P.  The 
effective  accelerating  force  acting  upon  the  mass  (m  -+-  m')  of 
Fig.  89  is  thus  (F  -  R)  =  (m'g  -  bP).  If  the  weight  m'g  of  the 
pulling  mass  m'  be  replaced  by  an  equivalent  constant  sliding- 
force  of  F  dynes  (where  F=m'^),  the  mass  moved  is  not 
(m  +  m'),  but  m  alone ;  and  the  acceleration  produced  is 
(F/m  -  R/m)  =  (F/m.  -  b  •  P/m)  =  (1  -  6P/F)  •  F/m.  If  there 
had  been  no  friction,  the  acceleration  would  have  been  F/m; 
the  negative  acceleration  ( —  b  -  P/m)  is  constant,  whatever 
may  be  the  sliding  force  F  —  within  a  pretty  wide  range  of 
values  —  and  is  therefore  independent  of  the  velocity ;  and  b  is 
the  Coefficient  of  Kinetical  Friction,  which  is  smaller  than 
/i,  the  coefficient  of  statical  friction. 

Influence  of  Duration  of  Contact.  —  There  are,  however,  even  for 
moderate  values  of  the  sliding  force  F  and  of  the  pressure  P,  slight  varia- 
tions in  the  values  of  /A  and  b.  When  two  bodies  have  been  in  contact 
for  a  long  time,  the  particles  of  each  develope  such  relations  to  one  another 
that  /x,,  the  coefficient  of  statical  friction,  increases  with  the  duration  of 
contact ;  it  is  more  difficult  to  make  a  body  slide  on  another  with  which  it 
has  been  long  in  contact  than  on  one  on  which  it  has  been  freshly  placed. 
When  one  surface  slides  on  another,  the  particles  seem  to  have  no  time  to 
assume  such  relations,  and  the  coefficient  of  kinetical  friction  is  compara- 
tively small ;  at  very  great  velocities  it  is  even  somewhat  smaller  than  at 
ordinary  velocities ;  but  when  the  velocity  of  sliding  is  very  small,  the  con- 
dition approximates  to  one  of  relative  rest,  the  coefficient  of  kinetical 
friction  approximates  to  that  of  statical  friction,  a  larger  proportion  of 


180  KINETICS.  [CHAP. 

energy  disappears  at  very  low  speeds  than  at  high,  and  a  body  which 
has  come  to  travel  very  slowly  soon  comes  to  rest. 

Transformation  of  Energy  by  Kinetic  Friction.  —  The 

sliding  force  F  has  apparently  lost  a  fraction  of  its  amount ; 
the  velocity  produced  is  diminished  in  proportion ;  Energy  is 
wasted,  but  not  destroyed,  by  being  transformed  into  molecular 
kinetic  or  potential  energy.  In  the  former  case  the  energy 
absorbed  may  perhaps  assume  the  form  of  the  energy  of  electric 
condition,  but  ultimately  it  takes  that  of  Heat,  which  warms  the 
machinery  and  the  air  surrounding  it ;  in  the  latter  case  it  cor- 
responds to  a  stress  between  the  particles  of  the  bodies,  to  pull 
which  asunder  requires  a  certain  amount  of  force. 

Kinetic  Friction  (—  R)  is,  accordingly,  not  a  Force ;  it  is 
a  Resistance  or  Reaction:  but,  like  static  friction,  —  F,  it 
enters  into  calculations  as  if  it  were  a  Force,  never  coming  into 
action  unless  Force  be  applied,  always  tending  to  prevent  or 
to  diminish  slipping,  and  always  proportional  to  the  pressure 
between  the  rubbing  surfaces. 

Negative  Acceleration  in  Kinetic  Friction.  —  Problems  of 
loss  of  momentum  through  Friction  may  be  dealt  with  by  using 
the  four  equations  of  p.  152,  writing  (  —  6-P/ra)  for  the  con- 
stant acceleration  a;  P  being  the  number  of  dynes  of  Total 
Pressure  and  m  the  mass,  in  grammes,  of  the  moving  body. 

Brakes.  —  The  function  of  a  Brake  is  to  modify  the  Total  Press- 
ure, P  dynes,  between  the  moving  mass  and  the  surface  against  which  it 
rubs.  This  may  be  done  by  clamping  the  brake  against  the  moving  mass 
to  any  desired  extent.  This  affects  the  value  of  the  negative  acceleration 
(— &.p/ra).  The  total  pressure  may  also  be  affected  by  multiplying  the 
surfaces  of  contact.  If,  for  example,  two  pamphlets  be  arranged  with  their 
leaves  alternately  interplaced,  it  is  surprising  how  small  a  weight  superim- 
posed will  lock  them  firmly  together. 

The  Critical  Angle  in  Kinetical  Friction.  —  This  angle,  correspond- 
ing to  x  m  statical  friction,  is  \f/,  where  b  =  tan  \j/.  There  is  no  work  done 
in  pulling  a  body  downhill  at  this  angle,  for  the  frictional  resistance  is  then 
exactly  balanced  by  the  component  of  the  Weight,  directed  down  the  slope. 
If  the  slope  be  steeper  than  this,  brakes  or  ropes  are  required  to  prevent 
acceleration.  If  b  =  ^,  as  in  most  railway  work,  with  good  lubrication, 
this  slope  is  such  that  tan  \j/  =  ^ ;  a  slope  of  1  in  250. 

The  mechanical  powers,  when  friction  is  taken  into  account,  give  rise 
to  several  problems ;  but  the  physical  principle  underlying  the  whole  subject 
is  the  same,  that  Friction  acts  in  the  same  way  as  a  Force  opposed  to  slid- 
ing, and  that  it  is  proportional  to  the  Total  Pressure.  As  an  example  let  us 
take  this  question :  A  copying-press  is  pressed  hard  down  on  the  copying- 
book;  the  hands  are  removed;  the  book  remains  under  pressure; — why 
does  the  screw  not  come  up  ?  The  reaction  of  the  book  has  a  component  up 
the  line  of  the  thread  of  the  screw ;  this  would  tend  to  send  up  the  screw,  but 


vi.]  FRICTION. 

it  is  counterbalanced  by  Friction,  acting  as  a  Resistance  in  the  opposite 
direction  down  the  thread.  It  may  be  left  to  the  reader  to  show  (1)  that 
the  better  the  screw  is  oiled  the  less  able  will  it  be  to  retain  its  hold ;  and 
(2)  that  a  screw  of  too  large  a  pitch  (one  the  turns  of  whose  thread  are  too 
far  apart)  may  fail  to  hold  the  book  down.  The  upward  pressure  is  resolved 
into  two  components,  of  which  the  one  along  the  thread  of  the  screw  must 
not  be  greater  than  /x  x  the  component  at  right  angles  to  the  thread ;  if  it 
be  greater  than  this  the  screw  will  slip  upwards  in  its  nut.  When  there  is 
actual  motion,  the  forces  acting  are  subject  to  deductions  equal  to  the  re- 
spective values  of  R  =  b  -  P. 

Work  done  against  Friction.  —  The  acceleration  (  -  5  •  P/m) 
being  constant,  the  work  done  against  Friction  =  Es  — 
m  -  bP/m  •  s  =  bPs ;  and  this  is  equal  to  b-mg-s,  when  gravity 
alone  determines  the  pressure  P.  To  this  would  have  to  be 
added  any  work  done  against  gravity  or  other  external  forces 
while  the  space  s  is  being  traversed. 

Resistance  to  Traction.  —  The  Frictional  Resistance,  -R  =  -&P, 
corresponds  numerically,  when  P  =  mg,  that  is,  when  gravity  is  the  only 
cause  of  the  pressure  P,  to  b  -  mg,  the  Weight  of  a  mass  bm,  where  b  is  a 
numerical  factor  less  than  unity.  Suppose  6  =  ^|7;  then  so  long  as  b 
remains  constant,  which  it  does  within  wide  limits,  the  work  to  be  done  in 
making  any  mass  m  move  horizontally  through  a  space  B  at  any  uniform 
velocity  is  -goings,  and  is  therefore  one  three-huiidred-and-twentieth  part 
^f  the  work  required  to  raise  the  same  mass  m  vertically  against  uniform 
terrestrial  gravity  through  the  same  space  in  the  same  time.  Engineers 
would  express  this  by  saying  that  the  f rictional  resistance  is  7  Ibs.  per  ton, 
or  3J-  kg.  per  tonne  of  1000  kg.  In  this  way  a  uniform  rate  of  motion 
might  be  maintained  in  a  train  weighing  100  tons  pulled  along  a  level  road 
by  an  engine  which  exerted  a  pull  equal  to  the  weight  of  700  Ibs.,  or  7  Ibs. 
per  ton  of  train-load ;  and  then  the  work  to  be  done  by  the  engine  would 
be  the  same  as  it  would  have  been  had  the  same  engine  been  set  to  pull 
700  Ibs.  vertically  upwards  with  the  velocity  at  which  the  train  is  travelling. 

The  resistance  is  said  to  be  so  many  Ibs.  per  ton,  nothing  being  said  as 
to  the  velocity.  This  is  because,  within  wide  limits,  the  coefficient  b  of 
kinetical  friction  is  practically  independent  of  the  velocity ;  though,  when 
the  speed  becomes  very  small,  the  waste  of  energy  occasioned  by  friction 
becomes  proportionately  large,  the  converse  holding  good  at  high  speeds. 

If  the  road  be  not  level,  but  go  uphill,  then  there  is  lifting  work  to  be 
done  as  well  as  work  done  against  friction  ;  and  if  the  slope  be,  say,  1  in 
100,  then  for  every  100  feet -of  horizontal  travel,  the  whole  load  must  also 
be  lifted  one  foot.  Therefore  an  engine,  moving  an  eighty-ton  train  along 
a  level  road  with  resistance  equal  to  7  Ibs.  per  ton,  would  have  to  do  work 
equivalent  to  raising  560  Ibs.;  whereas  when  it  begins  to  go  up  a  slope 
of  1  in  100,  it  has,  in  addition  to  these  560  Ibs.,  to  lift  0-80  ton,  making 
2352  Ibs.  in  all. 

When  a  man  walks  his  knee  is  straightened,  and  his  body  is  projected 
forwards  and  upwards  at  each  step.  The  impulse  may  be  resolved  into  two 
components  :  one  upward,  which  may  raise  the  centre  of  gravity  of  the  body 
about  an  inch  or  an  inch-and-a-quarter ;  one  forward,  which  has  to  overcome 


182  KINETICS.  [CHAP. 

the  intermittent  resistances  introduced  by  the  stoppage  occurring  at  the  end 
of  each  step,  when  the  foot  of  the  opposite  side  strikes  the  ground.  This  is 
an  intermittent  frictional  resistance. 

If  there  were  no  friction  between  the  wheels  of  a  railway  train  and  the 
rails  of  the  railroad,  there  would  be  slipping  but  no  progress.  Friction 
between  the  wheels  and  the  rails — friction  proportional  to  the  Weight  of 
the  vehicles — has  the  effect  of  preventing  slipping;  but  (as  in  the  case 
of  belting)  this  corresponds  to  the  maintenance  of  a  state  of  rolling  adhe- 
sion, under  which  each  wheel  is  turned  round  and  rolls  upon  the  rail. 

Rolling  Friction.  —  When  a  ball  is  set  to  roll  on  smooth  ice,  it  goes 
farther  than  it  can  on  a  wooden  floor ;  farther  on  that  than  on  a  carpet ; 
farther  on  a  carpet  than  on  grass.  The  rotation  is,  however,  at  length 
stopped.  To  produce  rotation  a  torque  is  needed  :  to  stop  rotation  a  torque 
is  also  required.  This  may  be  that  of  a  force  or  resistance  acting  at  any 
point  which  is  not  the  centre  of  mass.  The  greater  the  moment  of  the 
resistance  round  the  centre  of  mass  the  sooner,  for  a  given  momentum,  will 
the  rotation  be  stopped.  In  this  manner,  the  rotation  of  the  ball  is  stopped 
by  the  Resistance  of  Friction.  This  is  equivalent  to  a  small  force  acting  at 
the  circumference  of  the  ball,  and  bearing  a  constant  ratio  to  the  pressure 
produced  by  its  weight.  This  ratio  is  very  small.  The  resistance,  then,  to 
a  wheel  rolling  along  the  ground  may  be  very  much  less  than  the  resistance 
to  the  same  object  when  pressed  upon  by  a  brake.  It  is  very  much  easier  to 
move  the  trunk  of  a  tree  by  setting  it  on  logs  which  roll  on  the  ground  and 
under  the  trunk,  than  it  is  to  drag  it  along  the  ground.  There  is  less  fric- 
tion at  a  well-oiled  hinge  or  well-lubricated  joint  than  there  would  be  in  any 
other  contrivance  used  for  transferring  a  given  mass  from  one  position  to 
another.  If  a  wheel,  instead  of  having  its  axle  supported  in  bearings,  have 
it  supported  on  a  couple  of  pairs  of  Friction-wheels  which  are  free  to 
rotate,  the  axle  as  it  turns  does  not  rub  against  a  fixed  bearing,  but  the 
friction-wheels  yield  and  rotate,  so  that  the  rotating  axle  is  supported  by 
surfaces  which  travel  at  the  same  rate  with  it,  and  the  friction  is  accord- 
ingly very  small.  In  all  these  cases  the  frictional  resistance  has  some 
definite  moment  round  the  axis  of  rotation. 

The  friction  is  affected  by  the  relative  softness  of  the  surfaces  in  con- 
tact (Osborne  Reynolds).  An  iron  wheel  rolling  upon  an  indiarubber 
plane  will  raise  up  before  it  a  little  mound  of  indiarubber ;  and  if  it  stop, 
this  little  mound  will  recover  its  form  and  drive  the  wheel  backwards,  thus 
making  it  oscillate.  The  friction  of  iron  upon  indiarubber  is  thus  ten  times 
as  great  as  the  friction  of  iron  upon  iron.  Conversely,  an  indiarubber  tire 
is  deformed  in  the  same  way  against  a  hard  surface.  This  tendency  to 
thrusting  forward  the  contact-layer  of  both  substances,  but  particularly 
that  of  the  softer,  results,  in  the  case  of  iron  railway  rails,  in  the  wear  of 
the  rail  by  scaling  off  of  successive  laminae  of  iron.  A  similar  result  may 
influence  most  cases  of  ordinary  friction,  as  in  the  spreading  of  putty  with 
the  thumb,  to  take  an  extreme  example ;  or  as  in  the  transverse  wearing  of 
railway  rails  by  trains  rounding  a  curve. 

Belting.  —  There  is  a  very  interesting  and  familiar  case  in  which  fric- 
tion serves  as  a  means  for  the  transmission  of  energy  —  that  is,  transmission 
by  machine-belting.  A  rotating  wheel  has  a  belt  tightly  drawn  over  it,  as 
also  over  a  second  wheel,  not  too  near.  The  belt  must  be  tight,  so  that  there 
may  be  more  pressure  between  the  leather  and  the  iron.  If  the  wheel  be 
very  small  or  the  motion  be  very  rapid,  the  mutual  pressure  between  the 


vi.]  FRICTION.  183 

leather  and  the  iron  may  be  lessened  by  the  inertia  of  the  belt,  which  tends 
to  pass  the  wheel  and  to  be  carried  on.  The  friction  is  proportional  to  the 
pressure  between  the  wheel  and  the  belt,  for  the  relation  of  the  wheel  to 
the  belt  is  practically  one  of  rest,  though  the  surfaces  in  contact  are  changed 
from  instant  to  instant.  There  being  no  slipping,  the  friction  is  statical, 
and  is  proportional  to  the  pressure.  Though  the  belt  and  the  wheel  do  not 
move  relatively  to  one  another,  they  move  relatively  to  surrounding  objects, 
and  the  belt  is  set  in  motion.  If  the  second  wheel  be  free  to  rotate  on  an 
axis,  portion  after  portion  of  its  rim  tends  to  remain  at  rest  relatively  to  the 
leather,  and  the  second  wheel  is  set  in  motion  round  its  axis.  The  tension 
of  the  belt,  that  is,  the  Total  Tension,  is  greater  nearer  the  driving  power 
than  it  is  on  the  other  side  of  the  wheel  driven.  This  is  because  Energy 
has  been  taken  up  in  preventing  relative  motion  of  the  belt  and  the  driven 
wheel,  or — another  mode  of  expressing  the  same  thing  —  in  producing  abso- 
lute motion  of  the  latter.  This  difference  of  tension  is  equivalent  to  a  force 
directly  applied  to  the  rim  of  the  wheel.  Thus  the  kinetic  energy  of  the 
driving  wheel  is  in  part  imparted  to  the  leather  belt  and  the  rotating  wheel ; 
these  come  to  form  a  part  of  the  same  system  with  the  driving  wheel ;  the 
latter  cannot  rotate  so  fast  when  it  is  driving  a  second  wheel  as  it  can  when 
not  doing  so,  the  same  energy  being  supplied  to  it ;  and  thus  energy  is 
transmitted. 

Such  is  the  theory  of  belting  when  there  is  no  slipping ;  but  in  practice 
there  is  always  some  slipping.  The  part  of  the  belt  in  front  of  the  pulley  is 
under  greater  tension  than  the  part  behind ;  it  is  therefore  more  stretched 
out  and  assumes  a  greater  length ;  and  this  involves  slipping,  which  causes 
a  loss  of  energy  spent  in  deforming  the  belt,  and  ultimately  transformed  into 
heat  in  the  belt ;  a  loss  which  in  the  case  of  leather  belting  is  appreciable, 
but  which  in  the  case  of  indiarubber  belting  is  very  considerable  (Osborne 
Reynolds). 

The  efficiency  of  belting  is  greatly  increased  when  the  rim  of  the  wheel 
is  lined  with  leather,  hair  side  outwards,  the  hair  side  of  the  leather  belt 
being  inwards  :  or  when  the  rim  is  coated  with  paper. 

Activity  in  Belting.  —  The  transmission  of  energy  by  belting  is  sub- 
ject to  the  law  that  Activity  =  TV  ;  v  being  the  velocity  at  which  the  belt 
runs,  and  T  the  tension  (i.e.  total  tension)  along  the  belt ;  this  tension  being, 
essentially,  the  difference  of  tension  between  the  outgoing  and  the  incoming 
parts  of  the  belt.  Let,  for  instance,  the  speed  of  the  belt  be  400  feet  a 
minute  or,  say,  200  cm.  per  second ;  and  let  the  effective  total  tension  on 
the  belt  be  equal  to  the  weight  of  100  kilogrammes  or  to  98,100,000  dynes ; 
then  the  Activity,  or  rate  of  transmission  of  energy  =  TV  =  Tension  x  Veloc- 
ity =  98,100000  x  200  =  19620,000000  ergs  per  second,  or  about  2£  horse- 
power. 

If  the  velocity  be  very  great,  the  tension  may  be  small.  Thus  let  the 
velocity  be  6000  feet  per  minute  or,  say,  3000  cm.  per  second,  a  speed  which 
has  been  attained  in  practice ;  and  let  the  desired  activity  of  transmission  be 
100  horse-power,  or  745,948,005000  ergs  per  second ;  we  have  Activity  =  Fv 
=  TV  or  745,948,005000  =  3000  T ;  whence  T  =  248,649335  dynes,  and  the 
tension  T  is  therefore  equal  to  the  weight  of  248,649335/981  =  253,465 
grammes  or  253-465  kilogrammes.  Such  a  tension  would  (since  steel  has  a 
"  breaking-weight "  of  33  tons  per  square  inch)  be  barely  able  to  snap  a  steel 
wire  of  £  cm.  diameter ;  whence  a  slender  steel  wire-rope  may,  at  very  high 
speeds,  be  safely  used  to  transmit  large  amounts  of  energy;  a  conclusion 
which  experience  has  confirmed. 


184  KINETICS.  [CHAP. 

Friction-Dynamometers.  —  Friction  may  be  utilised  as  a  means  of 
measurement  of  Rate  of  Doing  Work.  Suppose  a  cord  passed  round  a 
revolving  pulley ;  the  two  ends  of  the  cord  pass  away  from  the  pulley,  both 
vertically  or  otherwise  in  line  with  one  another ;  the  lower  end  is  stretched 
by  a  weight  G  dynes ;  the  upper  end  pulls  upon  a  fixed  spring  and  imparts 
to  it  a  strain  which  indicates  a  total  tension  of  T'  dynes.  The  weight  G  is  so 
great  and  tightens  the  string  so  much  that  the  whole  of  the  energy  of  the 
pulley  is  spent  in  overcoming  friction,  and  the  pulley  stops  at  once  when  the 
driving  power  is  withdrawn.  The  string  wraps  round  the  circumference  of 
the  pulley,  i.e.  lirr  cm. ;  the  velocity  of  that  circumference  in  passing  any 
point  of  the  string  is,  if  the  pulley  rotate  n  times  per  second,  v  =  n  •  2?rr  cm. 
per  second ;  the  force  overcome,  F,  is  the  difference  between  G  and  T',  i.e. 
(G  —  T')  =  T  dynes;  the  product  TV  is  therefore  equal  to  n-27ir-(G  —  T'). 
This  product  TV  measures  in  ergs  the  work  done  per  second  by  the  revolving 
pulley,  the  Rate  of  Doing  Work  of  the  pulley  (p.  42).  Instruments  of  this 
class  may  be  graduated  so  as  to  indicate,  by  the  amount  of  distortion  of 
a  spring,  the  working  value  of  a  steam-engine  in  horse-powers :  the  whole 
power  of  the  engine  is  turned  on  to  the  dynamometer  for  a  brief  period,  and 
the  scale-reading  of  the  spring  observed,  as  well  as  the  speed  of  rotation  of 
the  pulley. 

Variations  in  Kinetical  Friction.  —  The  coefficient  of  ki- 
netical  friction  is  found  to  present,  at  high  values  of  F  or  P,  or 
at  high  velocities,  or  with  varying  lubrications  or  forms  of  sur- 
face, or  with  different  kinds  of  movement  (continuous  or  alter- 
nating), considerable  differences  from  the  simple  constant  value 
obtained  by  making  one  solid  slide  upon  another  under  moder- 
ate pressures  and  velocities.  For  example,  with  abundant  lubri- 
cation, the  coefficient  b  actually  varies  inversely  as  the  press- 
ure P,  and  also  varies  directly  as  the  square  root  of  the  velocity; 
so  that  —R  =  —  B  Vv,  where  B  is  a  coefficient  depending  on  the 
kind  of  lubricant,  and  R  becomes  independent  of  the  pressure  P. 
The  different  values  of  B,  this  varying  coefficient  of  kinetical 
friction,  or  Friction-factor,  under  different  circumstances, 
can  only  be  ascertained  by  laborious  direct  observation. 

Between  metal  and  metal,  at  ordinary  working  velocities  of  axles  in 
their  bearings,  the  coefficient  of  kinetical  friction  is  approximately  constant. 
If  castor  oil  be  used  as  a  lubricant,  this  coefficient  is,  at  low  speeds,  very 
small ;  but  it  increases  rapidly  as  the  speed  rises.  If  water  or  thin  petro- 
leum oil  be  used  as  a  lubricant,  the  friction  at  speeds  beyond  a  certain  limit 
is  very  small ;  but  at  speeds  whose  average  is  below  that  limit,  there  is  alter- 
nate "  biting "  and  slipping.  Hence  for  axles  at  low  speeds,  thick  oils  ;  for 
high  speeds,  thin  oils  or  water. 

Friction  of  Moving  Solids  against  Liquids  depends 
directly  upon  the  extent  of  surface  exposed.  Further,  when 
the  speed  is  very  small,  the  frictional  resistance  is  nearly  con- 
stant, and  the  power  required  to  overcome  the  resistance  there- 


vi.]  FRICTION.  185 

fore  varies  as  the  velocity,  as  in  the  case  of  kinetical  friction 
between  solids ;  as  the  speed  increases,  the  frictional  resistance 
itself  comes  to  vary  as  the  velocity,  and  the  power  required 
comes  to  vary  as  the  square  of  the  velocity. 

Perhaps  the  friction  of  sharp  skates  against  smooth  ice  may  be  found 
to  be  in  this  category,  the  ice  being  melted  as  the  skate  runs. 

Work  done  against  a  uniform  frictional  resistance  R  through  space  a  is 
Rs,  or  Ra/t  per  second.  This  value,  Rs/t  per  second,  is  the  Activity  or 
Power  required,  and  is  equal  to  Rv.  If  R  be  constant,  the  Power  required 
to  overcome  the  friction  cc  v ;  if  R  itself  vary  with  v,  and  if  it  be,  say,  &v, 
the  Activity  is  kv-s/t  or  &v2. 

Friction  on  a  raindrop.  —  A  raindrop  falling  in  vacuo  through  a  height 
h  feet  would  acquire  a  velocity  v  =  ^/'2gh  =  8-249  Vh  feet  per  second.  Its 
starting  point  might  easily  be  so  distant  that  a  blow  from  a  raindrop  travel- 
ling under  these  circumstances  would  be  fatal  to  any  living  being  struck  by 
it.  But  at  every  instant  of  its  course  it  is  subject  to  kinetic  friction  tend- 
ing to  reduce  its  velocity  at  the  instant ;  at  the  same  time  it  is  subject  to 
the  accelerating  force  of  gravity :  and  thus  there  must  be  a  certain  velocity 
at  which  the  retardation  of  friction  and  the  acceleration  due  to  gravity  will 
balance  one  another,  and  the  drop,  if  it  once  attained  that  speed,  would 
retain  it,  and  fall  with  a  constant  velocity.  This  happens  in  the  case  of  the 
raindrop,  and  also  in  the  case  of  a  stone  or  granule  falling  in  deep  water. 

Viscosity-resistances.  —  If  a  body  start  free,  with  initial 
velocity  V0,  in  a  viscous  medium  which  offers  frictional  resist- 
ance varying  as  the  velocity,  its  speed  will  diminish  in  geomet- 
rical progression  in  successive  equal  intervals  of  time,  and  it 
will  gradually  approach  a  condition  of  rest. 

The  retarding  acceleration  is  proportional  and  opposite  to  the  velocity ; 
a=v=  — kv;  this  is  a  Differential  Equation,  which  gives  the  result  that  at 
the  end  of  time  t,  the  velocity  v,  =  v0  •  e-  k<,  where  v0  is  the  initial  velocity, 
and  c  =  2-7183.  At  the  end  of  one  second,  log  vt  =  (log  v0)  —  k ;  at  the  end 
of  two  seconds  it  is  (log  v0)  —  2k ;  at  the  end  of  n  seconds,  it  is  (log  v0)  —  nk. 
Thus  during  each  second  the  (Naperian)  logarithm  of  v  is  altered  by  the 
numerical  quantity  k;  and  this  is  the  Naperian  Logarithmic  Decre- 
ment of  the  velocity. 

Friction  in  S.H.M.  —  If  a  Circular  Pendulum  be  set  to 
oscillate  in  a  viscous  medium  in  which  the  frictional  retardation 
is  proportional  to  the  velocity,  the  circle  described  by  it  will 
gradually  dwindle.  It  will  take  a  longer  time  to  go  round  360° 
than  it  would  in  a  frictionless  medium,  but  it  will  do  so,  on  its 
consecutive  rounds,  in  equal  times ;  and  its  path  will  be  the 
curve  known  as  a  logarithmic  spiral.  The  distance  between 
its  bob  and  the  point  of  ultimate  rest  always  diminishes  in  equal 
proportions  after  describing  equal  angles ;  so  that  this  distance 
diminishes  in  geometrical  progression,  for  equal  intervals  of 


186  KINETICS.  [CHAF.  vi.] 

time.  If  the  frictional  retardation  be  small,  the  bob  may  go 
many  times  round  the  midpoint  before  reaching  it ;  but  if  it 
exceed  a  certain  limit,  the  bob  will  travel  with  diminishing 
velocity,  by  a  more  or  less  indirect  path,  towards  the  midpoint, 
taking,  theoretically,  an  infinite  time  to  reach  that  point. 

If  this  conical  pendulum  be  looked  at  from  one  side,  the 
S.H.M.  which  it  describes  will  be  isochronous,  but  will  be 
slower  than  it  would  have  been  in  a  frictionless  medium ;  and 
the  amplitude  will  appear  to  diminish  in  geometrical  progres- 
sion; while  if  the  retardation  be  excessive,  the  displaced  bob 
will  simply  appear  to  return  to  its  median  position  with  dimin- 
ishing velocity.  Actual  instances  of  these  kinds  of  movement 
are  to  be  seen  in  vibrating  bodies  where  the  retardation  is  due 
to  the  resistance  of  the  surrounding  fluid  medium  or  to  an  equiv- 
alent resistance,  that  of  viscosity,  which  has  its  seat  within  the 
vibrating  substance  itself  ;  and  in  the  damping  of  oscillations 
of  a  moving  body  by  increasing  the  resistance  of  the  surround- 
ing medium. 

The  acceleration  o  =  M  consists  in  these  cases  of  two  parts ;  one,  =  —  n2s, 
proportional  to  the  displacement  B  and  oppositely  directed;  the  other, 
kv  =  —  ks,  proportional  and  opposite  to  the  velocity.  Then  &  =  —  (ks  +  n2s). 
This  is  again  a  Differential  Equation,  and  it  has  two  orders  of  solution. 
First,  when  n  >  k/2,  the  displacement  Bt  (that  is,  the  value  of  the  displace- 
ment at  the  end  of  time  t)  =(a-  e-W2)(cos  tVu'2  —  k2/4).  On  comparing 
this  with  the  equation  x  =  a  •  cos  <at,  on  p.  82,  we  see  that  it  represents  a 
S.H.M.  in  which  the  amplitude  a  -  f-W*  diminishes  in  geometrical  pro- 
gression, with  a  constant  log.  dec.,  from  a  to  0 ;  the  angular  velocity  is 
a/  =  Vn2  — k2/4,  instead  of  w  =  n,  and  is  constant,  so  that  the  motion  is 
isochronous.  If  on  the  other  hand  n<k/2,  the  equation  is  satisfied  by  the 
condition  that  at  the  end  of  time  t,  the  displacement  is  reduced  from  a  to 
a  -  e-a,',  where  n,  =  k/2  +  Vk2/4  —  n2 ;  and  the  displacement  accordingly 
diminishes  from  a,  its  maximum,  to  nothing,  without  oscillations. 


CHAPTER  VII. 

ATTRACTION  AND  POTENTIAL. 

WHEN  one  body  in  contact  with  others  forms  with  them  a 
system,  a  Conservative  System,  which  will  be  put  in  a  condi- 
tion of  stress  when  the  bodies  are  removed  from  contact  with 
one  another,  these  bodies  are  said  to  be  Attracted  towards  one 
another ;  and  if  they  be  fixed  in  such  a  position  that  the  stress 
of  the  system  is  permanent,  the  condition  is  one  of  statical 
equilibrium.  When  a  spring  is  drawn  out  and  fixed  by  a  catch, 
there  is  equilibrium  between  the  recoil  of  the  spring  and  the 
molecular  forces  within  the  catch,  which  resist  its  deformation ; 
when  a  heavy  stone  is  placed  on  a  wooden  table,  if  the  table  be 
strong  enough  to  support  the  stone,  there  will  be  equilibrium 
between  the  weight  of  the  stone  and  the  resistance  to  crushing 
offered  by  the  wooden  support.  If  the  spring  be  released  it 
will  fly  back :  if  the  supporting  table  be  removed  the  stone  will 
fall.  In  the  former  case  there  is  a  visible  medium,  the  spring, 
the  elasticity  of  which  comes  into  play ;  in  the  latter  case  there 
is  no  such  elastic  medium  visible. 

There  is  no  direct  analogy  between  the  two  cases.  In  the  former,  the 
greater  the  displacement  the  greater  the  stress  ;  in  the  latter,  the  greater  the 
mutual  distance  the  less  the  mutual  attraction. 

Let  us  consider  attraction,  which,  whatever  may  be  its 
cause,  obeys  the  particular  law  (the  so-called  "Law  of  Inverse 
Squares  ")  that  a  mass  m  and  a  mass  mt  are  attracted  by  each 
other,  along  the  line  joining  them,  with  a  force  which  depends 
on  each  mass  directly,  and  on  the  square  of  the  distance  between 
them  inversely.  Then  F  oc  mm^d2;  that  is,  F  =  k-mnij/d?. 

In  a  system  of  particles  of  this  kind  we  must  further 
assume  —  and  experience  warrants  us  in  so  doing  —  that  every 
particle  is  connected  with  every  other  particle  by  an  inde- 
pendent attraction ;  then  the  total  attraction  of  one  set  of  par- 
ticles for  another  set  of  particles  has  to  be  found  by  a  process 

187 


188  ATTRACTION  AND   POTENTIAL.  [CHAP. 

of  summation.  To  effect  this  summation  the  aid  of  the  Integral 
Calculus  has  in  general  to  be  called  in ;  the  process  is,  how- 
ever, of  this  kind :  —  The  mass  and  the  distance  of  each  par- 
ticle from  every  other  being  taken  into  account,  the  attraction 
between  each  particle  and  every  other  particle  is  to  be  sepa- 
rately found,  and  the  whole  attractions  are  then  to  be  summed 
up.  In  simpler  cases,  mutually  attracting  masses  may  be  con- 
sidered as  acting  at  their  centres  of  figure ;  then  the  mean  dis- 
tance between  two  such  masses  is  the  distance  between  their 
centres  of  figure,  and  each  mass  may  be  supposed  to  be  concen- 
trated at  its  centre  of  figure. 

Attraction  in  particular  cases.  —  (1.)  A  hollow  spherical  shell, 
whose  thickness  is  infinitesimal,  attracts  an  external  particle  as  if  all  its 
mass  were  gathered  at  its  centre.  Its  area  is  4?rr2 ;  the  amount  of  mass  per 
unit  of  surface  (its  "  surface-density  ")  =  <r ;  its  mass  m  is  therefore  47rr2<r. 
It  acts  on  mass  m,  placed  at  a  mean  distance  d  from  the  centre  of  the  shell 
as  if  the  whole  mass  47rr2o-  were  at  that  centre,  and  the  attraction 
F  =  k  -  mt  -  4:7rr2<r/d2.  If  the  attracted  particle  be  of  unit-mass,  mt  —  1,  and 
the  attraction. of  the  shell  on  a  unit-particle  is  k •  47rr2<r/rf2. 

If  the  external  particle,  of  mass  mt,  be  just  outside  the  shell,  so  that  its 
distance  d  from  the  centre  is  practically  equal  to  r  the  radius  of  the  shell, 
d  =  r  and  F  =  k  •  m,  -  47nr. 

(2.)  A  solid  sphere  and  an  external  particle  m,  will  in  the  same  way  act 
on  one  another  as  if  all  the  mass  of  the  sphere  were  gathered  at  the  centre. 
The  attraction  between  a  sphere  whose  radius  is  r  and  whose  amount  of 
mass  per  unit  of  volume  ("volume-density")  is  p,  and  a  particle  ml  at  a  dis- 
tance d  from  the  centre  of  the  sphere,  is  k-m^^Trr3)  p/d2  (for  the  volume 
of  a  sphere  of  radius  r  is  -f rrr8)  :  if  the  particle  be  just  on  the  surface  of  the 
sphere  the  attraction  is  k-  m,  -(-firr8)  p/r2  =  k-ml  •  fTrrp. 

(3.)  An  attracting  spherical  shell  of  any  thickness,  if  this  thickness  be 
uniform,  has  no  action  whatsoever  on  a  heavy  particle  contained  within  it. 
For  every  area  of  the  shell  on  one  side  of  the  particle  which  may  attract  it 
in  one  direction,  there  is  another  on  the  other  side  attracting  it  in  an 
opposite  direction  ;  and  the  one  exactly  balances  the  other,  for  what  advan- 
tage the  one  area  may  have  in  size  the  other  exactly  makes  up  for  in 
proximity.  Thus  it  is  not  possible  to  find  any  area  of  the  sphere,  the 
attracting  effect  of  which  on  the  particle  within  the  sphere  is  not  exactly 
counterbalanced  by  the  opposed  attracting  effect  of  an  opposite  area.  The 
particle,  attracted  equally  in  every  direction,  remains  at  rest. 

No  other  law  than  that  of  the  inverse  square  of  the  distance  will  give 
this  entire  absence  of  effect  within  a  hollow  spherical  shell  of  uniform  thick- 
ness, as  will  easily  be  found  on  trial. 

If  the  shell  have  any  other  form  than  the  spherical,  it  must,  in  order  to 
retain  this  absence  of  interior  effect,  have  a  thickness  which  is  other  than  a 
uniform  one.  For  example  :  an  ellipsoidal  shell  whose  inner  ellipsoidal 
surface  is  concentric  and  confocal  with  the  exterior  ellipsoidal  surface  has  a 
thickness  which  at  any  point  is  proportional  to  the  shortest  distance  between 
the  centre  and  the  tangent  to  the  ellipsoid  touching  the  point  in  question 
(see  Fig.  197) ;  and  such  a  shell  has,  under  the  law  of  inverse  squares,  no 


vii.]  ;  ATTRACTION.  189 

action  at  any  point  within  it.  Such  a  shell  is  thickest  at  the  extremities  of 
its  major  axis. 

If  a  particle  ra,  be  just  outside  such  a  shell,  at  a  point  where  the  surface- 
density  is  <r,  the  attraction  just  outside  is  F  =  k  -  mt  •  lira- ;  just  inside,  F  =  0 ; 
and  thus  the  particle,  if  it  pass  through  the  shell,  passes  from  a  field  where 
the  attraction  is  k-m^^Tra-  to  another  where  the  attraction  differs  from  its 
former  value  by  k  •  mt  •  47r<r. 

(4.)  Between  a  particle  ml  and  an  arc  of  a  circle  of  radius  r,  at  the  cen- 
tre of  which  the  particle  stands,  the  attraction  is  k  •  w,  -(r<r/r2)  x  2  sin  J  angle 
subtended  =  k-m^^/r}^  sin  £0;  as  if  a  mass  equal  to  that  of  a  chord  of 
the  arc,  with  the  same  density  as  the  arc,  were  concentrated  at  the  midpoint 
of  the  arc. 

(5.)  Between  a  semicircle  and  a  particle  ml  at  its  centre ;  the  angle  sub- 
tended is  180°;  the  force  isk-m,- (<r/r)  x  2 sin 90°  =  k-mr  2<r/r,  towards  the 
midpoint. 

(6.)  Between  a  definite  line  AB  and  a  particle  at  D  opposite  the  centre 
C  of  the  line  AB.  Draw  AD  and  BD.  With  centre  D  and  radius  DC  draw 
a  circular  arc  limited  by  the  lines  AD  and  BD.  The  line  AB  and  this  cir- 
cular arc  exert  the  same  attraction  on  a  particle  at  D :  and  the  attraction  of 
the  circular  arc  we  know  from  (4). 

(7.)  Between  an  indefinite  line  and  a  particle  at  a  distance  r  from  it; 
the  angle  subtended  is  180° ;  the  force  is,  as  in  (5),  equal  to  k-m^  2<r/r. 

(8.)  Between  a  hemispherical  shell  and  a  particle  m,  at  its  centre,  the 
attraction  is  k  •  m/  -  2ir<r,  and  is  independent  of  the  radius. 

(9.)  Between  an  indefinite  plane  and  a  body  of  mass  ml  at  a  finite 
distance  d  from  it,  the  attraction  is  the  same  as  in  case  8,  and  is  fc-m^Tnr, 
at  right  angles  to  the  plane.  If  the  particle  pass  through  the  infinite 
plane  it  passes  to  a  region  where  the  attraction  is  —  k-m^  27r<r,  because  it 
acts  in  the  opposite  direction,  and  therefore  differs  from  its  former  amount 
by  k'm,' 47nr. 

Since  the  force  acting  is  independent  of  the  distance  d  we  may  make 
d  —  0.  If  mt  =  1,  we  now  have,  as  the  body  acted  upon,  a  unit-mass  of  the 
substance  of  the  plane  itself,  and  it  is  acted  upon  by  a  force  k  •  27r«r.  The 
matter  distributed  over  a  sq.  cm.  is  <r,  and  this  is  acted  upon  by  a  force 
(k' 27T<r)- <r  =  k- 27Tcr2.  The  force  acting  on  the  substance  of  the  plane 
itself  is  therefore  k  •  27r<r2  per  sq.  cm.,  and  is  at  right  angles  to  the  surface. 

(10.)  In  a  spherical  shell,  the  inward  force  on  the  outmost  film  is  k  •  47r<r  per 
unit  quantity  (see  above,  (1.))  ;  on  the  inmost  film  it  is  zero  (see  above,  (3.))  ; 
the  average  inward  attraction  for  the  substance  of  the  shell  itself  is,  per  unit 
quantity,  k  •  27nr,  or  half  that  for  an  external  unit-mass ;  the  attraction  per 
unit  of  area  is  &•  27r<r2,  at  right  angles  to  the  surface,  the  same  as  in  (9.)  above. 

In  the  case  of  gravitation,  the  constant  k  in  the  above  examples  is  the 
gravitation-constant  y,  p.  202. 

Convention  as  to  Attraction  and  Repulsion.  —  A  force  of 
this  kind  is  conventionally  said  to  be  Positive  when  its  effect 
is  to  separate  (or  to  increase  the  distance  between)  the  bodies 
by  whose  relative  motion  it  is  manifested.  Thus  a  repulsive 
force  is  positive ;  an  attractive,  which  diminishes  the  distance 
between  two  masses,  is  negative.  This  convention  is  opposed 
to  the  ordinary  use  of  speech. 


190  ATTRACTION  AND  POTENTIAL.  [CHAP. 

Potential  Energy  in  case  of  Repulsion.  —  If,  as  the  phrase 
goes,  two  bodies  repel  one  another,  and  if  one  or  both  of  them 
be  free  to  move,  their  mutual  separation  may  be  carried  on  to 
an  infinite  distance.  So  long  as  it  is  still  possible  under  any 
specified  circumstances  for  the  bodies  to  become  still  farther 
separated  by  their  mutual  repulsion,  there  is  still  some  potential 
energy  in  that  system  which  consists  of  the  two  masses  (together 
with  the  intervening  medium,  if  there  be  any  such) ;  the  mutu- 
ally repelling  bodies  must  therefore  be  separated  to  an  infinite 
distance  from  one  another  before  their  repulsion  can  cease  to 
act,  before  the  Potential  Energy  of  the  system  becomes  reduced 
to  zero.  When,  on  the  other  hand,  the  bodies  which  repel  one 
another  are  in  contact,  the  Potential  Energy  of  the  system  is  as 
great  as  it  can  possibly  be. 

Work  done  by  Repulsion.  —  If  two  masses,  m  and  m,,  situ- 
ated at  a  distance  d  from  one  another  and  repelling  one  another 
with  a  force  =  k-mmJcP,  be  allowed  to  separate  through  a  little 
distance  Sd,  the  system  has  exchanged  a  configuration  in  which 
the  mutual  force  was  k-mmjd2  for  one  in  which  it  has  been 
diminished  to  ~k-mmj(d  -f  ScT)2;  and  the  work  done  by  the 
repulsion  is  the  product  of  the  mean  force  into  the  space  &d.  If 
$d  be  taken  small  enough  this  product  becomes,  with  an  indefi- 
nitely close  approximation  to  accuracy,  equal  to  (k-mmjd*) 
X  &d.  If  the  bodies  increase  their  distance,  making  the  distance 
(d  +  ScZ)  grow  to  (d  -|-  2&f),  the  work  done  in  this  stage  is 
Qt-mmJ(d  +  6d)z)  x  Bd.  Summing  up  by  means  of  the  Inte- 
gral Calculus  the  work  done  by  the  repelling  force  in  separating 
the  bodies,  stage  by  stage,  from  a  mutual  distance  d  to  a  dis- 
tance d1 ',  we  find  that  it  is  k'mmt(^./d  —  1/d'^). 

This  proposition  may  be  otherwise  presented  in  the  form  of 
a  positive  statement.  The  work  done  is  the  product  of  the 
Space  traversed,  (d' —  d),  into  the  Mean  Force;  but  the 
mean  force  in  question  is  not  the  arithmetical  but  the  geomet- 
rical mean  between  the  extreme  values;  that  is,  these  extreme 
values  being  k-mm^/d2  and  k-mmjd9'2',  the  mean  force  is  the 
square  root  of  their  product,  or  k'mmjdd';  the  latter  being 
multiplied  by  the  space  traversed,  d'  —  d,  gives  the  product 
k-mm'^d'  —  d)/dd'  or  k'mmt(\/d  —  \/d'^  as  the  work  done. 

Hence  the  work  done  by  a  repulsion  (which  at  any  distance 
d  is  equal  to  k-mmjd2)  in  separating  two  masses  from  a  dis- 
tance d  to  a  distance  2c?,  is  k-mmt(\/d  —  l/2c?)=  k'mmt/Zd; 
from  distance  d  to  an  infinite  distance  oo ,  the  work  done  is  equal 


vii.]  WORK  DONE   BY   REPULSION. 

to  k  •  mmt  (l/d  —  l/oo  )  =  &  •  mmt(l/d  -  0)  =  k  -  mmjd.  Hence  if 
a  certain  amount  of  work  be  done  by  the  repulsion  in  doubling 
the  distance  between  two  mutually  repelling  bodies,  the  repul- 
sion would  do  exactly  twice  as  many  units  of  work,  and  no  more, 
in  separating  the  two  bodies  to  an  infinite  distance  from  one 
another. 

The  Potential  Energy  unexhausted  at  any  given  distance.  —  To 

remove  a  mass  mt  from  a  point  at  a  distance  d  from  a  fixed  repelling  mass 
m  to  an  infinite  distance  would  involve  expenditure  (by  the  repulsion)  of 
work  =  kmm,(l/d  —  l/oo)  =  kmmjd:  this  work  not  having  been  done 
when  the  distance  between  the  bodies  is  finite,  the  potential  energy  of  the 
system  is  so  far  unexhausted.  At  an  infinite  distance,  d  =<x>,  and  the  unex- 
hausted potential  energy  is  kmm,/<x>  =  0. 

The  unexhausted  Potential  Energy  of  two  bodies  of  masses  m  and  mt, 
repelling  one  another  and  situated  at  a  mutual  distance  rf,  is  k  •  mmjd  ;  this 
is  called  the  Mutual  Potential  of  the  two  masses. 

If  mi  be  a  unit,  the  Mutual  Potential  is  k  •  m/d.  This  is  numerically 
equal  to  the  Potential  as  defined  in  the  next  paragraph. 

Direction  of  Movement.  —  At  an  infinite  distance,  where 
the  potential  energy  attributed  to  a  body  there  placed  would  be 
zero,  there  would  be  no  force  impelling  to  any  further  separa- 
tion. At  any  place  where  the  potential  energy  has  a  positive 
value,  it  will  tend  to  exhaust  itself,  and  a  body  there  placed 
will,  if  free  to  do  so,  move  away  towards  some  place  where  it 
would  have  less  potential  energy.  But  the  Potential  Energy 
which  a  Unit-mass  would  have  if  placed  at  a  particular 
Point  in  Space,  —  the  work  which  would  be  done  by  the 
repelling  force  in  removing  the  unit-mass  from  that  point  to  a 
place  of  zero-repulsion,  or  would  have  to  be  done  against  repul- 
sion in  conveying  the  unit-mass  from  such  a  place  to  that  point, 
—  may  be  stated  as  an  attribute  of  that  Point  in  Space 
and  may  be  called  its  Potential.  This  may  be  numerically  high 
or  low.  Then,  under  a  repelling  force,  a  body  tends  to  move 
from  a  place  of  high  potential  to  a  more  distant  place  of  low 
potential,  and  if  the  body  be  free  to  move  in  that  sense,  the 
force  will  do  work  ;  while  if  the  body  be  moved  from  a  place 
where  the  potential  is  low  to  one  where  it  is  high,  the  move- 
ment is  effected  against  repulsion  or  resistance,  and  work  is 
done  against  the  repelling  force. 

The  Direction  of  the  Force  is  opposed  to  the  direction  in  which  the 
potential  increases  most  rapidly  ;  and  its  amount  at  any  point  is  (per  unit- 
mass  acted  upon,  and  in  any  given  direction)  equal  to  the  mean  decrease 
of  potential  per  unit  distance  traversed  (in  that  direction),  that  is,  to  the 
potential-gradient  or  potential-slope  (in  that  direction).  The  product 


[TJIIVERSITT 

~   •*  -rt 


192  ATTRACTION  AND   POTENTIAL.  [CHAP. 

of  this  force  per  unit-mass  into  the  space  traversed,  Fs,  the  Work  Done  on 
a  unit-mass,  is  numerically  equal  to  the  whole  diminution  of  potential 
in  the  whole  distance  traversed. 

Potential  a  condition  at  a  point  in  space.  —  We  must  dis- 
tinguish between  the  Potential  Energy  which  a  mass  may  be 
said  to  have  in  virtue  of  its  position  at  a  certain  point,  and  of 
its  consequent  relation  to  neighbouring  masses  ;  and  the  Potential 
of  that  point  in  space.  The  condition  at  that  point  of  space  is 
such  that  if  a  body  of  mass  ml  were  placed  there,  the  forces 
acting  on  it  would  do  Vw,  units  of  work  in  conveying  it  to  an 
infinite  distance,  or  would,  on  the  other  hand,  have  Vm,  units 
of  work  done  against  them  if  the  mass  mt  were  forced  against 
them  from  an  infinite  distance  to  that  point :  and  this  is  a  prop- 
erty numerically  expressible  by  the  numerical  value  of  V  (the 
value  of  Vm,  when  m,  is  a  unit-mass),  but  independent  of  the 
actual  presence  or  absence  of  any  mass  at  that  point. 

At  a  point  situated  at  a  distance  d  from  a  mass  m  the  "  Potential "  V  is 
equal  to  km/d;  and  Vm,  =  k  •  mmjd. 

Work  done  against  Attraction.  —  If  a  body  m,  be  at  a  given  distance 
d  from  an  attracting  mass,  the  action  between  the  two  bodies  is  a  force 
tending  to  approximate  them :  work  is  done  by  the  attracting  force  in  doing 
this :  but  "  the  work  done  by  the  attracting  force  in  separating  the  bodies  to 
an  infinite  distance"  is  a  negative  quantity,  for  work  (  =  Vm/  units)  would 
have  to  be  done  against  the  attraction  in  producing  this  movement ;  and  the 
potential  at  a  distance  d  from  the  attracting  mass  is  —  V,  a  negative 
quantity. 

Potential  in  the  special  case  of  Gravitation.  —  It  would,  if  the 
Earth  were  a  sphere  of  radius  637,000000  cm.,  require  the  expenditure  of 
k-mmjd  =  (kmm,/d*)  x  d  =  mty  •  d  =  m,  x  981  x  637,000000  =  624897,- 
000000m,  ergs,  and  no  more,  to  remove  a  mass  m,  from  the  earth  to  an 
infinite  distance  against  gravitation  ;  and  therefore  any  point  on  the  earth's 
surface  would  be  at  a  negative  potential  V  =  -  624897,000000,  while  the 
potential  of  any  point  at  an  indefinitely  great  distance  would  be  zero.  By 
a  special  exception,  however,  the  Gravitation-Potential  of  a  point  at  the 
earth's  surface  is  considered  to  be  zero,  and  a  body  lying  on  the  earth's 
surface  has  no  potential  energy ;  while  a  mass  m,,  removed  to  an  infinite 
distance,  could  have  no  more  than  624897,000000m,  ergs  of  potential  energy 
stored  up  in  it ;  and  the  Gravitation-Potential  of  a  point  at  an  infinite  dis- 
tance is  +624897,000000  units  =  V. 

Absolute  Zero  of  Potential.  —  A  point  is  at  zero  poten- 
tial when  a  body  placed  there  would  have  no  potential  energy. 
This  is  the  condition  of  a  point  at  an  infinite  distance  from  all 
repelling  masses. 

Fields  of  Space  in  opposite  conditions.  —  If  there  be  two 
bodies,  the  one  attracting,  the  other  repelling :  a  unit-mass 
brought  near  the  former  will  on  the  whole  be  attracted ;  from 


vii.]  POTENTIAL.  193 

the  other  mass,  it  will  on  the  whole  be  repelled.  The  space  in 
the  neighbourhood  of  the  attracting  mass  will  be  a  field  of  space 
in  which  the  potential  is  negative ;  round  the  repelling  body 
there  will  be  a  field  of  force  of  positive  potential. 

Continuity  of  Potential  through  Zero  value.  —  A  parti- 
cle passing  from  a  region  of  positive  potential  into  one  of  nega- 
tive potential  must  pass  through  a  point  where  the  potential  is 
zero ;  for  if  it  were  possible  for  it  to  do  otherwise  there  would  be 
physical  discontinuity.  As  it  thus  moves,  the  positive  poten- 
tial energy  of  the  body  is  gradually  exhausted,  becomes  zero, 
and  then  becomes  a  negative  quantity. 

Arbitrary  Zero  of  Potential.  —  We  may  arbitrarily  assume 
any  point  or  surface  in  the  neighbourhood  of  attracting  or 
repelling  masses  as  one  whose  V  =  0 ;  then  those  places  which 
have  a  greater  potential  are  said  to  be  localities  of  positive 
potential,  and  those  at  which  the  potential  is  less  are  said  to  be 
localities  of  negative  potential.  This  is  convenient,  for  absolute 
zero  we  know  no  more  than  we  know  absolute  rest. 

Analogy  of  Sea-level.  —  Let  us  assume  that  the  surface  of  the  earth 
is  the  sea-level  taken  at  high-water  mark.  This  is  an  arbitrary  assumption, 
for  low-water  mark  might  just  as  well  have  been  chosen.  If  a  body  be 
placed  at  a  certain  height  above  sea-level,  gravitation  may  do  a  certain 
amount  of  work  in  bringing  it  down  to  that  level,  for  the  mass  placed  at 
that  height  has  a  certain  amount  of  potential  energy :  at  a  less  height  it 
has  less  potential  energy ;  at  the  sea-level  it  has  none ;  if  placed  below  the 
sea-level,  its  potential  energy  is,  on  this  assumption,  a  negative  quantity. 
Hence  the  gravitation-potential  above  sea-level  is  of  opposite  sign  to  that 
below  it,  if  the  gravitation-potential  at  sea-level  be  taken  as  zero. 

Obviously  it  would  be  possible,  instead  of  saying  that  a  point  is  so  many 
feet  above  or  below  sea-level,  to  say  that  a  mass  mt  there  placed  would  have 
Vw,  units,  +  or  —  ,  of  potential  energy  if  there  placed,  and  thus  to  define  the 
distance  between  that  point  and  sea-level  by  its  gravitation-potential  ±V. 

Equipotential  Surfaces.  —  In  Fig.  91,  O  is  a  repelling 
particle.  All  points  at  equal  distances  from  it  are  at  the  same 
potential.  If  these  be  joined  they  form  a  sphere.  The  poten- 
tial at  every  point  of  the  surface  of  one  of  these  imaginary 
spheres  is  the  same,  and  may  be  represented  by  Vr  This 
sphere  is  an  equipotential  surface  for  potential  Vr  Within  this, 
and  concentric  with  it,  lies  another  sphere,  the  potential  at 
every  point  of  which  is  V2.  Within  this  lie  successive  shells 
or  imaginary  spherical  surfaces,  over  each  of  which  the  poten- 
tial is  equal.  If  these  surfaces  be  chosen  such  that  their 
potentials  have  a  common  difference —7  that  is,  that  V2  —  V1 
=V3  — V2  =  V4  — V3,  etc.  —  and  if  these  differences  each  repre- 


194 


ATTKACTION  AND   POTENTIAL. 


[CHAP. 


sent  one  unit  of  work  done  per  unit-mass  moved  from  one  to 
the  next,  a  set  of  equipotential  surfaces  thus  obtained  is  called 
a  "  System  of  Equipotential  Surfaces." 

Motion  parallel  to  Equipotential  Surfaces  does  not  involve 
work  done  either  by  or  against  the  attracting  or  repelling  force. 

Motion  across  fiquipotential  Surfaces,  from  one  surface  to 
another,  implies  movement  from  a  place  where  the  potential  has 
one  value  to  a  spot  where  it  has  another.  A  unit-mass  moving 
away  from  the  second  to  the  first  surface  in  Fig.  91,  loses 
potential  energy  =  V2  —  Vj :  on  a  mass  mt  the  repelling  force 
would  do  work  =  mt  x  (Va  —  Vj).  A  mass  mt  moved  up  from 


Fig.91. 


equipotential  surface  No.  10  to  surface  No.  15  in  a  system  of 
such  surfaces,  whatever  be  their  form,  would  have  work  =  5my 
units  done  upon  it  against  the  repelling  force. 

The  work  done  would  be  the  same  whatever  be  the  points  of 
the  respective  surfaces  between  which  the  motion  is  effected. 
Any  transference  of  a  particle  from  one  equipotential  surface  to 
another  may  be  effected  by  a  vertical  translation  from  the  one  to 
the  other,  which  involves  work,  compounded  with  a  translation 
along  the  second  equipotential  surface,  which  involves  none. 

The  work  done  by  a  transference  of  a  particle  from  a  point 
A  on  one  equipotential  surface  to  a  point  B  on  another  is  also 
always  the  same,  by  whatever  path  the  transference  be  effected, 


vii.]  EQUIPOTENTIAL   SURFACES.  195 

provided  always  that  there  be  no  friction.  The  most  complex 
path  may  be  resolved  into  so  much  movement  at  right  angles 
to  the  equipotential  surfaces,  which  implies  work  done  by  or 
against  the  forces,  and  so  much  parallel  to  them,  which  con- 
sumes or  liberates  no  energy. 

This  may  also  be  proved  by  a  reductio  ad  absurdum.  If  in  Fig.  91  there 
were  two  possible  paths  between  A  and  B,  one  of  which,  ACB,  corresponded 
to  W  units  of  work  done  by  a  unit-mass  of  matter  traversing  it,  while  the 
other,  ADB,  corresponded  to  a  greater  amount,  W  units,  of  work ;  then  it 
would  be  possible  to  cause  a  body  to  fall  from  A  to  B  down  the  path  ADB, 
corresponding  to  the  greater  work,  and  by  falling  to  pull  directly  or  indi- 
rectly a  mass  equal  to  its  own  up  the  easier  path  BCA  :  it  would  itself 
acquire  kinetic  energy  corresponding  to  energy  =  W  —  W ;  the  body  thus 
pulled  up  along  BCA  might  in  its  turn  fall  down  the  path  ADB,  and  raise 
along  the  path  BCA  the  mass  which  had  previously  traversed  the  path  ADB, 
again  with  gain  of  energy  equal  to  W  —  W.  Thus  the  circuit  might  be 
kept  up  with  continuous  gain  of  energy,  and  this  contrivance  might  be 
utilised  as  a  perpetual  motor ;  but  this  is  an  impossibility ;  therefore  there 
is  an  equal  expenditure  or  liberation  of  energy,  so  far  as  the  attracting  or 
repelling  forces  are  concerned,  in  effecting  a  transference  along  every  pos- 
sible path  between  any  two  given  points  in  space. 

Analogy  of  Surfaces  of  equal  level.  —  Obviously  the  same 
propositions  apply  if  we  read  the  word  level  for  potential. 

Distances  between  Concentric  Equipotential  Surfaces.  — 

In  a  system  of  concentric  spherical  equipotential  surfaces,  the 
distance  between  every  pair  of  these  surfaces  is  proportional  to 
the  square  of  their  mean  distance  (i.e.  of  the  geometrical 
mean)  from  the  centre  of  the  single  attracting  or  repelling 
mass. 

Two  concentric  spherical  equipotential  surfaces  whose  potentials  are  V 
and  (Y  +  1),  and  whose  respective  radii  are  r  and  r' ;  we  wish  to  find 
the  value  of  r  -  r'.  Then  V  =  k  •  m/r,  and  (V  +  1)  =  k  •  m/r1 ;  whence 
r-r^fc-m-*- V(V+l)=rr7&ro;  .-.  (r  -  r')  oc  {VrP}2. 

Thus,  if  the  equipotential  surfaces  be  those  surrounding  the 
earth,  over  which  the  potential  due  to  gravitation  is  constant, 
and  if  the  distances  between  the  surfaces  be  such  that  transfer 
of  a  gramme-mass  from  any  one  surface  to  the  next  one  repre- 
sents one  erg  of  work  'done  :  then,  at  the  distance  of  one  earth's- 
radius  from  the  centre  of  the  earth  —  that  is  to  say,  on  the 
surface  of  the  earth  —  the  distance  between  two  equipotential 
surfaces  is  -^  cm. ;  twice  as  far  from  the  centre — that  is,  4000 
miles  (nearly)  from  the  surface  of  the  earth  —  the  distance* is  ^|T 
cm.,  and  the  same  amount  of  work  which  would  raises  gramme- 
mass  through  -g^  cm.  near  the  surface  of  the  earth  would,  at  a 
height  of  4000  miles,  raise  it  -gfy  cm. ;  and  similarly,  at  a  height 


196  ATTRACTION  AND   POTENTIAL.  [CHAP. 

of  8000  miles,  it  would  raise  it  ¥fT  cm.,  and  so  on.  Thus  at  a 
very  great  distance  exceedingly  long  paths  would  be  traversed 
by  a  gramme-mass  as  the  result  of  doing  a  single  erg  of  work 
on  it. 

A  mass  at  a  distance  of  240,000  miles  (=60  radii  nearly)  from  the 
earth's  centre  would  be  attracted  by  the  earth  with  a  force  which  bears 
to  the  attraction  at  the  earth's  surface  the  proportion  of  (1/60)2:(1)2 
=  1 :  3600.  Hence,  to  move  a  gramme-mass  through  one  cm.  —  that  is,  from 
an  equipotential  surface  by  any  path  to  any  point  on  an  equipotential  sur- 
face one  cm.  distant  from  it  —  at  a  distance  of  240,000  miles,  or,  roughly,  at 
the  distance  of  the  moon,  would  involve  the  expenditure  of  approximately 
erg  of  work. 


It  follows  that  if  the  equipotential  surfaces  be  chosen  at 
equal  distances  from  one  another,  the  amount  of  work  corre- 
sponding to  the  removal  of  a  mass  from  one  surface  to  the  next 
is  in  the  inverse  ratio  of  the  square  of  the  mean  distance  of  the 
two  surfaces  from  the  attracting  mass. 

Two  concentric  spherical  equipotential  surfaces  whose  potentials  are  V 
and  Vy,  and  whose  radii  are  r  and  (r  +  1)  ;  V  =  k  •  m/r :  V  =  k  -  m/(r+ 1)  ; 
V  -  V  =  &  •  ro  -s-  r  (r  +  1)  ;  .-.  (V  -  V)m/  oo  {  Vr(r+  I)}-2. 

Equipotential  Surfaces  of  Complex  Form.  —  If  A  and  B 

be  two  equal  particles,  A  attracting  and  B  repelling  external 
particles,  the  space  surrounding  A  will  be  a  region  of  negative 
potential,  while  the  potential  of  the  neighbourhood  of  B  is 
positive.  Over  a  plane  symmetrically  situated  with  respect  to 
A  and  B  the  potential  will  be  zero,  and  the  equipotential  sur- 
faces will  present  the  form  indicated  by  the  lines  marked  "  Lines 
of  Force  "  in  Fig.  234.  If  A  and  B  be  not  equal,  or  if  there 
be  more  than  two  masses  concerned,  the  form  will  be  still  more 
complex. 

Free  movement  always  at  right  angles  to  Equipotential 
Surfaces.  —  Whatever  be  the  form  of  any  equipotential  surface, 
it  always  happens  that  a  body  placed  on  such  a  surface,  and  free 
to  move,  will  tend  to  move,  under  the  influence  of  the  attracting 
or  repelling  forces,  in  a  direction  at  right  angles  to  that  surface. 
This  is  because  the  forces  of  attraction  or  repulsion  can  have  no 
component  tending  to  produce  motion  in  any  direction  along  a 
surface  of  equal  potential,  or  parallel  to  it. 

Lines  of  Force.  —  Thus,  if  the  equipotential  surfaces  be 
concentric  spheres,  as  those  of  Fig.  91,  a  body  repelled  from  O 
will  travel  along  radial  lines  such  as  are  exemplified  by  the 
dotted  lines  in  that  figure.  When  the  equipotential  surfaces 
have  a  more  complex  form,  the  lines  along  which  a  body  tends 


TIL] 


LINES   OF  FORCE. 


197 


Fig.  92. 


to  travel  are  more  complex,  as  is  shown  in  Figs.  234  and  235. 
These  lines,  always  at  right  angles  to  the  equipotential  surfaces 
which  they  cross,  are  called  Lines  of  Force. 

Space  in  the  neighbourhood  of  an  attracting  or  repelling 
body  may  be  conceived  to  be  pervaded  by  a  system  of  Lines  of 
Force,-  along  which  bodies  will  move  if  free  to  do  so.  The  work 
done  on  a  particle  thus  set  in  motion  by  an  attraction  or  repul- 
sion is  the  product  of  the  mean  force  into  the  space  traversed ; 
the  latter  must  be  measured  along  the  line  of  force  which  is  the 
body's  actual  path. 

Lines  of  Force  are  analogous  to  lines  of  steepest  fall  in  topography; 
water  poured  out  will  at  any  spot  run  in  the  direction  of  steepest  fall ;  and 
a  body  acted  upon  in  a  field  of  force  will  tend  to  fall  away  from  a  spot  of 
higher  potential  to  one  of  lower  potential,  following  the  direction  of  most 
rapid  potential-fall,  the  line  of  force. 

Tubes  of  Force.  —  Suppose  AB  to  be  a  portion  of  an  equipotential 
surface :  lines  of  force  pass  through  the  equipotential  surface :  some  of  these 
lines  graze  the  edge  of  the  area  AB  ; 
these  cut  off  an  area  A'B'  from  another 
given  equipotential  surface.  The  space 
comprised  between  these  equipotential 
areas  and  the  marginal  lines  of  force  is 
called  a  Tube  of  Force.  This  space  may 
be  supposed  to  be  filled  with  a  bundle 
of  lines  of  force,  extending  from  AB  to 
A'B'.  Such  a  tube  may  be  curved  in  form. 

Tubes  of  force  have  this  property, 
that  so  far  as  the  area  A'B'  cut  by  them 
from  one  equipotential  surface  is  greater 
than  the  area  AB  cut  oft'  from  another, 
so  does  the  intensity  of  the  force  act- 
ing across  any  unit  of  area  diminish ;  so 
that  if  A  and  A'  be  the  respective  areas 
of  AB  and  A'B',  and  /,  f  the  respective 
forces  per  unit  of  area  acting  across  these 
equipotential  areas,  the  product,  intensity 
of  force  x  area,  is  constant,  or^A  =f'Af. 

Thus  the  force  /  per  sq.  cm.  at  the  level  A'B'  is  less  than  that  at  AB,  in 
itiverse  proportion  to  the  relative  magnitude  of  the  area  A'B'  cut  off  by 
the  tube  of  force. 

Tubes  of  Force  drawn  in  such  fashion  as  each  to  contain  one  line  of 
force  are  called  Unit  Tubes  of  Force. 

Number  of  Lines  of  Force.  —  The  forces  at  any  two  points 
may  be  compared  by  stating  the  relative  numbers  of  the  lines  of 
force  which  pass  through  units  of  area  of  those  equipotential  sur- 
faces which  pass  respectively  through  each  of  the  points  com- 
pared; the  fewer  these  lines,  the  less  is  /,  the  local  intensity  of 


198  ATTRACTION  AND  POTENTIAL.  [CHAP. 

the  force,  in  the  direction,  at  any  point,  of  the  local  Lines  of 
Force.  Thus  in  Fig.  91  the  lines  of  force  which  cross  the 
outer  spheres  are  less  numerous,  per  unit  of  area  of  the  sphere, 
than  those  which  cross  the  inner  spheres,  and  the  force  per  unit 
of  area  is  there  correspondingly  less. 

Systems  of  Surfaces  and  Lines.  —  The  space  in  the  neigh- 
bourhood of  an  attracting  or  repelling  mass  or  system  of  masses 
may  thus  be  mapped  out  by  a  system  of  equipotential  surfaces 
and  lines  of  force,  and  such  a  region  of  space  is  called  a  Field 
of  Force.  The  system  of  surfaces  and  lines  may  be  so  con- 
structed that  (1)  the  work  done  in  passing  a  unit-mass  from  one 
equipotential  surface  to  the  next  is  always  the  same,  one  unit 
of  work;  and  (2)  the  lines  of  force  are  drawn  in  just  such  num- 
bers that  at  a  place  where  the  force  on  a  unit-mass  is  equal  to 
unity,  one  line  of  force  passes  through  the  corresponding  equi- 
potential surface  in  each  unit  of  area  of  that  surface,  and/  =  1. 
This  secures  the  following  advantages  :  — 

(1.)  The  potential  at  any  point  in  the  field  of  space  sur- 
rounding the  repelling  or  attracting  mass  or  masses  is  found  by 
determining  on  which  imaginary  equipotential  surface  that  point 
stands. 

(2.)  If  unit-length  of  a  line  of  force  cross  n  equipotential 
surfaces,  the  mean  force  (on  a  unit-particle)  along  that  line, 
along  the  course  of  that  part  of  it,  is  equal  to  n  units ;  for  the 
difference  of  potential  of  the  two  ends  of  that  part  of  the  line 
of  force  =  n;  it  is  also  equal  to/s,  because  it  represents  numeri- 
cally a  certain  amount  of  work  done  on  a  unit-particle  made  to 
travel  from  one  end  to  the  other  of  that  part  of  the  line  of  force; 
but  *  =  1;  whence  n  =  /,  where  /  is  the  mean  force  acting  on 
a  unit-particle,  along  the  line  of  force. 

(3.)  The  force  at  any  point  of  the  field  corresponds  to  the 
extent  to  which  the  lines  of  force  are  crowded  together ;  and 
thence  it  may  be  determined  by  the  number  (  =/)  of  lines  of 
force  which  pass  through  a  unit  of  area  of  the  corresponding 
equipotential  surface,  that  area  being  so  chosen  as  to  comprise 
the  point  in  question. 

Uniform  Field  of  Force.  —  If  the  equipotential  surfaces  be 
plane,  parallel,  and  equidistant,  the  lines  of  force  are  equally 
distributed,  parallel  to  one  another,  and  at  right  angles  to  the 
equipotential  surfaces,  and  the  Field  is  uniform.  The  force 
per  unit  of  area,  in  the  direction  of  the  lines  of  force,  is  the 
same  in  all  parts  of  such  a  field. 


TIL]  FIELD   OF  FORCE.  199 

Variations  in  Difference  of  Potential.  —  Any  movement  of 
a  body  across  the  surfaces  of  equal  potential,  if  these  surfaces  be 
not  equidistant,  alters  the  relative  difference  of  potential  between 
its  two  extremities,  because  a  bociy  approaching  a  repelling  or 
attracting  mass  meets  and  cuts  more  equipotential  surfaces  than 
it  quits,  as  may  be  seen  from  Fig.  91  ;  and,  vice  versd,  a  receding 
body  meets  fewer  surfaces  than  it  quits.  In  the  former  case  the 
movement  tends  to  cause  an  increase  in  the  difference  between 
the  potentials  of  the  extremities  of  the  body  moved  in  the  non- 
uniform  field  of  force  ;  in  the  latter  it  tends  to  diminish  it. 

An  increase  in  the  central  attraction  or  repulsion  has  the 
same  effect  as  an  approach  ;  a  diminution  the  same  as  a  recession. 

Theorem.  —  If  a  closed  surface  be  drawn  round  a  system  of 
attracting  or  repelling  masses,  the  number  of  lines  or  unit  tubes 
of  force  traversing  the  surface  is  numerically  equal  to  &-47rQ, 
where  Q  is  the  algebraical  sum  of  the  whole  matter  within  the 
closed  surface,  and  when  the  law  of  force  is  that  F  =  k-mmj 
distance2. 

Take  first  a  single  particle  q  at  the  centre  of  a  spherical 
surface.  The  force  per  unit  of  area  is  /  =  k  •  q/r2  ;  the  whole 
surface  is  47rr2;  the  force  over  the  whole  surface,  i.e.  the  num- 
ber of  lines  of  force  crossing  the  surface,  is  F  =  k  •  q/r2  x  4?rr2 


Take  next  any  particle  q  at  any  point  within  a  closed  surface  of  any  form. 
Any  small  area  8A  is  taken,  which  subtends  at  the  particle  a  solid  angle  o>, 
and  the  normal  to  which  is  inclined  at  an  angle  £  to  a  line  drawn  from  the 
particle  to  the  centre  of  8  A.  The  area  SA  is  equal  to  </2<o/cos  £,  where  d  is 
its  mean  distance  from  the  particle  ;  the  normal  force  per  unit  of  area  is 
k-  q-  cos  g./d2  ;  the  product  (</2w/cos  £)  •  k  •  (q  cos  £/d2)  =Tc-q^  represents  the 
number  of  lines  of  force  passing  through  the  element  of  surface,  a  number 
which  is  seen  to  be  independent  of  the  obliquity  or  the  distance  of  the  element 
of  surface  considered.  When  the  surface  completely  surrounds  the  particle, 
the  solid  angle  subtended  by  the  surface  is  <o  =  4?r,  and  the  force  due  to  this 
particle  and  acting  through  the  whole  surface  is  k  •  4irq.  So  for  every  par- 
ticle, wherever  situated,  within  the  closed  surface,  and  if  the  sum  of  the  q's 
be  Q,  the  number  of  lines  of  force  traversing  the  whole  surface  is  F  =  k-  4?rQ. 

If  some  of  the  q's  be  attracting,  some  repelling,  they  must  be  affected 
with  their  proper  signs,  and  Q  is  their  algebraic  sum. 

If  the  surface  be  one  which  is  repeatedly  indented  so  as  to  be  repeatedly 
traversed  by  lines  of  force,  the  exits  must  be  more  numerous  by  one  than 
the  entrances  ;  the  exits  and  entrances  in  any  region  of  the  surface  sub- 
tending the  solid  angle  to  must  compensate  one  another,  with  the  exception 
of  the  last  exit  ;  this  alone  contributes  to  the  aggregate  number  of  lines  of 
force  finally  issuing  from  the  surface.  The  closed  surface  may  thus  be  of 
any  degree  of  complexity,  without  affecting  the  numerical  value  of  the 
whole  system  of  lines,  as  enunciated  by  the  above  theorem. 


200  ATTRACTION  AND   POTENTIAL.  [CHAP.  VH.] 

Potential  of  a  Double  Sheet.  —  Take  a  very  small  sheet  of  repel- 
ling matter,  whose  quantity  is  Q,  uniformly  distributed  over  the  area  SA. 
A  similar  and  equal  sheet,  this  time  of  attracting  matter  —  Q,  is  brought 
up  parallel  to  the  former,  so  as  to  be  separated  from  it  only  by  the  very 
small  mutual  distance  L  Take  any  point  P ;  then  the  distance  between  that 
point  and  the  middle  of  the  small  area  8A  will  be  d,  and  its  direction  will 
make  some  angle  £  with  the  normal  to  that  area.  At  P  the  solid  angle  sub- 
tended by  that  area  will  be  co  =  SA-cos  £./d2.  Now,  the  potential  due  to 
the  charge  on  the  one  of  these  two  faces,  say  at  distance  d,  will  be  Q/ d ; 
that  due  to  the  other,  at  distance  d±8d,  will  be  —  Qi/(d±$d)  ;  the  algebraic 
sum  of  these  is  the  effective  potential  at  P,  and  it  is  Q.{l/d  —  l/(d  ±  &/)} 
=  ±Q-8d/d-(d  ±  8rf),  or,  when  the  mutual  distance  I  is  extremely  small, 
it  is  ±Q'8d/d2.  But  8d  is,  under  the  same  conditions,  equal  to  /-cos£; 
hence  the  potential  at  P  is  (±  Q/8A)- 1  -(SA  •  cos  £/d2)  =  (±  Q/SA) .  /  • «  =  <rZtu, 
the  product  of  the  Superficial  Density  <r,  or  the  quantity  of  matter  per  sq. 
cm.  of  either  of  the  two  opposed  sheets,  into  the  Distance  I  between  them, 
into  the  Solid  Angle  w  subtended  by  the  double  sheet  at  the  point  P.  On 
the  repelling  side  of  such  a  double  sheet,  the  potential  in  the  surrounding 
field  of  force  is  positive ;  on  the  other  side  it  is  negative  :  and  the  lines  of 
force  and  equipotential  surfaces  in  the  surrounding  field  are  disposed  as 
indicated  in  Fig.  234.  The  direction  of  each  line  of  force  is  outward  from 
the  repelling  or  positive,  and  back  (by  a  more  or  less  ample  sweep)  to  the 
attracting  or  negative  side  of  the  double  sheet. 

Isodynamic  Surfaces  and  "Lines  of  Slope."  —  If  all  those  points 
in  a  field  of  Force  be  connected,  at  which  the  force  is  equal,  we  have  a  set  of 
isodynamic  surfaces.  These  may  coincide  with  equipotential  surfaces,  as 
in  the  case  of  gravitation ;  but  they  may  have  a  totally  different  lie.  For 
example,  the  field  of  electromagnetic  force  surrounding  a  long  straight 
wire,  bearing  a  current  of  electricity,  is  permeated  by  equipotential  sur- 
faces, plane  and  radiating  from  the  straight  wire ;  the  lines  of  force,  cutting 
these  at  right  angles,  are  concentric  circles  round  the  wire ;  the  isodynamic 
surfaces  are  concentric  cylindrical  surfaces  surrounding  the  wire.  At  right 
angles  to  the  isodynamic  surfaces  we  may  imagine  lines,  the  so-called 
"Lines  of  Slope,"  or  of  Intensity-Slope,  which  trend  in  the  directions 
in  which  the  intensity  of  the  force  in  the  field  falls  away  most  rapidly.  In 
the  case  of  gravitation  these  trend  in  the  same  directions  as  the  lines  of 
force ;  in  the  case  of  the  straight  current  they  radiate  from  the  wire  at  right 
angles  to  it,  and  are  therefore  at  right  angles  to  the  lines  of  force. 

In  a  uniform  field  of  force  there  are  no  such  lines  or  surfaces,  for  the 
whole  region  is  isodynamic. 


CHAPTER  VIII. 

GRAVITATION  AND  THE  PENDULUM. 

Law  of  Gravitation.  —  Every  particle  of  matter  in  the 
Universe  is  attracted  directly  towards  every  other 
particle  with  a  force  varying  directly  as  the  mass 
of  each  particle,  and  inversely  as  the  square  of  the 
distance  between  them. 

We  have  already  seen  that  the  Weight  of  a  body  is  a 
synonym  for  the  Force  with  which  it  is  attracted  by  the  earth. 
The  law  just  enunciated  indicates  that  the  weight  of  a  double 
mass  is  twice  that  of  a  single  mass,  and  so  on.  This  seems  a 
truism ;  but  it  is  an  experimental  result,  not  a  truism,  that  the 
weight  of  a  mass  of  lead  is  equal  to  that  of  an  equal  mass  of 
wood.  This  might  have  been  otherwise.  The  mass  of  a  given 
piece  of  wood  is  known  to  be  equal  to  that  of  a  certain  piece  of 
lead  by  the  experimental  fact  that  equal  forces  acting  on  each 
for  equal  times  produce  equal  velocities :  F  =  ma;  these  veloci- 
ties being  those  of  short  horizontal  trajectories,  which  are  inde- 
pendent of  gravitation.  Now  a  piece  of  iron  and  a  piece  of 
cork,  whose  masses  are  thus  found  to  be  equal,  will,  if  placed  in 
the  neighbourhood  of  a  magnet,  be  found  to  be  by  no  means 
affected  by  equal  accelerations  towards  the  magnet;  yet  they 
are  both  equally  attracted  by  the  earth,  have  both  the  same 
weight  in  the  balance,  and,  if  caused  to  fall  through  a  vacuum 
(the  friction  of  the  air  being  thus  removed),  are  found  to  fall 
with  concurrently  equal  velocities. 

It  is  remarked  that  horizontal  trajectories  are  independent  of  gravita- 
tion. A  cannon  ball  at  the  moon,  if  it  had  weighed  60  Ibs.  on  the  earth, 
would  weigh  less  than  10  Ibs.  there :  and  so,  as  has  been  said,  it  might  be 
dropped  on  the  toes  of  the  observer  there  without  serious  consequences; 
but  it  would  be  found  exactly  as  difficult  there  as  here  to  heave  the  ball 
horizontally,  for  it  would  still  contain  60  pounds  mass. 

Problem. —What  would  a  pound-mass  weigh,  half-way  between  the 
Earth's  surface  and  centre?  Ans.  —  £  lb..  See  prop.  3,  p.  188. 

201 


202  GRAVITATION  AND   THE   PENDULUM.  [CHAP. 

Again,  a  heavy  and  a  light  mass  of  any  substance  fall  at  the 
same  rate  through  a  vacuum.  It  was  long  believed  that  the 
heaviest  bodies  fall  fastest ;  but  Galileo  experimentally  disproved 
this.  The  attraction  of  the  earth  for  a  large  mass  is  greater 
than  for  a  small  one,  but  the  mass  to  be  moved  increases  in  the 
same  proportion  as  the  attraction;  and  thus  the  acceleration 
produced  is  the  same  in  all  cases,  and  is  independent  of  the 
amount  as  well  as  of  the  substance  of  the  falling  mass. 

Cavendish's  Experiment.  —  This  was  a  direct  measure- 
ment of  the  attraction  of  masses  for  one  another.  Small  balls 
of  lead  were  poised  on  a  rod  and  their  position  carefully  noted : 
large  balls  of  lead  were  carefully  brought  near  them :  the  light 
balls  were  attracted  by  the  heavy  masses,  and  their  displacement 
measured.  Great  experimental  precautions  were  necessary, 
such  as  the  observation  of  the  position  of  the  balls  with  a  tele- 
scope placed  at  a  distance,  the  avoidance  of  draughts  of  air  and 
of  vibrations,  etc. ;  the  result  showed  that  if  lead  balls  had  been 
employed  as  large  as  the  earth,  the  attraction  of  such  balls 
would  have  been  greater  than  the  actual  attraction  of  the  earth 
in  the  ratio  of  11-35  to  5-67 :  but  lead  is  11-35  times  as  heavy 
as  water ;  hence  the  earth  as  a  whole  is  5-67  times  as  heavy  as 
an  equal  bulk  of  water,  or  the  density  of  the  earth  is  5-67. 

If  two  masses  be  respectively  m  and  mfl  and  their  distance  d,  the  gravita- 
tion-attraction between  these  masses  oc  mnij/d2,  or  G  =y  -  mmt/d^  where  y 
is  a  constant,  the  Gravitation-Constant.  The  earth  being  approximately 
spherical  attracts  falling  bodies  of  mass  m,  as  if  its  own  mass,  m,  were  gath- 
ered together  at  the  centre,  about  637,000,000  cm.  from  the  surface.  Its 
mass  m  is  6140,000000,000000,000000,000000  grammes.  A  mass  m,  =  1 
gramme  is  attracted  by  the  earth  with  a  force  equal  to  981  dynes  :  hence,  in 
this  case,  G  =  981  dynes  =  y.mmy/J2  =  y-614-1025  x  1  •*•  (637,000)2:  and 
y  =  981  x  (637,000)2  -4- (614  - 10»)  -  T^sWo-  Hence  G  =  T5TSW  mm,/d* : 
and  the  gravitation-attraction  between  two  gramme-masses  whose  centres 
are  1  cm.  apart  is  T-rcsWr  dyne. 

The  astronomical  unit  of  mass  is  15,430000  grammes,  and  the  corre- 
sponding unit  of  force  15,430000  dynes.  With  such  units  the  constant 
y  =  1,  and  G  =  mmjd^  simply. 

Prof.  C.  Vernon  Boys,  by  means  of  Cavendish's  experiment  conducted 
according  to  methods  described  in  Nature,  Aug.  2,  9,  and  23,  1894,  finds 
•y  =  15,020400,  and  the  density  of  the  Earth  =  5-5270. 

Accelerated  Motion  under  Gravity.  —  A  body  free  to  fall 
in  vacuo  would  be  subject  to  constant  downward  acceleration  of 
about  981  cm.-per-sec.  or  32-2  ft.-per-sec.  —  that  is,  of  a  =g  units 
of  velocity  —  per  second,  and  its  movement  would  be  described 
by  the  four  formulae  of  page  152 ;  the  +  sign  being  used  when 


viii.]  ACCELERATED   MOTION   UNDER   GRAVITY.  203 

the  attraction  of  gravity  acts  in  the  same  sense  as  the  original 
velocity  v0 ;  the  —  sign  when  it  acts  in  an  opposite  sense. 

When  bodies  fall  through  the  air  there  is  friction  between 
the  air  and  the  falling  body.  This  is  found  to  vary  as  the  radius 
of  the  sphere  if  the  falling  body  be  an  exceedingly  small  sphere; 
and  generally  it  increases  with,  but  is  not  proportionate  to,  the 
surface  exposed.  Thus  a  feather,  which  presents  much  surface, 
falls  more  slowly  than  a  similar  feather  rolled  into  a  ball. 

In  a  cloud,  the  minute  droplets  fall  extremely  slowly,  for  the  weight  of 
each  is  proportional  to  the  cube  of  the  radius,  while  the  resistance  is  pro- 
portional to  the  radius  itself ;  and  thus,  as  the  size  of  the  droplets  dimin- 
ishes, their  Weight  diminishes  more  rapidly  than  the  Resistance. 

Path  of  a  Projectile.  —  We  have  already  seen  that  combi- 
nation of  a  uniform  rectilinear  movement  with  a  uniformly 
accelerated  movement,  not  in  the  same  direction,  results  in 
movement  in  a  parabolic  path.  This  is  the  theoretical  course 
of  a  bullet  flying  in  vacua  ;  but  the  actual  course  of  a  shell  or 
bullet  in  the  air  differs  widely  from  this  on  account  of  friction, 
its  path  being  at  first  somewhat  straight  and  ending  with  a 
somewhat  sudden  fall. 

If  a  shot  were  fired  horizontally  in  vacuo  at  such  a  rate  v  (about  26,077 
feet  per  second),  that  a  =  g,  the  acceleration  earthwards  (=  32'2  ft.-per-sec. 
per  second)  would  be  y2/r,  r  being  the  distance  of  the  earth's  centre  from 
the  bullet's  path,  the  shot  would  never  fall  to  the  ground,  but  would 
travel  round  the  earth  at  the  level  of  the  gun's  mouth. 

The  most  general  case  of  continuous  motion  of  a  body  round  a  point, 
under  the  influence  of  an  attraction  towards  that  point,  varying  inversely 
as  the  square  of  the  distance,  is  motion  in  an  Ellipse.  When  this  occurs, 
the  following  three  propositions  hold  good:  —  (1)  The  point  towards 
which  the  acceleration  is  directed  is  at  one  of  the  two  "  foci "  of  the  ellipse ; 
(2)  a  line  ("radius-vector")  drawn  from  that  point  to  the  moving  body 
sweeps  over  equal  areas  in  equal  times  as  the  body  moves;  and  (3)  the 
time  taken  to  perform  a  complete  revolution  in  the  elliptical  path  is  pro- 
portional to  the  square  root  of  the  cube  of  the  mean  distance  from  the 
central  point.  These  propositions  had  been  empirically  established  by 
Kepler  as  a  statement  of  the  actual  relative  movements  of  the  planets  with 
reference  to  the  Sun ;  and  they  are  known,  in  that  connection,  as  Kepler's 
Laws.  Sir  Isaac  Newton  showed  that  they  are  all  a  consequence  of  the 
law  of  inverse  squares,  and  that  the  same  law  of  Gravitation  is  accordingly 
adhered  to  throughout  the  Solar  System. 

Some  Comets  move  in  ellipses,  these  being  sometimes  so  long  that 
thousands  of  years  must  be  taken  in  completing  one  revolution;  others 
sweep  round  the  sun,  are  deflected  by  its  attraction  into  a  hyperbolic 
path,  and  again  disappear  into  the  depths  of  space. 

s 

Universal  Gravitation.  —  The  fact  of  terrestrial  gravitation 
and  many  of  its  laws  were  well  known  before  Newton's  time ; 


204  GRAVITATION  AND   THE  PENDULUM.  [CHAP. 

he  stated  the  law  of  gravitation  as  a  universal  one  :  "  Gravitatem 
in  corpora  universa  fieri,"  etc.  —  Principia,  Bk.  III.  Prop.  vii. 
and  Corol.  2. 

The  moon  makes  a  revolution  round  the  earth  *  in  about  2,360,000  sec- 
'  onds,  in  an  orbit  whose  mean  radius  is  59-964  times  the  earth's  equatorial 
•    radius.     The  formula  a  =  v'2/r  shows  that  this  corresponds  to  an  actual  fall 
of  the  moon  towards  the  earth  of  £a  =  0-136  cm.  per  second:   this,  coin- 
pounded  with  the  tangential  velocity  at  every  instant,  keeps  the  moon  in  its 
orbit.     This  acceleration,  due  to  the  attraction  of  the  earth,  is  l/(59-964)2 
of  981 ;  thus  the  moon  is  under  the  influence  of  terrestrial  attraction  which 
obeys  the  law  that.F  oo</~2.     Newton  made  similar  deductions  from  other 
astronomical  phenomena,  particularly  those  of  the  satellites  of  Jupiter,  and 
ultimately  asserted  the  universality  of  the  law  of  gravitation. 

The  average  attraction  of  the  sun  for  a  gramme-mass  of  the  earth's  sub- 
stance is  G={y  x  (Sun's  mass  x  1  gramme)  -f-  (Sun's  distance)2}  =  {g  x  (Sun's 
mass  H- Earth's  mass) -i-( Sun's  distance  in  earth-radii)  2}={g  x  316000-^-232002} 
=  0-00058710. 

At  the  equator,  at  an  equinox,  this  would,  if  the  earth  and  the  sun  were 
kept  at  a  fixed  distance  apart,  cause  a  loose  unit-mass  on  the  earth's  surface 
to  have  an  apparent  weight  of  0-9994:1290  at  midday,  and  one  of  1-00058710 
at  midnight ;  a  variation,  during  the  24  hours,  of  0-00117420,  or  3-63  Ibs. 
per  ton :  while  the  rising  or  setting  sun  would  displace  a  plumb-line  or 
mercury-mirror  to  an  extent  reaching  a  maximum  of  0°  2'  1"  east  or  west ; 
arid  the  pendulum  would  oscillate  at  correspondingly  varying  rates.  No 
such  effects  are  observed.  The  earth  and  a  stone  lying  loose  on  its  surface 
both  fall  freely  towards  the  sun :  if  the  stone  is  the  nearer,  the  Earth 
follows  it  closely,  and  there  is  apparently  no  tendency  to  separation  between 
the  two  bodies;  but  yet  there  is  a  certain  small  difference.  The  stone,  if  it 
face  the  sun,  is  at  23199  earth-radii  from  its  centre ;  the  earth  is  at  23200 
radii :  the  solar  gravitation  on  the  stone  is  mtg  x  316000  -*-  231992  :  the  dif- 
ference between  this  and  the  average  solar  gravitation  on  the  earth  is 
0-000000,05150  per  gramme.  In  the  case  of  the  moon,  though  the  lunar 
gravitation  is  much  smaller  than  that  of  the  sun,  this  difference  is  greater, 
being  0-000000,130.  To  these  differences,  small  though  they  be,  the  Tides 
are  due  :  the  water  nearer  the  sun,  or  moon,  is  more  attracted  than  the 
bulk  of  the  earth,  and  is  heaped  up:  but  the  bulk  of  the  earth  is  more 
attracted  than  the  more  remote  water,  and  that  water  is  left  in  a  heap  at 
the  farther  side  of  the  earth.  The  tides  are  highest  when  the  sun  and 
moon  co-operate  in  their  effects ;  lowest  when  the  sun  and  moon  lie  at  right 
angles  to  each  other.  The  actual  tides  are  modified  by  friction,  so  as  to  be 
always  belated.  If  the  sun  and  earth  (or  the  moon  and  earth)  had  been 
fixed  in  their  relative  position,  there  would  have  been  but  one  heap  of 

*  This  is  only  a  rough  statement  of  the  fact.  The  moon,  as  it  runs  in  its  "  orhit 
round  the  earth,"  never  touching  it,  describes  a  sinuous  path  in  space  along  the 
course  of  the  earth's  orbit.  It  happens  that  Fig.  49  may  serve  to  give  an  approxi- 
mate notion  of  this  sinuous  path.  The  earth  is  also  equally  attracted  by  the  moon,  - 
and  moves  towards  it,  in  its  varying  positions,  with  smaller  accelerations  corre- 
sponding to  its  own  greater  mass  (1 : 0-0114) ;  its  path  in  its  orbit  is  therefore  also 
affected  with  small  sinuosities,  of  opposite  phase  to  those  of  the  moon's  motion ;  but 
its  mean  distance  from  the  moon,  and  the  moon's  mean  acceleration  towards  the 
earth,  remain  on  the  whole  unaltered ;  and  the  result  is  as  before. 


vin.]  GRAVITATION.  205 

water,  on  the  side  nearer  the  sun  (or  moon),  towards  which  the  water  would 
flow;  but  the  free  fall  of  the  earth  towards  the  sun  (or  moon)  prevents 
this:  and  then  this  free  fall,  compounded  with  the  tangential  velocity, 
results  in  the  orbital  motion. 

Gravity  acts  instantaneously.  Laplace  showed  that  if  it  were  propa- 
gated through  space  with  a  speed  less  than  9300,000,000000  miles  a  second, 
which  is  itself  a  minimum  limit,  there  would  be  a  rapid  shortening  of  the 
year.  It  must  travel,  therefore,  at  least  50,000000  times  as  fast  as  Light 
does.  There  is  also,  so  far  as  appears,  no  loss  by  diffusion  in  Gravitation. 

Variations  of  the  acceleration  of  gravity  on  the  earth's 
surface.  —  At  the  equator  the  earthward  acceleration  of  gravity 
is,  at  the  sea-level,  978-1028  kines  per  second :  at  the  pole  it  will 
be  983-1084,  if  the  law  of  variation  in  accessible  regions  of  the 
earth's  surface  be  obeyed  there.  This  law  is,  that  at  any  place 
whose  latitude  is  X,  the  local  acceleration  of  gravity  is,  in  kines, 
g  =(980-6056  -  2-5028  cos  2x  -  -000003A),  where  h  is,  in  cm., 
the  height  of  the  observing  station  above  the  sea-level.  This 
diminution  of  gravity  —  equal  masses  weighing  less,  and  there- 
fore distorting  spring-balances  less,  in  regions  nearer  the  equa- 
tor—  is  due  to  two  concurrent  causes:  (1)  That  the  mean 
equatorial  radius  is  greater  than  the  polar ;  the  polar  radius  is 
635,639,000  cm. ;  the  longest  equatorial  radius  (from  lat.  14°  23' 
E.  to  lat.  165°  37'  W.)  is  637,839,000  cm.;  the  shortest  equa- 
torial radius,  at  right  angles  to  the  former,  is  637,792,000  cm.*' 
(2)  The  rotation  of  the  earth.  If  the  earth  came  to  rest,  the 
earthward  acceleration  of  gravity  would,  at  the  equator,  be 
increased  by  <7/289  or  3-3908  kines  per  second,  and  the  weight 
of  bodies  would  be  increased  in  the  ratio  of  289  to  290.  If  the 
earth  rotated  17(=  V289)  times  as  fast  as  it  does,  loose  objects 
would,  at  the  equator,  have  no  weight;  and  if  it  rotated  faster, 
they  would  fly  off  its  surface  at  a  tangent. 

The  acceleration  due  to  gravity  is  in  Paris  980-94,  at  Greenwich  981-17, 
at  Manchester  981-30,  at  Edinburgh  981-54  kines  per  second. 

The  velocity  of  rotation  at  the  equator  is  456,510  cm.  per  second ;  whence 
vz/r  =  3-3908  :  at  Greenwich  it  is  2-100. 

Local  Variations.  —  In  the  neighbourhood  of  a  high  range  of  moun- 
tains a  plumb-line  inclines  towards  the  mountains.  The  ebb  and  flow  of 
water  in  the  Firth  of  Forth  affects  the  apparent  latitude  of  Edinburgh  by 
about  TTJ-0-o-  degree,  for  when  the  water  is  at  high  tide,  plumb-lines  are 
inclined  towards  it,  and  the  mercury  used  as  a  means  of  producing  perfectly 
level  mirrors  is,  in  the  vessels  containing  it,  heaped  up  towards  the  mass  of 
sea-water.  The  water  of  oceans  is  slightly  heaped  up  towards  the  continents. 

*  Col.  A.  R.  Clarke's  values  are  635,638,756  cm.,  637,837,929  cm.,  arfd  637,791,478; 
the  longest  equatorial  axis  being  from  8°  15'  W.  to  171°  45'  E.  of  Greenwich  (Phil. 
Maga.,  1878). 


206  GRAVITATION   AND   THE   PENDULUM.  [CHAP. 


At  sea  the  effect  of  gravity  is  less  than  it  is  on  land,  because  the  mass 
of  water  under  the  spring-balance  is  lighter  than  a  corresponding  amount  of 
rock  would  have  been.  The  depth  of  the  sea  may  be  determined  by  a 
graduated  instrument  of  the  nature  of  a  spring-balance,  sufficiently  sensitive 
to  take  account  of  these  variations. 

Measurement  of  the  Local  Force  of  Gravity.  —  The  force 
of  gravity  must,  like  all  other  forces,  be  measured  by  its  accelera- 
tion. This  may  be  done  directly  by  Attwood's  machine,  already 
described.  Observation  of  a  single  fall  cannot,  however,  give 
accurate  results,  and  the  value  of  g  is  best  determined  by  the 
oscillations  of  a  pendulum. 

There  are  at  the  basis  of  this  determination  four  main  facts: 
(1)  that  a  pendulum  of  a  given  length  will  oscillate  through 
small  arcs  in  equal  times,  of  whatever  substance  it  be  made  — 
this  last  experimental  result  being  due  to  Newton  ;  (2)  that  the 
relation  between  I  the  length  of  a  simple  pendulum,  T  its  time 
of  complete  to-and-fro  oscillation,  and  g  the  local  acceleration  of 
gravitation,  is  given  by  the  formula  g  —  4?r2l/T2  presently  to  be 
proved  (p.  212)  ;  (3)  that  the  length  I  of  a  simple  pendulum 
may  be  very  accurately  observed,  for  in  practice  it  is  equal 
(p.  213)  to  the  distance  between  two  points  on  a  solid  rod,  called 
a  compound  pendulum  ;  and  (4)  that  the  time  of  one  oscillation 
may  be  very  accurately  observed  by  counting  the  number  of 
oscillations  in  a  sufficiently  long  period  of  time.  Hence  g  can 
be  found  to  any  nicety. 

Centre  of  Gravity.  —  The  earth  is  approximately  spherical, 
and  bodies  on  the  surface  have  all  their  particles  drawn  approxi- 
mately towards  its  centre.  But  the  centre  is  so  distant  that, 
within  the  limits  of  ordinary  terrestrial  objects,  the  gravitation 
forces  acting  on  the  several  particles  of  a  body  are  nearly  par- 
allel to  one  another,  and  their  resultant  acts  on  the  Centre  of 
Gravity,  or  Centre  of  Mass.  This  centre  is  the  Centre  of 
Figure  of  a  uniform  body  attracted  by  the  earth. 

The  centre  of  gravity  of  any  plane  figure  may  be  found  by 
cutting  that  figure  out  in  cardboard,  and  suspending  the  card 
first  from  any  one  point  and  then  from  any  other.  A  line  drawn 
vertically  downwards  from  the  first  point  of  suspension  when 
the  body  is  suspended  from  it,  and  another  line  drawn  in  the 
same  way  from  the  second  point  of  suspension,  will  cross  one 
another  at  the  centre  of  gravity. 

Whatever  be  the  form  or  the  arrangement  of  matter  in  a 
body,  if  it  be  suspended  from  any  point  arbitrarily  chosen,  the 


viii.]  CENTRE   OF   GRAVITY.  207 

centre  of  gravity  is  in  a  line  vertically  drawn  through  the  point 
of  suspension  —  vertically  here  meaning  at  right  angles  to  the 
free  horizontal  surface  of  liquid  at  that  place.  If  the  centre  of 
gravity  be  found  by  two  suspensions,  the  vertical  lines  drawn 
from  any  other  points  of  suspension  will  all  pass  through  the 
same  centre. 


Centre  of  Gravity  of  Two  Masses.  —  In  Fig.  93  the  two  bodies,  A 
and  B,  whose  masses  are  m  and  mfl  will  have  their 
centre  of  gravity  at  a  point  C,  which  is  deter-  Fig.93. 

mined  by  the  equation  in  x  AC  =  mt  x  BC.     The 

whole  mass  m  +  ml  may,  as  regards  other  bodies,  be  jj  -  J"1  -  1' 
considered  as  if  it  were  aggregated  at  the  point  C. 

Centre  of   Gravity  of  a  System  of  Masses.  —  This  is  found  by 
taking  account  of  each  seriatim.     In  Fig.  94  let  the  bodies  be  A,  B,  C,  D,  E, 
whose  respective  masses  are  mfl  mn,  mul,  muil,  and  mv      First  the  centre 
of  gravity  of  any  two,  say  A  and  D,  is  found  at  O  ; 
A  and  D  are  supposed  to  be  replaced  by  a  mass  Fig.94. 

mt-\-  mnil  at  O.  Next  the  centre  of  gravity  between 
another  of  the  masses,  say  E,  and  the  imaginary 
mass  mt  +  mull  at  O  is  found  to  be  at  O,  ;  at  this 
point  O;,  there  is  supposed  to  be  placed  a  body 
whose  mass  is  mt+  mllu  +  mv.  In  the  same  way, 
the  centre  between  this  imaginary  mass  and 
another,  say  mn  at  B,  may  be  found  at  O/7. 
Finally  the  centre  between  this  and  the  mass  min 
is  found  at  Oy//,  and  this  is  the  last  operation,  for,  as  regards  external  masses, 
the  system  ABCDE  acts  as  if  it  were  a  mass  (m,  +  my/+  min  -f  mlltl+  mv} 
concentrated  at  the  point  O//7  ;  this  point  is  therefore  the  centre  of  gravity 
of  the  system.  The  same  point  will  be  found  whatever  the  order  in  which 
the  masses  are  considered. 

Since  the  resultant  of  parallel  external  forces  acting  upon 
the  several  particles  of  a  body  acts  upon  the  centre  of  gravity, 
that  centre  moves  as  if  it  were  a  single  particle  under  the  action 
of  a  single  translatory  force.  But  there  may,  in  addition,  be 
rotations  round  the  centre  of  gravity,  and  rotatory  oscillations 
which  can  not  displace  that  centre  ;  and  these  are  independent 
of  the  former. 

Thus  a  ball  of  unsymmetrical  density,  thrown  through  the  air,  will 
swerve  and  gyrate  in  a  puzzling  way  ;  but  its  centre  of  gravity  describes  a 
smooth  trajectory.  The  moon  and  the  earth  have  sinuous  paths  in  space  ; 
but  their  common  centre  of  mass,  which  lies  between  their  respective 
centres  (see  Fig.  93,  C  being  about  2700  miles  from  the  earth's  centre,  and 
therefore  within  the  earth's  mass),  describes  a  smooth  path. 

Overturning  a  body.  —  Let  ABCD  be  a  block  of  material 
supported  on  a  base  CD.  How  great  a  force  applied  at  E,  in 
the  direction  EF,  is  necessary  to  overturn  the  block?  The 


208  GRAVITATION  AND  THE   PENDULUM.  [CHAP. 

question  is  really  one  of  moments  round  C,  for  if  the  force  along 
EF  prevail  over  the  weight  of  the  block,  it  will  do  so  by  turning 

it  over  the  point  C.  From  that  point 
C,  CG  is  the  shortest  line  drawn  to 
meet  the  line  EF,  and  CH  is  the 
shortest  distance  to  the  line  MH, 
along  which  the  force  of  gravity  may 
be  considered  to  act.  At  the  instant 
when  overturning  is  just  going  to 
commence,  the  moments  round  C 
must  be  equal,  and  CG  x  force  along 
EF  =  CH  x  wt.  of  body.  Therefore,  if  the  force  along  EF 
be  greater  than  weight  of  body  x  CH/CG,  the  body  will  be 
overturned.  The  greater  CH  is,  the  greater  must  be  the  force 
exerted  along  EF  in  order  to  overturn  the  body ;  the  smaller 
CH  is,  the  less  need  that  force  be.  When  GH  =  0  —  i.e. 
when  the  centre  of  gravity  M  is  vertically  above  C  —  any  force, 
however  small,  will  upset  the  block  ABCD ;  while,  if  H  be 
on  the  other  side  of  C,  the  block  cannot  stand  unless  propped 
up.  In  this  way  a  body  resting  on  a  wide  base  is  less  easily 
upset  than  one  standing  on  a  narrow ;  one  in  which  a  vertical 
line  drawn  from  the  centre  of  gravity  falls  outside  the  base  of 
support  cannot  stand  unsupported;  while  one  in  which  the 
centre  of  gravity  stands  over  the  very  edge  of  the  base  of  sup- 
port is  upset  by  the  least  disturbance. 

A  microscope,  then,  ought  for  the  sake  of  steadiness  to  have  a  wide  base ; 
and  since  a  tripod  stand  is  the  most  steady  form  of  support,  for  reasons 
already  stated  (see  Spherometer),  instruments  of  this  class  should  be  sup- 
ported on  broad  tripod  stands.  An  old  man  using  a  staff  widens  his  basis 
of  support  by  virtually  converting  his  two  legs  and  the  staff  into  a  broad 
tripod  stand. 

It  amounts  to  the  same  thing  whether  the  base  of  an 
object  be  relatively  broad  or  its  centre  of  gravity  be  relatively 
low.  If  the  centre  of  gravity  be  relatively  high  or  the  base 
relatively  narrow,  —  as  in  the  case  of  young  animals  learning 
to  walk,  children  learning  to  walk,  persons  learning  to  move  on 
skates,  or  on  stilts,  or  on  a  narrow  rail,  or  rope,  or  wire,  or  a 
bicycle,  or  a  person  standing  on  one  foot  or  on  his  heels,  —  a 
relatively  small  displacement  of  the  object  will  readily  cause 
the  centre  of  gravity  to  be  placed  vertically  over  a  point  beyond 
the  base  of  support;  then  the  object,  if  it  be  not  propped  up, 
or  if  the  centre  of  gravity  be  not  brought  over  the  base  of  sup- 


V1II-1  CENTRE   OF   GRAVITY.  209 

port,  or  the  base  of  support  not  brought  up  under  the  centre 
of  gravity,  will  topple  over. 

If,  on  the  other  hand,  the  base  be  relatively  wide,  or  the 
centre  of  gravity  be  relatively  low,  as  in  the  case  of  a  lampstand 
loaded  at  its  base,  the  task  of  upsetting  such  an  object  is  greater, 
since  the  centre  of  gravity  is  in  such  a  case  less  easily  induced 
to  pass  to  a  position  vertically  beyond  the  base. 

Curious  positions  may  be  assumed  by  objects  when  they  are 
so  balanced  that  the  centre  of  gravity  is  vertically  over  some 
point  in  the  basis  of  support.  A  man's  centre  of  gravity  is  at 
a  point  about  the  front  of  his  last  lumbar  vertebra.  If  he  carry 
a  burden,  then,  in  order  to  bring  the  centre  of  gravity  of  the 
conjoined  mass  of  his  body  and  the  burden  borne  by  him  into  a 
position  vertically  over  some  part  of  the  narrow  basis  of  sup- 
port furnished  by  his  feet  (heels,  and  balls  of  great  toes,  and 
lines  joining  these),  he  must  stoop ;  if  the  burden  be  towards 
the  front  of  the  body,  as  in  the  case  of  obese  persons,  the  gait 
becomes  very  erect. 

When  a  body  is  suspended  from  any  point  in  its  own  sub- 
stance and  set  a-swinging,  its  centre  of  gravity  ultimately  finds 
its  way  into  the  lowest  position  possible. 

If  a  disc  roll  round  a  curve,  its  centre  of  mass  will  have  an  outward 
horizontal  Acceleration  vz/r,  and  the  disc  will  be  overturned  unless  it  can 
be  made  to  incline  towards  the  centre  of  curvature  so  that  the  Weight  of 
the  disc,  acting  vertically  through  the  centre  of  mass,  will  also  have  an 
overturning  moment  round  the  point  of  support,  equal  and  opposite  to  that 
of  the  horizontal  Force  mv2/r.  This  takes  place  at  an  angle  £  of  inclination, 
such  that  tan  £  =  v2/rg.  On  this  principle,  skaters,  circus-riders,  etc.,  round- 
ing a  curve,  incline  inwards:  and  the  permanent  way  of  a  railway  is 
adjusted  by  superelevating  the  outer  rail  of  a  curve,  so  as  to  incline  the 
train  at  the  angle  £. 

If  the  superelevation  of  the  outer  rail  be  e,  and  the  width  between  the 
rails  be  w,  e/w  =  tan  £ ;  and  e  =  w  •  tan  £=wv2/rg.  If  e  be  measured  in  inches, 
w  and  r  in  feet,  and  v  in  miles  per  hour,  this  becomes  E  =  0  •  173  WV2/R. 
On  English  railways,  the  practice  is  to  make  E  =  0  -8  WV2/R,  and  thus  to 
allow  for  a  speed  of  a  little  more  than  twice  the  expected  maximum. 

Work  done  in  overturning  a  body.  —  If  there  be  rotation 
round  the  point  C  of  Fig.  95,  so  far  that  M,  the  centre  of  gravity  of 
the  body,  comes  to  be  immediately  over  that  point  and  overturn- 
ing is  effected,  the  centre  of  gravity  is  raised  through  a  certain 
height  h.  The  weight  of  the  body,  mg,  x  that  height,  h,  is  the 
work  which  must  be  done  before  overturning  can  be  accomplished. 

s 

Angle  of  Overturning.  —  If  the  force  applied  at  E  in  Fig.  95  be 
applied  in  a  direction  too  nearly  parallel  to  BD,  its  moment  may  be  too 

p 


210  GRAVITATION  AND  THE   PENDULUM.  [CHAP. 

small  (its  arm  CG  being  too  short)  to  produce  overturning.  At  a  certain 
definite  angle,  BEK,  there  will  be  equilibrium,  the  arm  CG  being  of  exactly 
such  length  as  will  make  the  moment  of  EF  equal  to  that  of  the  weight  of 
the  body.  If,  then,  this  angle  BEK  be  less  than  ^,  the  angle  of  repose,  the 
body  will  overturn  before  sliding ;  if  BEK  be  greater  than  ^,  the  body  will 
slide  before  overturning. 

Equilibrium,  Stable,  Unstable,  and  Neutral.  —  If  the  cen- 
tre of  gravity  be  so  situated  in  a  body  that  work  has  to  be  done 
in  disturbing  it  —  that  is  to  say,  if  the  centre  of  gravity  be 
already  at  its  lowest  possible  position  —  the  equilibrium  is 
stable.  If  a  ball  lie  in  a  bowl,  work  must  be  done  in  order  to 
effect  any  displacement  of  it,  for  no  displacement  can  be  effected 
without  raising  the  centre  of  gravity  of  the  ball,  and  thus 
imparting  potential  energy  to  it.  When  the  ball  is  let  go,  it 
rolls  back  and  oscillates  in  the  bowl  until  it  comes  to  rest. 
The  same  thing  is  seen  in  a  swing,  a  cradle,  a  rocking-horse,  a 
pendulum,  or  a  ship  well  ballasted,  which  are  all  in  Stable 
Equilibrium;  in  the  last  case  the  oscillations  somewhat 
resemble  those  of  a  pendulum  whose  point  of  suspension  and 
whose  length  both  vary. 

If  a  body  have  its  centre  of  gravity  placed  above  its  point 
of  support,  so  that  any  displacement  lowers  its  centre  of  gravity, 
then  the  body  already  has  potential  energy,  which  it  is  disposed 
to  convert  into  kinetic  by  the  fall  of  its  centre  of  gravity  to  the 
lowest  possible  point.  Hence  in  bodies  thus  in  Unstable 
Equilibrium,  a  very  slight  disturbance  may  cause  a  very 
great  displacement,  disproportionate  to  the  disturbance,  but 
depending  on  the  potential  energy  stored  in  the  system.  In 
this  case  are  boats  in  which  people  stand,  high  chairs  in  which 
children  are  seated,  cars  which  are  heavily  loaded  atop,  deck- 
loaded  ships,  and,  in  short,  everything  which  is  "  topheavy." 

Where  no  work  is  done  upon  or  by  an  object,  so  far  as  the 
attracting  forces  are  concerned,  when  it  is  displaced,  the  Equi- 
librium is  Neutral.  A  uniform  sphere  may  be  displaced  and 
assume  a  new  position  without  either  raising  or  depressing  its 
centre  of  gravity. 

A  sphere  floating  in  water  is  in  neutral  equilibrium ;  a 
plank  floating  in  the  usual  way  is  in  stable,  while  a  plank  float- 
ing with  its  edges  vertical  is  in  unstable  equilibrium. 

Simple  Pendulum.  —  This  is  an  ideal.  It  is  a  heavy  part- 
icle suspended  by  a  weightless  cord.  An  approximation  to  a 
simple  pendulum  is  obtained  by  suspending  a  small  bullet  by 
a  very  thin  wire.  The  length  of  this  pendulum  is  the  dis- 


VIII.} 


SIMPLE   PENDULUM. 


211 


Fig.90. 


tance  between  the  point  of  suspension  and  the  centre  of  the 
bullet. 

If  in  Fig.  96  a  simple  pendulum  of  length  I  =  AC  be  repre- 
sented as  displaced  from  the  vertical  position  through  the  angle 
6,  m  being  the  mass  of  the  bob,  and  mg  consequently  its  Weight, 
when  the  bob  is  at  C  the  force  of  gravity  may 
be  resolved  into  two  components  :  one  =  mg  sin  6 
in  the  direction  of  the  tangent  at  C,  and  tend- 
ing to  bring  the  bob  towards  the  middle  line 
with  acceleration  =  g  sin  6  ;  and  a  radial  com- 
ponent =  mg  cos  0,  which  renders  the  cord 
tense.  The  displacement  of  the  bob,  the  dis- 
tance between  B  and  C  measured  along  the 
arc,  is  equal  to  16.  Now,  so  long  as  6  is  small, 
some  2°  or  3°  at  most,  sin  6  and  the  angle  B  are 
nearly  equal,  and  the  tangential  acceleration,  which  is  equal  to 
g  sin  0,  bears  to  the  displacement  W  an  almost  constant  ratio, 
for  g  sin  O./IO  =  (approximately)  g/\.  But  we  know  that  when 
a  body  after  displacement  is  subject  to  a  force  tending  to  bring 
it  back,  which  produces  an  acceleration  proportional  to  the  dis- 
placement, the  result  is  a  S.H.M. ;  and  thus  a  pendulum  very 
slightly  displaced  oscillates  in  S.H.M. 

Simple  Harmonic  Motions  experimentally  performed.  —A 
simple  pendulum  approximately  describes  a  S.H.M.  A  pendu- 
lum whose  bob  consists  of  a  flask  containing  coloured  fluid  or 
ink,  which  pours  from  Fig.97. 

a  narrow  orifice  as  the 
flask  oscillates,  will 
record  the  path  of  the 
pendulum.  A  quan- 
tity of  sand  may  be 
used  instead  of  ink. 
The  apparatus  shown 
in  Fig.  97  (Black- 
burn's  pendulum) 
may  be  used  for  the 
description  of  the 

compounded     H.M.'s      

exemplified    in    Figs. 

35-40.  If  the  bob,  whose  path  is  recorded  by  sand,  be  displaced 
in  a  line  making  an  angle  of,  say,  45°  with  the  line  of  the  cross- 
bar, there  will  be  two  simultaneous  and  independent  oscillations 


i 


212  GRAVITATION  AND   THE   PENDULUM.  [CHAP. 

set  up  when  it  is  liberated  :  one  from  the  cross-bar  and  at  right 
angles  to  it ;  one  from  the  point  P  and  at  right  angles  to  the 
other  oscillation.  By  adjusting  the  relative  lengths  of  the  effec- 
tive long  and  short  pendulum,  by  means  of  a  peg  A,  round  which 
a  certain  quantity  of  cord  is  wound,  or  by  shifting  a  ring  at  P, 
an  indefinite  variety  of  such  figures  may  be  produced. 

The  Time  taken  by  a  single  pendulum  to  effect  one  com- 
plete oscillation  —  one  "  swing-swang  "  —  depends  on  the  square 
root  of  its  length  I  and  varies  inversely  as  the  square  root  of  #, 
the  local  acceleration  of  gravity.  It  is  equal  to  2?r Vl/V/.  Thus 
a  clock  will  go  slower  at  the  equator  than  in  polar  regions ;  a 
clock  will  go  slower  when  its  pendulum  has  been  lengthened  by 
heat :  a  clock  with  a  ten-inch  pendulum  will  tick  twice  as  often 
as  one  with  a  forty-inch  pendulum. 

In  S.H.M.  the  angular  velocity  o  in  the   circle  of  reference  is  equal 
acceleration  at  any  point  of  the  S.H.M. 
displacement  at  that  point 

That  is,  o)  =  V<jr  sin  6./W  =  V^/l. 

But  o)  =(the  angular  path  traversed  in  time  t  -=-  the  time  t)  ;  and  if  the 
time  be  so  chosen  that  in  it  the  body  describing  the  S.H.M.  would  perform 
exactly  one  revolution  in  the  circle  of  reference  —  that  is,  if  t  —  T,  the 
period  of  one  complete  oscillation  back  and  fore,  or  swing-swang  of  the  pen- 
dulum, o)  =  2ir/T ;  whence  T  =  2?r  Vl/^  and  g  =  4?r2l/T2. 

The  value  of  T  may  be  otherwise  written.  The  moment  of  inertia 
N  =  wu2  =  ml2 ;  the  weight  G  =mg ;  whence  T  =  2ir  y/\/g  =  2?rVK/Gl. 


We  see,  from  the  equation  g  =  (4?r2/T2)l,  that  of  those  pen- 
dulums which  oscillate  at  equal  rates  in  different  places,  the 
lengths  are  proportional  to  the  local  intensities  of  gravity. 

The  leg  acts  partly  as  a  pendulum,  and  in  natural  locomotion  a  person 
with  short  legs  has  a  tendency  to  take  shorter  and  quicker  steps  than  a 
person  with  longer  limbs. 

Isochronous  Oscillations  of  a  Simple  Pendulum.  —  The 
equation  T  =  27rVl/#  shows  that  so  long  as  (g  •  sin  0./10)  may 
be  considered  to  be  equal  to  gfi,  —  that  is,  so  long  as  the  angle 
6  is  not  too  wide,  —  the  period  of  oscillation  does  not  depend  on 
the  amplitude  of  oscillation,  for  0  does  not  enter  into  that  equa- 
tion ;  and  thus  within  certain  limits  a  pendulum  swings  in  equal 
periods  through  comparatively  large  or  comparatively  small  arcs. 

Length  of  the  Ideal  Simple  Seconds-Pendulum.  —  The 
seconds  pendulum  performs  one  "complete  oscillation  "  in  two 
seconds.  The  equation  g  =  4?r2l/T2,  when  T  =  2  sec.,  gives 
I  =  TT*  =  3-2616083  feet  or  39-1393  inches  at  the  latitude  of 


VIII.] 


SIMPLE   PENDULUM. 


213 


London,  at  sea-level,  and  when  the  barometer  and  thermometer 
are  at  standard  heights  (30  inches  of  mercury,  60°  F.). 

Work  done  in  moving  a  Simple  Pendulum.  —  In  Fig.  96  suppose 
the  bob  to  be  displaced  from  C  to  D ;  its  angular  displacement  0  is  increased 
to  0,.  When  at  C  its  vertical  height  above  B  is  Be  —  that  is,  AB  —  Ac,  or 
I  —  I  cos  0.  When  at  D  its  vertical  height  above  B  is  I  —  I  cos  Ot ;  and  its 
vertical  height  above  C  is  cd  =  Ac  —  Ad  =  Icos0—  icos  0r  When  the 
bob,  whose  weight  is  mg,  is  raised  from  C  to  D,  the  work  done  against  gravity 
is  mg  x  I  (cos  6  —  cos  07). 

Compound  Pendulum.  —  Let  a  body  of  any  shape  be  poised  on  a  point 
or  axis  of  suspension,  as 
from  the  point  A,  or  an 
axis  passing  through  the 
point  A  in  Fig.  83rf  or 
Fig.  98;  let  the  radius  of 
inertia  of  the  body  with  re- 
spect to  this  point  A  be  t ; 
and  h  the  distance  of  the 
centre  of  gravity  B  below 
A.  If  two  positions  be 
taken,  as  in  Fig.  98,  respec- 
tively corresponding  to  dis- 
placements through  angles 


Fig.98. 


(h  cos  0  —  h  cos  0,)  =  mgh(cos  0  — 


6  and  0,,  the  vertical  dis- 
tances between  the  centre 
of  gravity  and  the  point 
of  suspension  are  respec- 
tively h  cos  0  and  h  cos  0,. 
The  work  done  by  the  body 
during  a  movement  from 
the  higher  position  to  the  lower  is  mg  x 
cos  0,)  =  (by  page  164)  £Xo>2  =  $wu2G>2. 

.».  <o2  =  2gh  (cos  0  -  cos  <9,)  /t2.      (1.) 

The  work  done  by  the  bob  of  a  simple  pendulum  during  a  similar  dis- 
placement is  mgl  (cos  6  —  cos  #,).  In  this  fall  it  acquires  angular  velocity 
to, ,  and  kinetic  energy  =  iNw,2  =  $ml2  •  a)/2.  The  kinetic  energy  acquired  is 
equal  to  the  potential  energy  lost,  whence  — 

jwil2  •  a>;2  =  mgl  (cos  0  -  cos  0,}. 
w/2  =  2g  (cos  0  -  cos  0,)/l.  (2.) 

If  the  simple  pendulum  were  of  such  a  length  as  to  oscillate  at  the  same 
rate  as  the  compound  one  -*-  that  is,  if  o>,  the  angular  velocity  of  the  one  =  <«>„ 
that  of  the  other  —  we  find  from  (1)  and  (2)  that 

2gh  (cos  0  -  cos  0,)/i2  =  2g  (cos  0  -  cos  0,)/l. 
.-.  I  -  t/h. 

Hence  the  compound  pendulum  oscillates  at  the  same  rate  as  a  theoret- 
ical simple  pendulum  of  length  1  =  i*/h ;  that  is,  its  period  is  T  =  27r  V*2  A<7- 

This  length,  i2/A,  the  length  i  of  the  equivalent  simple  pendulum,  is  the 
distance  between  the  centre  of  suspension  and  the  centre  of  oscillation.  Let 
mi02  be  the  moment  of  inertia  of  the  compound  pendulum  round  its  centre 


914  GRAVITATION  AND   THE   PENDULUM.  [CHAP. 

of  mass.  Then  round  the  centre  of  suspension  the  moment  of  inertia  is 
m(i02  +  h2),  and  the  length  I  of  the  equivalent  simple  pendulum  is  i2  /  h  = 
(i02  +  A2)  +  h  =  (t02/^)  +^«  Similarly,  if  the  body  be  suspended  from  the 
centre  of  oscillation,  the  distance  between  this  new  point  of  suspension  and 
the  centre  of  mass  being  h,,  the  length  of  the  equivalent  pendulum  is  now 
\t  =  if/h,  =  (i02//O  +  hr  But  (Fig.  83rf)  the  centre  of  suspension  A  and 
the  centre  of  oscillation  C  are  interchangeable  ;  whence  I  =  lr  Therefore 
(i02/A)  +  h  =  (i02A)  +  V  whence  i02  =  hh,,  and  I  =  I,  =  7*  +  h,  =  AB  +  BC 
(Fig.  83d)  =  AC  ;  that  is,  the  distance  AC  between  the  interchangeable 
centres  of  suspension  and  of  oscillation  is  equal  to  the  length  I  of  the 
equivalent  simple  pendulum.  Therefore  — 

If  a  body  of  any  form  be  suspended  at  a  certain  point  and 
be  found  to  oscillate  at  a  certain  rate  ;  if,  after  trial,  another 
point  in  the  body  be  found,  situated  on  the  other  side  of  the 
centre  of  gravity,  and  such  that  the  body  will,  when  suspended 
from  it,  oscillate  at  the  same  rate,  —  then  the  distance  between 
these  interchangeable  points  of  suspension  is  the  true  length  of 
the  ideal  simple  pendulum  oscillating  at  the  observed  rate  ;  it 
being  observed,  however,  that  in  the  case  of  a  symmetrical  bar, 
these  two  points  must  not  be  at  equal  distances  from  the  centre 
of  gravity,  for  in  that  case  any  two  such  equidistant  points  are 
interchangeable. 

Problem.  —  Prove  that  a  cylindrical  rod  will  swing  at  the  same  rate, 
whether  it  be  suspended  from  its  extremity  or  from  a  point  one-third  of  the 
length  from  the  extremity.  —  Am.  I,  =  i?/ht  =  /2/3  -*-  1/2  (see  p.  162,  No.  1) 
=  ¥•  I//  =  */;2A/  5  ^  =  V2  +  V;  tf  -  /2/12  (see  p.  162,  No.  2)  ;  hn  =  l/S; 


Ballistic  Pendulum.  —  Suppose  a  heavy  mass  M  to  be  suspended  from 
a  point.  Into  this  heavy  mass  let  a  bullet  of  mass  m  be  horizontally  fired, 
striking  the  centre  of  percussion  with  velocity  y,  and  let  the  bullet  sink  into 
it  so  as  to  form  a  conjoint  mass  M  +  w,  whose  centre  of  mass  is  at  a  dis- 
tance h  below  the  point  of  suspension.  The  energy  of  the  striking  bullet  is 
7wt>2/2;  this  energy  is  wholly  imparted  to  the  conjoint  mass  before  that 
mass  has  had  time  to  become  appreciably  displaced.  In  virtue  of  this 
energy  imparted  the  whole  is  displaced  so  far  that  the  suspending  cord 
comes  to  make  an  angle  9  with  its  original  position,  work  being  thus  done 
against  gravity,  equal  to  (M  +_  m)gh(l  -  cos  0)  or  (M  +  m)gh  •  2  sin2  (0/2)  ; 
the  whole  then  falls  back  and  thereafter  oscillates  in  ordinary  pendulum 
fashion.  The  energy  imparted  and  the  work  done  against  gravity  are 
equal;  whence 

mvz/2  =  (M  +  m)gh  •  2  sin2  (0/2), 

sinW2> 

i.e.  sin  (0/2)  <x  v. 

The  throw  is  such  that  the  sine  of  half  the  angle  of  deflection  is  propor- 
tional to  the  velocity  of  the  impinging  bullet,  and  therefore  to  the  square 
root  of  its  energy. 


VIII.] 


BALLISTIC   PENDULUM. 


215 


Fig.  98  a. 


If  o>  be  the  angular  velocity  imparted  to  the  swinging  mass,  the  energy 
imparted  may  also  be  written  as  ^No>2;  whence 

No>2  =  2(M  +  m)gh  -  2  sin2  0/2 
and  o>  =  2V(M  +  m)gh/N •  sin  J/2. 

Stability  of  a  Ballistic  Pendulum.  —  The  amounts  of  energy  which 
must  be  imparted  in  order  to  jerk  such  a  pendulum  through  15°,  30°,  45°, 
60°,  75°,  90°,  105°,  120°,  135°,  150°,  165°,  or  180°,  are  ttie  products  of 
(M.  +  m)gh  into  0-0341,  0-134,  0-2929,  0-5,  0-7412,  1-0000,  1-2588,  1-5, 
1-7071,  1-866,  1-9659,  or  2-0000  respectively.  From  these  figures  we  see 
that  the  Stability  of  a  pendulum  in  any  position  —  the  amount  of  energy 
which  must  be  imparted  to  it  in  order  to  throw  it  farther  through  one 
degree  —  is  greatest  when  the  throw  is  already  about  90°.  Thereafter  it 
diminishes ;  and  when  the  energy  imparted  exceeds  2(M  +  m)gh,  or  when  the 
velocity  of  the  impinging  bullet  exceeds  a  certain  limit(v  =  2 
the  pendulum  is  thrown  right  over,  and  describes  a 
somersault.  Problems  of  this  nature  are  of  great 
importance  in  connection  with  the  stability  of 
ships. 

Bifilar  Suspension.  —  If  two  masses,  A,  B,  at 
the  extremities  of  a  weightless  rod  AB,  be  suspended 
by  the  parallel  cords  aA  and  6B,  and  if  these  be  \ 

displaced  so  that  round  O,  the  central  part  of  AB, 
there  is  rotation  through  an  angle  0,  there  is  neces- 
sarily a  lifting  up  of  both  A  and  B,  and  there  will 
be  components  of  gravitation  tending  to  stretch  each 
suspending  string,  and  components  tending  to  restore 
A  and  B  to  their  original  positions.  We  need  only  I 

consider  the  cord  a  A.  Its  lower  extremity  is  swung 
horizontally  through  an  angle  0  with  respect  to  O  ; 
the  whole  cord  is  deflected  from  its  vertical  position 
through  an  angle  ^  such  that  a  A  •  tan  {j/  =  the  chord 
of  the  arc  of  0  =  OA-2  sin  0/2.  The  component 
of  gravitation  tending  to  restore  A'  to  A,  acting 
towards  A,  is  equal  to  mg  tan  \j/.  Its  moment  round 
O  is  (mg  tan  i^)  •  (O  A  •  cos~#72).  The  whole_moment 
of  the  couple  is  2mg  tan  \f/  •  OA  •  cos  0/2  =  2mg  A« 
(0 A2/a A)  2  sin  072  •  cos  0/2  =  2mg  (O A2/aA) sin  0. 
The  moment  of  the  restoring  force  is  thus  propor- 
tional to  the  sine  of  the  angle  of  deflection,  and  the  oscillations  of  such 
a  system  are  approximately  simple  harmonic. 


B 


.0x0 


B 


CHAPTER  IX. 

MATTER. 

THE  essential  nature  of  matter  —  its  substratum  —  is  unknown 
to  us  ;  we  only  know  matter  by  those  of  its  properties  which  we 
perceive  by  our  senses.  These  properties  are  subject  to  our 
direct  observation  and  to  our  study,  and  from  them  we  may 
infer  as  to  the  constitution  of  matter  much  which  we  cannot 
directly  perceive. 

THE  PROPERTIES  OF  MATTER. 

Some  of  these  properties  of  matter  are  general,  so  that  if 
they  were  other  than  they  actually  are,  the  nature  of  our  uni- 
verse would  be  totally  different ;  and  thus,  in  relation  to  the 
matter  of  the  existent  universe,  these  properties  are  with  suffi- 
cient appropriateness  said  to  be  essential.  For  instance,  all 
experience  leads  us  to  say  that  matter  must  necessarily  exist  in 
definite  or  measurable  Quantity ;  and,  since  quantity  of  matter 
is  expressed  briefly  by  the  word  Mass,  we  say  that  every  body 
must  have  a  definite  mass,  for  it  is  to  us,  with  our  range  of  ideas, 
impossible  to  conceive  of  a  definite  body  having  a  physical 
existence,  but  consisting  of  an  indefinite  quantity  of  matter. 
There  are  many  bodies  of  which  we  do  not  definitely  know  the 
mass,  but  every  body  must  have  some  definite  mass,  great  or 
small.  If  the  mass  of  a  body  be  great,  the  body  is  said  to  be 
massive ;  if  its  mass  be  small,  it  is  usually  said  to  be  light, 
though  that  adjective  is  properly  antithetical  to  heavy,  a  per- 
fectly distinct  idea.  A  massive  gate  is  difficult  to  move,  not 
because  it  is  heavy,  for  gravity  does  not  affect  the  horizontal 
swing  of  a  gate  on  its  hinges,  except  indeed'  by  affecting  the 
friction  at  the  hinges ;  it  is  difficult  to  move  because  its  mass  m 
is  great ;  and,  since  F  =  ma,  in  order  to  produce  a  given  accel- 
eration a,  if  the  mass  m  be  large  the  force  applied  must  be 
great.  It  would,  in  theory,  be  equally  considerable  were  the 

216 


[CHAP,  ix.]  PROPERTIES  OF  MATTER.  217 

gate  and  its  hinges  removed  to  a  region  where  the  effect  of 
gravity  vanished,  and  the  gate  had  therefore  not  even  a  feather's 
Weight. 

As  regards  Quality  of  Matter,  experience  shows  us  that 
every  substance  with  which  we  are  acquainted  is  made  up  of  one 
or  other  or  more  or  fewer  of  about  seventy  different  kinds  of 
matter.  These  kinds  of  matter  are  called  elements.  They  are 
considered  to  be  distinct  kinds  of  matter,  and  are  called  sepa- 
rate elements,  simply  as  a  confession  of  our  relative  experimental 
impotence,  and  of  our  complete  failure  up  to  this  time  to  break 
up  any  one  of  them  into  simpler  substances,  or  to  build  any  one 
up  by  any  synthetic  process.  A  piece  of  brass  may  be  by  ana- 
lytic processes  resolved  into  its  component  copper  and  zinc, 
and  when  copper  and  zinc  are  fused  together  in  proper  propor- 
tions, brass  of  a  similar  quality  may  be  made ;  but  no  one  has 
broken  up  either  zinc  or  copper  into  any  simpler  components, 
neither  have  these  metals  been  made  by  causing  any  simpler 
substances  to  combine.  These  metals  are,  then,  Elements ;  and 
the  substances  which  they  form  by  entering  into  combination 
with  other  elements,  as  well  as  the  circumstances  under  which 
these  combinations  are  effected,  form  the  subject-matter  of  the 
Science  of  Chemistry.  The  description  of  any  given  substance 
—  chlorine,  nitrogen,  calcium  —  as  an  element  is  thus  seen  to  be 
entirely  provisional.  The  experience  of  1807  may  possibly  be 
repeated  when  least  expected.  Before  that  date  lime,  soda,  and 
potash  were  enumerated  in  the  list  of  elements,  though,  from 
their  strong  likeness  to  metallic  oxides,  it  was  vehemently  sus- 
pected that  they  were  really  not  elements  at  all,  but  oxides. 
When  Sir  Humphry  Davy  brought  the  galvanic  battery  of  the 
Royal  Institution  to  bear  upon  masses  of  these  substances,  he 
resolved  them  into  oxygen  and  into  metals  never  seen  till  then ; 
and  thus  the  list  of  elements  suffered  a  profound  modification. 
Now  evidence  of  a  speculative  character,  on  the  one  hand, 
based  (de  Chancourtois,  Newlands,  and  Mendelejeff)  upon  the 
remarkable  relations  existing  between  the  chemical  properties  of 
the  elements  and  their  atomic  weights,  and  also  (Gladstone) 
upon  the  relations  between  tliese  chemical  properties  and  the 
extent  to  which  such  of  the  elements  as  are  transparent  or  form 
transparent  compounds  possess,  either  when  pure  or  in  combina- 
tion, the  power  of  refracting  a  beam  of  incident  light ;  and  evi- 
dence, on  the  other  hand,  of  a  directly  observational  char- 
acter, based  (Lockyer)  upon  the  results  of  spectrum  analysis  as 


218  MATTER.  [CHAP. 

applied  to  the  stellar  bodies,  results  which  seem  to  show  that 
many  elements  are  decomposed  by  intense  heat  into  simpler 
elements ;  this  mass  of  evidence  lends  cumulative  support  to  a 
belief,  which  is  rapidly  gaining  ground,  that  all  the  elements 
differ  from  one  another  only  in  their  intimate  structure,  and 
have  a  common  basis  which  may  not  impossibly  be  the  element 
helium  which  exists  in  the  Sun;  or  in  other  words,  that  all 
the  elements  are  structural  modifications  of  one  form  of  Matter. 
Thus  even  the  alchemist's  dream  of  the  transmutation  of  metals 
cannot  now  be  treated  with  such  unmitigated  contempt  as  it 
received  forty  or  even  twenty  years  ago,  though  it  may  con- 
tinue to  be  a  dream  to  the  realisation  of  which  no  approach  is 
possible,  on  account  of  the  necessary  limitations  of  our  experi- 
mental appliances. 

In  reference  to  Space :  every  mass  of  matter  must  at  any 
instant  occupy  a  definite  volume  of  space  :  it  must  have  some 
Extension  in  tridimensional  space  —  it  must  have  dimensions 
expressible  in  terms  of  length,  breadth,  and  thickness.  As  a 
natural  consequence,  every  mass  of  matter  must  have  some 
definite  form,  whether  that  form  be  imposed  on  it  by  surround- 
ing matter  or  not  —  whether,  like  a  solid  rock,  it  have  a  form  of 
its  own,  or,  like  water  in  a  basin,  it  have  a  form  which  depends 
partly  on  the  form  of  the  vessel  containing  it,  or,  like  gas  con- 
fined in  a  gas-holder,  its  form  as  well  as  its  volume  depend  on 
that  of  the  vessel  in  which  it  is  enclosed. 

Among  the  properties  of  matter  which  are  said  to  be 
essential  we  usually  find  mentioned  that  known  as  impene- 
trability. This  means  that  two  masses  of  matter  cannot  occupy 
the  same  space.  In  view  of  the  peculiar  phenomena  attending 
the  solution  of  substances  in  water  —  a  very  large  quantity  of 
different  salts  being  soluble  in  water  without  materially  increas- 
ing its  bulk —  we  cannot  state  this  absolutely.  But  matter  is 
believed  to  be  composed  of  minute  masses  called  Molecules, 
and  of  these  it  is  held  to  be  true  that  two  cannot  coincide  in 
position.  But  these  are  not  in  contact  even  in  solids,  and  so  a 
body  is  always  free  to  shrink  in  size  —  as,  for  instance,  when  it 
is  cooled  down  or  compressed  —  because  the  distance  between 
its  molecules  is  capable  of  diminution ;  and  thus  a  quantity  of 
water,  which  is  not  a  continuous  substance,  may  receive  between 
its  own  molecules  a  number  of  molecules  of  other  substances, 
and  so  form  a  solution,  without  entirely  sacrificing  that  freedom 
of  movement  past  one  another  which  its  own  molecules  possess 


ix.]  PROPERTIES  OF  MATTER.  219 

—  that  is,  without  entirely  losing  its  fluidity.  The  impene- 
trability of  matter  is,  then,  a  property  of  molecules,  not  neces- 
sarily of  masses. 

If  a  certain  bulk  of  metallic  potassium  contain  200  atoms  or  half- 
molecules  of  potassium,  an  equal  bulk  of  caustic  potash  will  contain  331 
atoms  of  potassium,  331  of  hydrogen,  and  331  of  oxygen. 

In  respect  of  Time :  Lapse  of  time  brings  about  no  change 
either  in  the  quantity  of  matter  or  in  its  quality  —  that  is, 
matter  is  indestructible  both  in  regard  to  its  total  quantity  and 
to  the  quantity  of  each  element.  Such  is  the  ordinary  belief ; 
the  former  statement  is  in  accord  with  the  universal  experience 
of  Chemistry ;  but  he  would  be  bold  who,  from  the  experience 
of  mankind  on  the  surface  of  the  earth,  should  venture  to  deny 
that  in  the  interior  of  this  planet  there  may  even  now,  as  the 
earth  is  cooling,  be  an  increase  taking  place  in  the  quantity  of 
the  more  complex  at  the  expense  of  the  more  simple  elements ; 
not  to  speak  of  the  positive  probability  which  spectrum  analysis 
lends  to  a  belief  that  this  kind  of  action  is  actually  going  on  in 
the  fixed  stars.  Be  that  as  it  may,  within  our  experimental 
range  of  knowledge  there  is  no  transmutation  of  elements  and 
no  destruction  or  creation  of  matter.  Matter  changes  its  forms 
and  its  combinations  incessantly,  but  it  can  always  be  traced  up 
by  chemical  analysis.  A  closed  glass  tube  containing  oxygen 
and  powdered  charcoal  weighs  exactly  the  same  after  the  char- 
coal has  been  induced  to  burn  in  the  oxygen,  and  thus  to  dis- 
appear, as  it  did  before  that  action ;  the  gaseous  carbonic  acid 
produced  is,  though  invisible,  equal  in  total  weight,  and  there- 
fore in  mass,  to  the  sum  of  the  carbon  and  oxygen  which  com- 
posed it. 

The  general  properties  of  matter. — By  a  distinction  which 
is  somewhat  arbitrary,  the  preceding  properties  are  said  to  be 
essential,  while  those  of  inertia,  weight,  divisibility,  and  porosity 
are  said  to  be  general,  because  found  to  be  possessed  by  all 
matter.  The  statement  that  inertia  is  a  general  property  of 
matter  simply  means  that  Newton's  first  law  of  motion  is  a 
universal  result  of  experiment. 

All  bodies  possess  weight  at  the  earth's  surface  and  within 
experimental  or  observational  limits.  A  mass  placed  on  the 
earth's  surface  is  attracted  by  the  earth,  by  the  sun,  the  moon, 
the  planets  of  the  solar  system,  and  in  a  less  degree, —  the  attrac- 
tion being  so  small  that  we  have  no  direct  evidence  of  its  exist- 
ence—  by  the  distant  fixed  stars;  the  resultant  differs  so  very 


220  MATTER.  [CHAP. 

little  from  the  direct  attraction  of  the  earth  that  the  latter 
alone  may  be  considered  as  pulling  the  mass  downwards  towards 
its  centre ;  but  this  is  only  a  first  approximation,  for  a  more 
careful  discussion  of  the  intensity  and  direction  of  the  resultant 
force  helps  us  to  explain  the  phenomena  of  the  Tides. 

All  masses  are  divisible  ;  the  only  question  which  here 
emerges  is  that  as  to  indefinite  divisibility.  Is  a  given  mass  — • 
say  of  chalk  —  divisible  to  infinity,  or  would  we  after  division 
effected  with  sufficient  frequency  obtain  a  small  mass  of  chalk 
which,  if  further  divided,  would  be  no  longer  chalk,  but  might 
perhaps  be  broken  up  into  lime  and  carbonic  acid?  The  facts 
of  chemical  equivalence,  as  ascertained  by  the  balance,  seem 
to  be  susceptible  of  no  natural  explanation  other  than  that 
matter  is  made  up  of  such  ultimate  particles,  and  hence  matter 
is  concluded  not  to  be  indefinitely  divisible.  We  shall  recur  to 
this  subject. 

Nobert  engraved  parallel  lines  on  glass  at  a  mutual  distance  of  1/40,000 
cm.,  half  the  wave-length  for  bluish-green  light.  No  microscope  made  can 
show  these  as  distinct  lines. 

All  matter  is  porous  or  possesses  porosity.  Hydrogen  gas 
leaks  through  white-hot  iron  under  pressure  ;  cold  water  can  be 
pressed  through  iron,  as  may  be  seen  in  Bramah's  hydraulic 
press,  or  through  lead,  as  in  Francis  Bacon's  famous  experiment, 
in  which  he  took  a  shell  of  lead  filled  with  water  and  com- 
pressed it ;  the  water  oozed  through  the  lead  and  stood  in  drops 
and  beads  on  the  surface  of  the  shell. 

Contingent  properties  of  matter.  —  Some  of  the  properties 
of  matter  are  contingent,  and  depend  on  the  particular  kind  of 
matter  considered  and  on  the  surrounding  circumstances.  As 
examples,  we  may  take  the  facility  with  which  a  body  is  heated, 
the  rate  at  which  heat  can  run  along  it,  the  ease  or  difficulty 
with  which  light  can  pass  through  it,  and  so  on. 

The  Quantity  of  matter  per  unit  of  volume  is  denned  as  the 
density  of  the  mass  occupying  that  volume.  Thus  a  gramme  of 
water  occupies  a  cubic  centimetre,  and  according  to  the  C.G.S. 
or  centimetre-gramme-second  system  of  measurement,  the  den- 
sity of  water  is  (1  gramme/1  cub.  cm.)  =  1.  More  accurately 
(Kupffer),  1  cub.  cm.  water  at  3°-9  C.  weighs  1-000,013  stan- 
dard grammes,  and  its  density  is  1-000,013.  In  the  same  way  the 
density  of  lead  is  11-35,  because  1  cub.  cm.  weighs  11-35  grammes. 

In  general,  if  m  be  the  mass  contained  in  volume  ij,  and  p  the  density, 
m/ij  =  p,  or  m  =  ty>. 


ix.]  DENSITY.  221 

Every  kind  of  matter,  simple  or  compound,  has  a  special 
density  of  its  own;  thus  a  given  bulk  of  lead  contains  11-35 
times  as  much  mass  as  the  same  bulk  of  water.  Water  is  taken 
as  the  standard  of  density ;  its  specific  density  is  said  to  be  =  1, 
though  sometimes,  in  estimating  the  density  of  liquids,  it  is,  in 
order  to  avoid  decimal  fractions,  reckoned  as  1000.  In  the 
same  way,  the  specific  density  of  lead  is  11-35,  and  those  of  all 
substances  may  be  experimentally  found  and  recorded  in  a 
table  of  specific  densities,  or,  as  it  is  more  commonly  called, 
a  table  of  Specific  Gravities.  These  are  experimentally  found 
by  taking  advantage  of  the  fact  that  Weight  is  proportional 
to  Mass ;  G  =  mg.  The  piece  of  lead  which  will  occupy  a 
given  space  not  only  contains  11*35  times  as  much  mass,  but  also 
weighs  11-35  times  as  much  as  the  quantity  of  water  which  would 
fill  the  same  space.  Thus  the  density  of  a  body,  as  compared 
with  that  of  water,  presents  a  ratio  —  its  specific  density  —  which 
is  numerically  identical  with  the  ratio  —  its  specific  gravity  — 
of  its  weight  to  that  of  an  equal  bulk  of  water. 

If  m  and  m,  be  the  masses  of  equal  volumes  of  the  body  and  of  water, 
to  these  equal  volumes ; 

density  of  the  body      m/b      m 

Sp.  density  =  — ^ — i- 5 —    — ^  =  — L  .  - —  ; 

density  ot  water         wyb     m/ 

~  .  weight  of  the  body  G  _  mg      m 

"weight  of  equal  bulk  of  water"  G,     mtg  ~  m, 
Hence  sp.  density  and  sp.  gravity  are  numerically  identical  ratios. 

Sketch    of    the    Experimental    Methods     of    finding    the    Specific 
Density  of  Bodies. 

(a)  Of  Solids.  —  1.  Weigh  the  body  in  air  (properly  in  vac  wo);  measure 
its  bulk  by  dipping  it  (suspended  by  a  thin  string)  into  water  contained  in 
vessel  A,  and  observing  the  rise  of  level  in  that  vessel;  take  it  out :  out  of  a 
known  quantity  of  water  in  vessel  B  pour  enough  water  into  vessel  A  to 
produce  an  equal  rise  of  level  in  the  vessel  A  ;  find  the  weight  of  the  water 
that  has  been  poured  out  of  vessel  B.  Then  the  weight  of  the  body  -4-  the 
wt.  of  the  equal  bulk  of  water  poured  out  of  B  =  the  sp.  density  of  the 
body.  The  practical  objection  to  this  method  is,  that  the  body  when  taken 
from  vessel  A  removes  some  of  the  water. 

2.  Weigh  the  body  in  air  ;  put  it  in  a  vessel  —  a  "  specific-gravity  flask  " 
—  marked  distinctively  at  a  certain  level;  fill  with  water  up  to  the  marked 
level ;  weigh.  Empty  the  vessel  and  fill  with  water  alone  up  to  the  mark  ; 
weigh. 

Then  Weight  of  vessel,  body  and  water  up  to  level,  =  V  +  B  +  w. 

"  vessel  and  water  alone  up  to  level  =  V  +  W. 

.•.  the  Weight  of  the  water  which  replaces  the  body  is  W  —  w, 

and  the  Weight  of  the  body  is  B. 
_  Weight  of  the  body  _       B 

~  Weight  of  an  equal  bulk  of  water  ~  W  —  w 


222  MATTER.  [CHAP. 

3.  Take  advantage  of  the  following  proposition  in  Hydrostatics  :  —  A 
body  suspended  in  a  liquid  is  buoyed  up  by  that  liquid  to  such  an  extent  as 
to  diminish  in  apparent  weight  by  an  amount  equal  to  the  weight  of  the 
bulk  of  the  liquid  which  it  may  be  considered  as  displacing.     If  a  body  of 
exactly  the  same  density  as  water  be  suspended  in  water,  it  will  neither  sink 
nor   rise ;  its  apparent  weight  will  =  0 ;  its   sp.   gr.  =  (wt.  in  air  -=-  loss  in 
water)  =  1/1  =  1.     If  it  be  more  dense  it  will  sink,  but  slowly,  for  while  its 
mass  is  unaltered,  the  force  acting  on  that  mass  is  apparently  diminished. 
If  it  be  less  dense  than  water,  it  will  rise ;  its  weight  will  appear  to  be  less 
than  nothing,  negative. 

A  cork  of  volume  to  and  density  -8  will  have  a  mass  to  x  '8  =  -8to.  An  equal 
bulk  of  water  would  have  a  mass  to  x  1  =  to.  The  weight  of  this  mass  of 
water  would  be  mass  x  g  =  to# ;  the  weight  of  the  mass  of  cork  is  similarly 
•8to#.  The  apparent  weight  of  the  mass  of  cork  will  be  •S'ag  —  to<7  =  —  •  2to<?  ; 
i.e.,  its  downward  acceleration  will  be  negative,  or  the  cork  will  move 
upwards.  This  upward  accelerating  force,  =  -2to//,  acting  on  a  mass  -8to,  will 
produce  an  acceleration  (since  a  =  F/m)  =  -2to#/-8to  =  \g,  and  but  for  fric- 
tion in  the  water  the  cork  would  rise  with  an  upward  acceleration  of  8-05 
ft.-per-sec.  per  second. 

Therefore  weigh  a  body  in  air ;  suspend  it  by  a  fine  thread  from  the  pan 
of  a  balance,  so  that  it  just  sinks  wholly  into  water.  It  will  appear  to  weigh 
less.  Then  the  sp.  gr.  =  (weight  in  air  -^-  apparent  loss  of  weight  in  water), 
or,  accurately,  (weight  in  vacua  -t-  apparent  loss  of  weight  in  water).  If  the 
body  be  lighter  than  water,  attach  to  it  a  piece  of  heavy  substance  —  say  lead 
—  of  known  weight  and  known  density  (lead  =  H'35).  Then  the  weight  of 
the  lead  is  ll-35to/7 ;  that  of  the  light  substance  is  p  to,  g ;  together  they  weigh 
ll-35to<7  +  /oto,#.  In  water  they  weigh  10-35to/7  +  (/o  —  1)  to^.  The  loss  of 
the  lead  alone  must  be  to//,  or  l/ll'35th  of  its  weight;  whence  the  loss  of  the 
light  body  in  water  is  easily  determined  by  difference ;  and  its  density,  the 
fraction  (weight  in  air  -=-  apparent  loss  in  water),  found. 

4.  If  the  solid  be  soluble  or  be  otherwise  acted  on  by  water,  some  other 
fluid  is  made  use  of,  such  as  naphtha  or  turpentine,  the  specific  density  of 
which  is  known.     If  the  specific  density  of  the  solid  with  reference  to  the 
liquid,  found  by  any  of  the  above-mentioned  methods,  be  s/l,  and  the  sp.  d. 
of  the  liquid  with  reference  to  water  be  l/w,  then  the  product  of  these  spe- 
cific densities  =  s/l  x  l/w  =  s/w  is  the  sp.  d.  of  the  solid  as  compared  with 
water. 

(b)  Of  Liquids.  —  1.  Fill  a  weighed  vessel  up  to  a  certain  mark  with 
the  liquid :  the  whole  weighs  so  much  :  by  difference  find  the  weight  of 
the  liquid  alone.  Empty  the  vessel  and  repeat  the  process  with  water ;  the 
water  which  fills  the  vessel  up  to  the  same  mark  weighs  so  much.  Then  the 
sp.  gr.  of  the  liquid  =  (wt.  of  given  bulk  of  liquid  -=-  wt.  of  same  bulk  of 
water). 

2.  A  body  immersed  in  water  appears  to  lose  1/xth  of  its  weight;  im- 
mersed in  the  liquid  to  be  tested  it  appears  to  lose  l/;yth  of  its  weight.     This 
body  is  x  times  (x  being  a  whole  number  or  a  fraction)  as  dense  as  water,  y 
times  as  dense  as  the  liquid.     The  density  of  the  liquid  is  to  that  of  water  as 
the  apparent  loss  \fy  in  the  liquid  is  to  the  apparent  loss  \/x  in  the  water  — 
that  is,  as  x  :  y ;  and  the  sp.  d.  of  the  liquid  is  x/y. 

3.  By  the  use  of  Hydrometers,  Alcoholometers,  and  the  like.     The  prin- 
ciple of  these  instruments  is  the  following :  —  A  body  which  floats  in  water 
without  being  wholly  submerged  is  in  equilibrium  under  the  action  of  two 


ix.]  DENSITY.  223 

forces  —  (1)  the  weight  of  the  whole  body  ;  (2)  the  buoyancy  of  the  water, 
equal  to  the  weight  of  the  part  of  the  water  displaced.  Thus  ice  floats  in 
water  with  -f^\  of  its  bulk  submerged.  If  the  volume  of  a  mass  of  ice  be 
ij  and  its  specific  density  p,  the  weight  of  the  mass  is  bpg,  and  the  weight  of 
the  mass  of  water  equal  in  volume  to  the  submerged  part  of  the  mass  of  ice 
is  -918b  x  g.  These  are  equal ;  bpg  =  -918ij#;  whence  p  =  -918,  the  specific 
density  of  ice  as  compared  with  water  =  1.  Thus  the  sp.  d.  of  a  body  float- 
ing in  water  =  (part  immersed  -4-  whole  volume).  This  method  might  be 
used  to  determine  the  sp.  d.  of  solids  were  it  not  for  practical  difficulties  of 
measurement,  which,  for  the  sake  of  explanation,  we  here  suppose  overcome. 
If  a  mass  of  ice  be  placed  in  chloroform,  it  will  float  with  -6125  of  its  mass 
submerged;  the  sp.  gr.  of  ice  in  reference  to  chloroform  is  -6125.  The 
problem  comes  to  be  this  :  Water  is  -VrV°<r  times  as  heavy  as  ice,  chloroform 
is  Vr°A°-  times  as  heavy  as  ice  —  What  are  the  relative  densities  of  water  and 
chloroform  ?  Chloroform  is  heavier  than  water  in  the  ratio  of  VA0/  :  VrVA 
or  9180  :  6125,  or  1497 : 1 ;  that  is,  its  sp.  d.  is  1497.  We  See,  then,  that 
the  comparative  sp.  d.  of  liquids  could  be  ascertained  by  finding  to  what 
depth  bodies  floating  in  them  will  sink,  if  we  could  perform  the  necessary 
measurements  with  the  requisite  accuracy.  But  instruments  which  have 
been  graduated  by  the  instrument-maker,  who  observes  and  marks  on  them 
with  more  or  less  care  the  depth  to  which  they  sink  in  various  liquids  of 
known  specific  gravity,  and  marks  the  corresponding  positions  by  proper 
figures,  are  in  common  use  for  promptly  ascertaining  the  density  of  various 
liquids.  They  are  usually  made  with  large  bulbs,  generally  loaded  with 
mercury  or  lead-shot,  in  order  that  they  may  float  vertically,  and  they  have 
a  narrow  stem  which  is  graduated.  They  are  caused  to  float  in  the  liquid 
whose  density  is  to  be  found  :  the  graduated  stem  stands  more  or  less  out  of 
the  liquid  ;  the  figure  upon  it  which  corresponds  most  nearly  with  the  general 
level  of  the  surface  of  the  liquid  is  read  off  and  recorded.  It  is  convenient 
in  effecting  such  readings  to  arrange  a  piece  of  black  paper  to  serve  as  a 
background,  and  to  place  the  liquid  to  be  tested  in  a  glass  vessel. 

Rousseau's  Densimeter  bears  at  its  summit  a  little  cavity.  In  this  a 
cubic  centimetre  of  the  fluid  is  placed:  according  to  the  depth  to  which 
this  makes  the  instrument  sink  in  water  is  the  density  of  the  fluid  deter- 
mined, according  to  the  graduation  performed  beforehand  by  the  instrument- 
maker. 

Fahrenheit's  Areometer  consists  of  a  similar  instrument  provided  at  its 
summit  with  a  little  platform  or  pan.  It  is  placed  in  water  at  3°-9  C.,  and 
loaded  by  small  additional  masses  placed  in  the  pan,  until  the  areometer 
sinks  in  the  water  just  so  far  that  the  level  of  the  water  coincides  with  a 
certain  mark  on  the  instrument.  Then  the  sum  of  the  known  weight  of  the 
areometer  -f-that  of  the  additional  masses  =  G=  the  weight  of  the  bulk  of 
water  displaced.  It  is  removed,  dried,  and  placed  in  the  liquid  to  be  tested, 
and  again  loaded  till  it  stands  at  the  same  level  as  before.  The  weight  of  the 
instrument  +  the  weights  now  added  =  G,  =  the  weight  of  an  equal  bulk  of 
the  liquid  to  be  tested.  Then  G7/G,  the  ratio  of  these  weights,  is  the  specific 
density  of  the  liquid  in  question.  If,  for  example,  the  instrument  weigh 
800  grs.  and  float  in  water  at  the  proper  level  when  loaded  with  200  grains ; 
loaded  with  80  grains  it  floats  at  the  same  level  in  an  aqueous  solution  of 
ammonia:  the  total  weights  are  1000  and  880;  the  density  of  ammonia 
solution  is  -880. 

Nicholson's  Areometer,  which  is  sometimes  used  for  determining   the 


224  MATTER.  [CHAP. 

density  of  solids,  is  a  modification  of  these  instruments.  It  bears  two 
platforms  —  one  at  the  summit,  out  of  the  water,  and  a  lower  one  in  the 
water.  The  body  whose  density  is  to  be  found  is  placed  on  the  upper  pan 
along  with  masses  just  sufficient  to  sink  the  areometer  to  a  certain  mark. 
The  body  is  transferred  to  the  lower  pan,  beneath  the  level  of  the  water. 
More  masses  must  be  placed  in  the  upper  pan  to  cause  the  areometer  to  sink 
to  the  same  level ;  the  weight  of  these  masses  is  equal  to  the  apparent  loss  of 
weight  suffered  by  the  body  when  placed  in  the  water  on  the  lower  pan. 
Then  (weight  of  body  —  the  additional  weight)  =  density  of  the  solid. 

4.  By  Specific-gravity  Bulbs.  Bulbs  are  sold  which  are  known  to  float 
without  rising  or  sinking  in  liquids  of  the  sp.  gr.  marked  in  numbers  upon 
them.  A  number  of  these  are  thrown  into  the  liquid ;  those  which  bear  too 
high  a  number  sink,  those  which  are  too  light  rise ;  the  one  exactly  corre- 
sponding, if  there  be  one,  is  at  rest  anywhere  in  the  liquid. 

(c)  Of  Gases.  —  The  density  of  a  gas  is  found  by  an  application  of  the 
same  principles  as  those  employed  in  determining  that  of  a  liquid.  A 
copper  or  glass  vessel,  as  light  as  is  consistent  writh  adequate  bulk  and 
strength,  has  the  contained  air  extracted  from  it  by  means  of  a  good  air- 
pump  ;  it  is  then  weighed  empty ;  it  is  very  slowly  filled  with  the  gas  and 
left  for  some  time  in  communication  with  a  reservoir  of  it ;  its  stopcock  is 
closed,  and  the  vessel  thus  filled  with  the  gas  is  again  weighed  :  the  weight 
of  the  volume  of  gas  which  fills  the  vessel  is  thus  ascertained.  It  is  again 
emptied  by  means  of  the  air-pump,  and  then  air  is  allowed  to  enter  it. 
After  standing  for  some  time  in  order  to  acquire  the  temperature  of  the 
room,  it  is  again  closed  and  weighed :  thus  the  weight  of  that  volume  of  air 
which  fills  the  vessel  is  found.  Then  the  weight  of  the  gas  -=-  wt.  of  equal 
vol.  of  air  =  density  of  the  gas  in  reference  to  air  as  a  standard. 

The  Dimensions  of  Density,  [m/ij]  =  [M/L8]  ;  those  of  sp.  gr.  and 
sp.  d.  =  0  =  [Numbers  merely]. 

It  is  convenient  for  chemical  purposes  to  take  the  rarest 
gas  —  that  is,  the  least  dense  gas,  Hydrogen  —  as  a  standard 
of  density,  and  then  we  say  that  the  specific  density  or  sp.  gr.  of 
Hydrogen  is  1,  that  of  air  14-439,  that  of  Oxygen  16,  and  so  on. 

A  cubic  centimetre  of  Hydrogen  weighs  -0000895682 
grammes  ;  a  cub.  cm.  of  air  weighs  (at  Paris)  -0012932  grammes 
at  the  freezing  point  of  water  and  at  the  barometric  pressure  of 
76  cm.  of  mercury.  A  cub.  cm.  of  liquefied  marsh-gas 
(which  liquefies  at  —  73°-5  C.,  if  the  pressure  be  raised  to  56-8 
atmospheres)  weighs,  according  to  Wroblewski,  0-37  gramme ; 
the  same  volume  of  water  weighs  1  gramme  at  3°-9  C.,  or  more 
accurately,  it  weighs  1-000013  standard  platinum  grammes ;  a 
cub.  cm.  of  liquefied  oxygen  weighs  1-124  grammes  (Olszewski) ; 
the  same  volume  of  lithium,  the  least  dense  solid  metal,  weighs 
•5936  grammes,  and  of  hammered  platinum  it  weighs  21-25 
grammes.  Thus  we  see  that,  bulk  for  bulk,  solid  platinum  is 
nearly  240,000  times  as  dense  as  gaseous  hydrogen. 

For  the  estimation  of  Vapour-density,  see  p.  391. 


ix.]  THE   STATES   OF   MATTER.  225 


THE  STATES  OF  MATTER. 

We  know  that  in  popular  language  there  are  said  to  be 
three  states  of  matter,  the  solid,  the  liquid,  and  the  gaseous. 
Closer  observation  shows  us  that  these  merge  into  one  another, 
and  that,  in  addition  to  these,  matter  exists  in  conditions  more 
recondite;  thus  we  shall  have  to  pass  in  review  the  follow- 
ing conditions :  —  (1)  Rigid  solid,  (2)  soft  solid,  (3)  viscous 
liquid,  (4)  mobile  liquid,  (5)  vapour,  (6)  critical  state,  (7)  gas, 
(8)  radiant  matter.  These  may  be  all  classified  under  the  two 
heads  of  Solid  and  Fluid,  which  are  not,  however,  separated 
from  each  other  by  any  distinct  line  of  demarcation. 

A  perfect  solid  is  an  ideal  body  which, -when  brought  into 
a  condition  of  stress,  if  it  do  become  deformed,  becomes 
deformed  to  a  definite  extent,  and  then  retains  its  newly- 
acquired  bulk  or  shape  for  an  indefinite  period  of  time,  so  long, 
that  is,  as  the  same  condition  of  stress  is  continuously  kept  up 

.     .  .         ,.        deformation          strain 

in  it ;  thus  the  ratio  j— -„ : ^ or  —    — •  =  const.     A  steel 

deforming  force       stress 

spring  becomes  stretched  by  the  weight  of  a  mass  hung  upon  it : 
it  is  stretched  to  a  definite  extent,  and  it  then  retains  the  same 
length  for  any  length  of  time,  or  rather  it  appears  to  do  so 
when  we  do  not  use  sufficiently  accurate  measurements ;  if  it 
did  so  perfectly,  steel  would  be  a  perfect  solid.  A  fluid,  on  the 
other  hand,  is  a  substance  which,  if  continuously  acted  upon  by 
deforming  force,  continuously  yields  with  continuously  increas- 

deformation 

ing  deformation  ;  and  thus  the  ratio  -5— ^ : ~ is  not  con- 
deforming  force 

stant,  but  increases  with  the  lapse  of  time. 

The  force  so  applied  to  a  fluid  must  not  be  one  which  is 
kept  up  equably  over  its  whole  surface  —  in  other  words,  there 
must  not  be  a  "Hydrostatic  Stress"  —  but  it  must  act  more 
effectively  on  one  part  of  the  mass  of  the  fluid  than  it  does  on 
another. 

A  rigid  solid  is  one  which,  when  a  stress  is  applied  to  it, 
experiences  no  deformation,  no  strain ;  and  therefore  in  a  per- 
fectly rigid  solid  the  fraction  (strain/stress)  =  0,  and  continues 
so  for  an  indefinite  period  of  time.  This  is  an  ideal;  no  sub- 
stance is  absolutely  rigid ;  but  we  may  form  a  sufficient  idea  of 
•a  rigid  solid  by  considering  an  anvil  on  which  a  nail  is  placed ; 
the  anvil  appears  undeformed  by  the  weight  of  the  nail  —  it 
appears  to  be  absolutely  rigid.  There  is,  then,  no  body  abso- 

Q 


226  MATTER.  [CHAP. 

lately  solid,  no  substance  absolutely  rigid ;  the  bodies  which 
we  call  Rigid  Solids  are  found  to  yield,  and  to  yield  continu- 
ously if  we  experiment  upon  them  in  sufficiently  small  masses 
(such  as  thin  wires),  and  act  upon  them  with  sufficient  forces 
for  adequate  periods  of  time.  In  practice  a  body  is  said  to  be 
Solid  if  its  deformation  remain  practically  or  sensibly  constant 
for  a  long  time ;  it  is  said  to  be  Rigid  if  the  deformation  asso- 
ciated with  a  given  stress  be  exceedingly  small,  or,  in  other 
words,  if  the  fraction  (stress/strain),  the  Coefficient  of  Rigidity, 
be  very  large.  A  soft  solid  is  one  in  which  the  coefficient  of 
rigidity  is  very  small  —  that  is,  it  is  a  solid  in  which  a  small 
stress  accompanies  great  deformation.  A  mass  of  jelly  is  very 
easily  deformed ;  but  if  a  moderate  force,  such  as  the  weight  of 
a  caraway  seed  or  currant,  be  applied  to  jelly  even  of  very  thin 
consistency,  the  deformation  produced  by  it  is  approximately 
constant;  the  jelly  does  not  flow  over  the  small  load,  arid  is 
therefore  a  true  solid,  though  it  is  soft. 

Very  considerable  rigidity  can  be  temporarily  conferred  on  a  body  by 
rotating  it;  e.g.,  a  sheet  of  tissue-paper  spun  on  a  wheel  becomes  quite 
stiff,  the  whole  paper  being  under  centrifugal  tension. 

It  would  seem  at  first  sight  rather  an  abuse  of  terms  to 
declare  that  thin  jelly  is  a  solid,  and  that  such  substances  as 
sealing-wax,  pitch,  and  cobbler's  wax  are  fluids ;  yet  these  latter 
are  fluids  because  they  flow,  because  they  suffer  continuous 
deformation  under  the  action  of  a  continuous  force.  A  stick  of 
sealing-wax  supported  at  its  ends  yields  continuously  to  its  own 
weight;  a  mass  of  sealing-wax  or  pitch  will  flow  down  hill;  a 
cake  of  cobbler's  wax  of  a  definite  form  will  soon  lose  its  sharp- 
ness of  outline  ;  a  cake  of  this  material  placed  in  water,  with 
bullets  on  it  and  corks  under  it,  will  be  traversed  by  the  bullets 
at  the  rate  of  about  a  quarter  of  an  inch  per  month,  and  by  the 
corks  at  a  somewhat  slower  rate;  the  wax  slowly  flows  round 
them  as  they  sink  or  rise  under  the  influence  of  their  relative 
weights,  just  as  water  would  much  more  rapidly  do.  These  sub- 
stances are,  then,  Fluids ;  but  their  flow  is  extremely  slow,  or,  in 
other  words,  their  viscosity  or  resistance  to  flow  is  extremely 
great.  Treacle  is  another  example  of  a  viscous  fluid ;  strong 
syrup,  weak  syrup,  very  weak  syrup,  cold  water,  alcohol,  hot 
water,  ether,  are  examples  of  liquids  whose  flow  is  successively 
more  rapid,  whose  viscosity  is  less ;  and  the  last  mentioned  are 
said  to  be  mobile  liquids,  though  perfect  mobility,  the  perfect 
absence  of  viscosity,  is  an  ideal  attribute  not  possessed  by  any 
actual  fluid. 


ix.]  VISCOSITY.    ?  227 

The  Coefficient  of  Viscosity.  —  In  Fig.  99  let  ABCD  represent  a 
volume  of  fluid  comprised  between  two  planes  passing  through  AB  and  CD, 
and  let  a  shearing  force  represented  by  arrows 
act  on  the  fluid  so  as  to  produce  continuous 
deformation,  as  there  shown.  The  deformation, 
since  the  substance  is  fluid,  goes  on  increas- 
ing ;  let  Acr/B  represent  a  form  assumed  through 
the  different  layers  (imagined  separate  as  in 
Fig.  25)  slipping  over  one  another,  the  lowest, 
AB,  being  relatively  at  rest.  If  the  velocity 
imparted  to  each  stratum  be  proportional  to  its  distance  from  AB,  the  ratios 
of  their  displacements  II',  mm',  nn',  oo',  to  their  respective  distances,  A/,  Am, 

IV       mm'      nn'       oo'         Cc 
etc.,  will  be  equal;  i.e.,  —  =  — — ==~r~  =  ~T~=~r~\  ~  tanCAc  ;  or, mother 

words,  this  is  a  proper  Shear.  If  the  displacement  of  the  stratum  CD  in 
time  t  be  Cc,  and  the  angle  CAc  =  0,  where  0  is  the  angle  through  which  a 
line  AC,  at  right  angles  to  AB  and  CD,  has  been  rotated,  then  the  total 
shear  is  tan  CAc  =  tan  0;  and  the  shear  per  unit  of  time  is  tan  O./t. 

This  slipping  of  one  imaginary  stratum  over  another  is  retarded  as  by 
friction,  for  each  stratum  rubs  against  or  is  delayed  by  the  one  next  to  it ; 
and  thus  the  retardation  acts  as  a  force  would  do,  opposed  to  that  which 
tends  to  produce  movement  —  i.e.,  to  the  shearing  force ;  and  it  has  to  be 
measured  over  every  unit  of  surface  over  which  the  retardation  is  effected. 
The  shearing  force  itself  is  measured  as  a  force  exerted  parallel  to  the  planes 
passing  through  AB  and  CD  (and  therefore,  in  the  general  case,  tangential 
to  the  surfaces  of  the  different  strata),  and  acting  on  each  unit  of  area  of 
these  surfaces  (or,  equally,  on  each  unit  of  cross-sectional  area  of  AC  or  BD). 

Shearing  Force  per  unit  of  Area      f  •  t 

The  ratio ^  ..     .  „. —      —  =- 2  =  -n,    the    Coeffi- 

Shear  per  unit  of  Time  tan  0 

cient  of  Viscosity.  When  0  =  0,  there  is  no  deformation  in  unit  of  time, 
and  the  viscosity  is  infinite ;  when  tan  O./t  is  very  small,  the  deformation  is 
very  small  in  unit  of  time  —  that  is,  it  is  very  slow,  and  the  viscosity  is 
very  great.  When  the  viscosity  is  very  small,  the  deformation  produced  in 
unit  of  time  is  very  great,  or  the  obstruction  offered  to  flow  is  very  small. 

If  the  distance  AC  =  1,  and  the  displacement  of  CD  in  unit  of  time 
(i.e.,  its  velocity)  also  —  1,  then  tan  O./t  =  1,  and  the  shearing  force  per 
unit  of  area  is  equal  to  the  coefficient  of  viscosity ;  whence  the  Coefficient  of 
Viscosity  may  be  measured  in  terms  of  a  force  —  that  is,  by  "  the  tangential 
force  on  the  unit  of  area  of  either  of  two  horizontal  planes  at  the  unit  of 
distance  apart,  one  of  which  is  (relatively)  fixed,  while  the  other  moves  with 
the  unit  of  velocity,  the  space  between  being  filled  with  the  viscous 
material"  (Maxwell).  The  "  kinematical  coefficient  of  viscosity"  is  rj/p, 
and  measures  the  corresponding  shearing  force  required  per  unit  of  mass. 

Relation  between  Rigidity  and  Viscosity.  —  The  measure  of  Rigid- 
ity in  the  ideal  perfect  solid,  the  ratio  of  the  deforming  Force  to  the 
total  Deformation  produced  by  it  (that  is,  using  Fig.  99,  the  quotient 
n  =  Deforming  Force  f  per  sq.  cm.  -f-  tan  0)  remains  constant  for  any  period 
of  time  during  which  the  deforming  force  applied  remains  constant.  The 
viscosity  rj  in  a  fluid,  on  the  other  hand,  is  the  ratio  of  the  deforming  force  per 
unit  of  area  to  the  deformation  produced  in  a  unit  of  time. ,  Whether  the 
time  of  experiment  be  long  or  short,  the  numerical  value  of  the  fraction 
Force  f  applied  x  Time  t  =  M  ^  ^  ^  same  Yet  -t  ig  f ound 

shear  produced  in  Time  t 


228  MATTER.  [CHAP. 

if  a  very  small  period  of  time  be  taken,  the  coefficient  of  viscosity  may 
have  some  experimental  value  different  from  that  found  when  the  period  of 
time  is  longer.  If,  for  instance,  Canada  balsam  be  stirred  with  a  spatula,  it 
will  be  found  (Maxwell)  to  be  doubly  refracting.  This  shows  that  it  is  for 
the  instant,  and  locally,  in  a  condition  like  that  of  a  stretched  or  a  com- 
pressed solid,  and  that,  though  it  is  liquid  if  sufficient  time  (time  of  relax- 
ation) be  allowed  it,  yet  under  the  action  of  impulsive  forces  its  coefficient 
of  viscosity  is  so  high  that  it  practically  behaves  after  the  fashion  of  a  solid. 

There  is  no  substance  perfectly  solid,  and  hence  when  an 
apparently  solid  substance  in  a  suitable  form  —  for  convenience, 
that  of  a  wire  —  is  exposed  to  a  constant  distorting  force,  though 
its  form  is  found  to  be  affected  to  a  certain  extent  (say  by 
lengthening  or  by  twisting),  and  it  appears  soon  to  reach  the 
limit  of  its  distortion,  which  may  then  be  measured,  the  appar- 
ent rigidity  of  the  substance  being  thus  ascertainable ;  yet  in 
general  it  will  be  found  that  the  prolonged  application  of  a  con- 
stant force  induces  a  slow  constantly-increasing  additional  dis- 
tortion; and  the  substance  then  acts  as  a  fluid  of  exceedingly 
great  viscosity.  Further,  we  now  know  by  the  experience  of 
manufacturing  industry  that  lead  and  even  iron,  when  exposed 
to  sufficient  pressure,  can  be  made  to  flow  slowly  ;  and  the 
operation  of  wire-drawing  involves  among  the  particles  of  the 
metal  drawn,  as  they  pass  through  the  aperture  in  the  plate,  a 
relative  movement  which  is  similar  to  that  of  the  particles  of  a 
flowing  fluid.  Amorphous  solids,  such  as  glass,  are  particu- 
larly apt  to  shade  off  into  liquids,  and  to  soften  gradually  by 
heat  instead  of  presenting  a  definite  melting  point. 

When  the  viscosity  of  a  fluid  is  infinite,  there  is  no  differ- 
ence between  that  fluid  and  a  rigid  solid.  It  is  supposed  by 
some  that  the  matter  at  the  centre  of  the  earth  approximates  to 
this  condition.  If  the  earth  have  a  crust  about  25  or  30  miles 
thick  floating  on  a  melted  magma,  at  increasing  depths  the  press- 
ure (which  near  the  centre  must  amount  to  about  45,000000 
pounds  per  square  inch)  would  compress  the  liquid  so  much 
that,  though  melted,  it  would  have  a  viscosity  so  extreme  that 
the  mass  would  have  the  same  relation  to  extraneous  forces  as 
a  very  rigid  solid  body  would  have.  The  earth  would,  in  conse- 
quence of  this,  comport  itself  as  if  it  had  a  solid  nucleus  float- 
ing in  and  merging  by  gradations  of  relative  softness  into  a  thin 
liquid  layer  on  which  the  crust  floats  (Osmond  Fisher).  The 
crust  of  the  earth  is,  however,  more  rigid  than  is  consistent 
with  this  theory. 

Substances  in  the  solid  and  liquid  form  are  broadly  dis- 


ix.]  STATES  OF   MATTER.  229 

tinguished  from  those  in  the  gaseous  condition  by  the  two 
following  characteristics.  In  the  first  place,  the  former  have  a 
free  surface,  while  the  latter  cannot  permanently  retain  a 
free  bounding  surface  independent  of  the  vessel  containing 
them.  Secondly,  solids  and  liquids  tend  (apart  from  volatilisa- 
tion) to  assume  a  definite  limited  bulk  and  density ;  while  gases 
always  tend  to  assume  an  infinite  volume  and  a  corre- 
spondingly small  density.  If  any  definite  quantity  of  a  gas  be 
confined  within  a  limited  space,  it  will  always  fill  that  space  and 
press  against  the  sides  of  the  containing  vessel ;  and  it  will  sub- 
ject that  vessel  to  stress  or  pressure,  which  the  vessel  must  be 
strong  enough  to  withstand.  Thus  a  gas,  with  its  tendency  to 
indefinite  expansion,  has  Elasticity  of  Volume:  work  has 
to  be  done  in  order  to  compress  it,  and  when  compressed  it  tends 
to  restore  the  work  done  upon  it.  Under  ordinary  circumstances 
gases  are  prevented  from  expanding  to  an  indefinite  extent  by 
the  pressure  of  the  mass  of  air  which  lies  upon  the  earth's  sur- 
face. This  air  is  drawn  down  towards  and  pressed  upon  that 
surface  by  the  downward  attraction  of  gravity,  and  consequently 
compresses  itself  and  all  objects  at  the  earth's  surface  with  an 
"  atmospheric  pressure  "  of  about  15  Ibs.  per  square  inch.  Hence 
a  closed  flask  containing  air  is  subject  to  two  equal  and  opposite 
pressures,  whose  resultant  is  nil ;  the  air  in  the  flask  tends  to 
burst  it  outwards ;  the  air  external  to  it  tends  to  make  it  col- 
lapse: between  the  two  pressures  the  flask  has  no  need  for 
strength.  If,  however,  the  air  be  in  any  way  extracted  from  the 
flask,  the  external  pressure  will  alone  act,  and  the  flask  may 
collapse.  In  a  steam  boiler  the  internal  pressure,  the  tendency 
of  the  steam  to  expand,  is  greater  than  the  external  pressure  ; 
and  the  boiler  must  be  of  sufficient  strength  to  provide  for  the 
difference. 

The  pressure  exerted  by  a  confined  gas  is  equal  over  the 
whole  internal  surface  of  the  vessel  containing  it ;  and  a  press- 
ure equal  to  the  weight  of  a  Ibs.  per  sq.  in.,  exerted  on  a  piston 
forced  into  a  cylinder  of  gas,  is  communicated  to  every  sq.  in. 
of  the  inner  surface  of  the  cylinder,  for  the  face  of  the  in-mov- 
ing piston  is,  at  any  instant,  a  part  of  the  inner  surface  of  the 
cavity  containing  the  gas ;  and  if  any  other  part  of  the  wall  of 
the  cavity  —  e.g.,  the  face  of  a  second  piston  —  be  at  the  same 
time  movable  outwards,  the  excess  of  internal  pressure  will 
cause  it  to  yield.  If  its  area  be  the  same  as  that  of  'the  in-mov- 
ing piston,  the  out-moving  piston  would  be  acted  on  by  an  equal 


230  MATTER.  [CHAP. 

total  pressure,  and  would  move  to  exactly  the  same  extent  as 
the  in-moving  one. 

There  is  an  apparent  paradox  in  the  statement  that  a  single- 
inch  piston,  pressed  in  with  a  pressure  equal  to  the  weight  of 
60  Ibs.  per  square  inch,  will  communicate  a  pressure  of  60  pounds 
to  every  square  inch  of  cylinder,  however  large ;  but  there  is 
no  law  of  Constancy  or  Conservation  of  Force :  there  is  one  of 
Conservation  of  Energy,  which  in  this  instance  is  rigorously 
respected.  The  work  that  can  be  done  by  a  second  piston 
allowed  to  move  outwards  is  (friction  not  being  considered) 
equal  to  that  done  by  the  compressing  piston,  whatever  be  their 
respective  areas :  the  range  of  movement  and  the  area  vary 
inversely :  a  smaller  second  piston  may  be  pushed  through  a 
longer  stroke  than  the  first,  a  larger  one  through  a  shorter. 

In  this  way  machinery  may  be  driven  at  the  distance  of  a  mile :  air  is 
driven  into  a  tube  ;  at  the  other  end  of  the  tube  there  is  a  cylinder  connected 
with  the  tube,  and  by  the  alternating  admission  of  the  pressure  to  one  and 
the  other  face  of  a  piston  movable  in  this  cylinder,  the  piston  is  caused  to 
move,  and  the  energy  is  thus  applied  at  a  distance. 

If  a  uniform  pressure  p  (i.e.,  p  dynes  per  sq.  cm.  of  the 
bounding  surface)  will  keep  a  certain  quantity  of  gas  within  a 
space  ft,  pressure  2p  is  found  to  be  required  to  confine  within  an 
equal  space  twice  as  much  gas :  pressure  xp  will  keep  x  times 
the  quantity  of  gas  within  the  same  space ;  that  is  to  say,  on  the 
assumption  that  the  temperature  does  not  vary.  The  densities 
in  these  examples  are  in  the  ratio  1 :  2 :  #,  and  the  Pressures  are 
proportional  to  the  Densities  (the  same  kind  of  gas  being  sup- 
posed to  be  used  throughout)  or  to  the  quantities  of  gas  forced 
into  a  given  space. 

The  pressures  being  proportional  to  the  densities  acquired  under  their 
action,  pec  p ;  but  m  =  top ;  whence  p  varies  as  ra/to;  or  />to  varies  as  m:  or  joij 
is  equal  to  a  constant  x  m ;  or,  for  a  given  mass,  p  varies  inversely  as  the  vol- 
ume, or  the  volume  varies  inversely  as  the  pressure  applied ;  and  these  last 
are  the  usual  forms  of  Boyle's  law. 

This  statement  may  be  otherwise  expressed  in  the  form  of 
Boyle's  law,  that  the  pressure  exercised  by  a  given  mass  of  gas 
varies  inversely  as  the  volume  of  the  space  within  which  it  is 
confined ;  or  that  the  volume  occupied  by  a  given  quantity  of  gas 
varies  inversely  as  the  pressure.  Thus,  if  a  'quantity  of  gas 
occupying  one  cubic  foot  at  a  pressure  of  15  pounds  per  sq.  inch 
were  compressed  by  a  piston  forced  down  with  an  additional 
pressure  of  15  pounds  per  sq.  inch,  the  total  pressure  being 
doubled  the  volume  would  be  halved;  if,  on  the  other  hand, 


ix.]  BOYLE'S  LAW.  231 

the  pressure  were  diminished  to  half  by  the  piston  being  pulled 
out  with  a  force  equivalent  to  7J  Ibs.  per  sq.  in.,  the  volume  of 
the  gas  contained  in  the  cylinder  would  be  doubled.  There  are 
no  perfect  gases  which  absolutely  obey  this  law  at  all  tempera- 
tures and  pressures  ;  but  a  gaseous  substance,  at  a  temperature 
and  pressure  far  removed  from  those  at  which  it  will  be  con- 
densed into  a  liquid,  approximates  to  this  condition. 

Now  suppose  that  a  quantity  m  of  a  liquid  is  placed  in  a 
vessel  which  it  does  not  completely  fill,  every  other  substance, 
such  as  air,  being  removed  from  the  vessel,  which  is  then  closed. 
The  liquid  does  not  fill  the  whole  vessel ;  it  has  a  free  surface. 
Above  this  free  surface  there  is  a  space,  which  becomes  filled  with 
part  of  the  liquid  substance,  volatilised  into  a  gaseous  form. 
Let  the  quantity  of  liquid  which  has  assumed  the  gaseous  form 
be  represented  by  mt :  the  remainder,  m  —  m(,  is  still  in  the  liquid 
form.  The  proportion  volatilised  (mjm  of  the  whole)  depends 
on  the  temperature  as  well  as  on  the  space  which  has  to  be 
filled.  At  another  temperature  some  different  proportion  will 
be  volatilised.  When  the  liquid  is  heated  this  proportion  rap- 
idly increases.  But  as  we  have  already  seen,  the  pressure 
exerted  by  a  confined  gas  on  the  vessel  containing  it  depends 
directly  on  the  amount  of  it.  Hence  in  this  case  the  pressure 
exerted  rises  rapidly  as  the  temperature  rises.  Gas  having  this 
relation  to  the  liquid  form  of  the  same  substance,  confined  with 
it  in  a  vessel  otherwise  empty,  and  in  contact  with  it,  is  called 
the  vapour  of  that  substance ;  if  it  be  compressed  or  cooled,  it 
partly  condenses  into  liquid.  Even  though  not  in  contact  with 
the  liquid,  if  the  gaseous  form  of  a  substance  be  compressed  or 
cooled  just  so  far  that  any  further  condensation  or  cooling  will 
cause  the  deposition  of  some  of  it  in  the  liquid  form,  it  is  said 
to  be  a  Vapour.  In  some  cases  a  vapour  condenses  directly  into 
a  solid ;  e.g.,  arsenious  acid. 

The  term  Vapour  is  often  applied  in  a  wider  sense  to  the 
gaseous  form  of  a  liquid  or  solid  substance  —  as,  for  instance, 
ether-vapour,  chloroform-vapour,  the  vapour  of  arsenious  acid ; 
and  then  those  vapours  which  are  on  the  point  of  condensation 
are  called  saturated  vapours,  while  those  which  can  suffer  a 
certain  amount  of  compression  or  cooling  without  condensation 
are  called  unsaturated  vapours.  In  this  sense  (with  the  possible 
exception,  as  yet,  of  hydrogen)  all  gases  are  unsaturated  vapours; 
for  they  can  all  be  condensed  by  the  simultaneous  application  of 
sufficient  cold  and  sufficient  pressure.  Oxygen  has  been  con- 


232  MATTER.  [CHAP. 

densed ;  at  a  pressure  of  300  atmospheres,  and  at  the  tempera- 
ture of  —  39°  C.  (Cailletet),  there  is  no  condensation,  but  when 
the  gas  is  liberated  it  becomes  foggy ;  according  to  Pictet  it  is 
liquefied  at  320  atmospheres  and  —  140°  C.;  and  then,  upon  allow- 
ing a  jet  of  this  liquid  to  escape  into  the  air,  the  escaping  jet  of 
liquid  oxygen  becomes  extremely  cold  and  is  partly  solidified, 
while  the  remaining  oxygen  in  the  vessel  becomes  cloudy.  When 
oxygen  has  been  compressed  into  a  liquid,  this  liquid  must  be  in 
some  state  of  molecular  aggregation  distinct  from  that  of  the 
gas,  for  it  has  no  effect  on  alkali-metals  or  phosphorus,  while  it  is 
powerfully  magnetisable.  Ozone  is  with  comparative  readiness 
condensed  by  Cailletet's  method  to  a  bright-blue  liquid.  The 
term  Vapour  is  used  in  still  another  sense  —  that  is,  a  gas  at 
such  a  temperature  that  by  the  application  of  pressure  alone 
it  can  be  condensed  into  a  liquid.  In  this  sense  carbonic  acid 
below  300>92  C.  is  a  vapour;  above  that  temperature  it  is 
properly  a  gas,  for  no  amount  of  pressure  alone  will  liquefy  it. 

The  Critical  State.  —  When  a  liquid  and  its  vapour  are 
together  in  a  tube,  otherwise  unoccupied,  and  are  exposed  to  heat, 
there  arrives  a  temperature  at  which  the  singular  phenomenon 
of  a  blending  or  continuity  of  the  liquid  and  gaseous  (or  vapor- 
ous) states  is  observed.  If  a  capillary  tube,  for  instance,  be 
filled  with  liquid  CO2  and  slightly  heated,  some  of  the  carbonic 
anhydride  will  escape  :  the  tube  may  then  be  sealed  up,  and  will 
now  contain  nothing  but  liquid  CO2  and  saturated  vapour  of 
CO2.  If  these  be  heated  to  30°-92  C.,  and  if  there  be  sufficient 
CO2  present  to  produce  a  pressure  above  73  atmospheres,  the 
free  surface  of  the  liquid  becomes  blurred  and  merges  into  the 
superjacent  gas :  above  this  temperature  the  tube  is  full  of  what 
appears  to  be  nothing  but  gas :  if  cooling  be  permitted  there  is 
a  flickering  seen  in  the  tube,  and  the  liquid  and  the  gas  again 
separate. 

Some  say  that  the  liquid  and  the  gas  mutually  dissolve  each  other; 
others  (Ramsay)  point  out  that  the  liquid  CO2  rapidly  becomes  lighter, 
while  the  confined  vapour  of  CO2  becomes  denser,  at  higher  temperatures, 
and  that  at  the  critical  temperature  and  under  sufficient  pressure  these  two 
states  meet  and  become  undistinguishable.  Both  the  liquid  and  the  gas 
would,  under  these  conditions,  have  the  same  volume. 

If  CO2  gas  be  exposed  to  any  temperature  above  30°-92  C. 
and  be  subjected  to  any  pressure  above  73  atmospheres,  it  will 
still  be  a  gas:  allow  it  to  cool,  the  pressure  being  kept  up, 
arid  it  will  be  a  liquid  after  it  passes  30°-92  C. ;  and  yet  the 


ix.]  CRITICAL  STATE.  233 

transition  is  unobservable.  If  pressure  and  temperature  be 
allowed  to  fall  together,  the  flickering  already  mentioned  is 
produced. 

If  the  liquid  originally  fill  the  tube  and  then  be  heated  to 
the  critical  temperature,  the  tube  becomes  filled  with  gas,  but 
the  precise  mode  of  transition  from  the  one  state  to  the  other 
cannot  be  observed.  If  the  tube  thus  containing  gaseous  CO2 
at  a  high  pressure  be  locally  cooled,  there  is  local  condensation 
and  flickering. 

This  temperature,  30°-92  C.  for  carbonic  acid,  is  called  the 
Critical  Temperature.  If  the  temperature  of  the  gas  be  above 
30°-92  C.,  no  pressure  can  condense  it  into  a  liquid;  if  it  be  just 
below  that  point,  a  pressure  of  73  to  75  atmospheres  is  competent 
to  effect  its  liquefaction :  and  this  pressure  is  called  the  Critical 
Pressure  of  carbonic  acid.  (See  Van  der  Waals'  Law,  p.  375.) 

The  volume  occupied  by  unit  mass  of  the  liquefied  gas  at 
the  crit.  temp,  and  crit.  pr.  is  called  the  Critical  Volume. 

The  critical  temperature  of  oxygen,  nitrogen,  and  other 
gases  formerly  known  as  permanent  gases,  is  very  low  in  the 
thermometric  scale  (oxygen,— 113°  C.,  Wroblewski),  and  ex- 
ceeding cold  is  a  necessary  condition  of  their  condensation 
under  pressure. 

Water  filling  a  sufficiently  strong  boiler  might  be  exposed 
to  a  low  red  heat,  720°-6  C.,  and  would  then  be  transformed 
into  a  gas  exercising  such  enormous  pressure  as  to  make  any  ex- 
periments upon  it  excessively  difficult ;  yet  it  is  believed  (see 
p.  390)  to  present  this  phenomenon  at  that  temperature. 

At  the  critical  temperature,  matter  under  sufficient  pressure 
is  in  the  Critical  State ;  if  heated  a  little  more  it  is  undoubtedly 
gaseous ;  if  allowed  to  cool  a  little  it  is  undoubtedly  liquid,  and 
is  far  less  compressible  ;  and  if  the  pressure  be  kept  up,  the  tran- 
sition is  unrecognisable.  By  no  optical  test  can  the  liquid  just 
below  the  critical  temperature  and  the  gas  just  above  that  tem- 
perature be  distinguished. 

If  a  solid  be  dissolved  in  a  liquid,  and  if  the  whole  be  heated  under 
sufficient  pressure  to  a  temperature  above  the  critical  point,  the  liquid  is  now 
gas,  and  yet  the  solid  remains  dissolved  in  it  (Hannay).  Iodide  of  potassium, 
for  instance,  or  chlorophyll,  if  dissolved  in  alcohol  and  treated  in  this  way, 
will,  it  is  said,  remain  in  solution  in  gaseous  alcohol  at  350°  C. 

When  gaseous  matter  has  been  rarefied  to  a  very  great 
degree  it  assumes  remarkable  properties,  of  which  the  most 
striking  is  that  such  exceedingly  rarefied  gas  or  ultragaseous 


234  MATTER.  [CHAP. 

matter  can  be  induced  —  as  we  shall  see  under  Electricity, 
p.  656  —  to  exercise  pressure  specially  on  localised  areas  of  the 
walls  of  the  containing  vessel,  and  by  this  concentrated  pressure 
to  produce  mechanical  and  luminous  effects  characteristic  of  the 
so-called  Ultragaseous  or  Radiant  Matter  (p.  252). 

The  Ether.  —  We  have  already  said  that  we  can  know 
matter  only  by  those  of  its  properties  which  we  perceive  by 
means  of  our  senses.  The  existence  of  any  form  of  matter  is  to 
us  only  an  inference  from  the  phenomena  to  which  it  gives  rise ; 
and  if  a  large  group  of  phenomena  find  their  best  or  their  only 
explanation  in  the  assumed  existence  of  a  form  of  matter  of  an 
unfamiliar  kind,  the  evidence  for  its  existence  is  of  exactly  the 
same  character  as  that  on  the  ground  of  which  we  believe  our- 
selves entitled  to  assert  the  existence  of  any  kind  or  form  of 
matter  whatsoever.  The  phenomena  of  Light  are  best  explained 
as  those  of  undulations ;  but  undulations  —  even  in  the  most 
extensive  use  of  the  term,  as  signifying  any  periodic  motion  or 
condition  whose  periodicity  obeys  the  laws  of  wave-motion  — 
must  be  propagated  through  some  medium.  Heat  while  passing 
through  space  presents  exactly  the  same  undulatory  character, 
and  requires  a  medium  for  its  propagation.  Electrical  attraction 
and  repulsion  are  explained  in  far  the  most  satisfactory  way  by 
considering  them  as  due  to  local  stresses  in  such  a  medium. 
Current  electricity  seems  due  to  a  throb  or  series  of  throbs  in 
such  a  medium  when  released  from  stress.  Magnetic  phenom- 
ena seem  due  to  local  whirlpools  set  up  in  such  a  medium.  And 
the  assumption  of  the  existence  of  a  single  medium,  with  proper- 
ties of  great  simplicity,  will  explain  these  varied  phenomena  and 
even  co-ordinate  them ;  thus  the  crest  or  the  trough  of  a  light- 
wave or  a  heat-wave  is  a  point  of  maximum  displacement  due 
to  transverse  tension  —  exactly  the  condition  of  the  medium 
during  the  persistence  of  electric  attraction  or  repulsion,  that 
is,  the  Electrostatic  condition ;  the  middle  point  of  the  wave 
is  changing  its  position  with  rapidity  —  exactly  the  condition 
of  the  medium  during  the  passage  of  a  current,  the  Electro- 
kinetic  condition  ;  thus  Light  and  Radiant  Heat  are  explicable  as 
electromagnetic  disturbances  of  rapidly-alternating  character; 
and  this  leads  to  the  conclusion,  sustained  by  experiment,  that 
the  velocity  of  light  should  be  equal  to  the  rate  of  propagation 
of  an  electric  disturbance  through  a  medium  of  this  kind.  We 
are  led  to  infer,  therefore,  that  there  is  such  a  medium,  which 
we  call  the  Luminif erous  Ether,  or  simply  the  Ether ;  that  it  can 


ix.]  THE  ETHER.  235 

convey  energy ;  that  it  can  present  it  at  any  instant  partly  in 
the  form  of  kinetic,  partly  in  that  of  potential  energy ;  that  it  is 
therefore  capable  of  displacement  and  of  exercising  pressure  or 
tension  ;  and  that  it  must  have  rigidity  and  elasticity.  Calcula- 
tion leads  us  to  infer  that  its  density  is  936/1000,000000,000000,- 
000000  that  of  water  (Clerk  Maxwell),  or  equal  to  that  of  our 
atmosphere  at  a  height  of  about  210  miles,  a  density  vastly 
greater  than  that  of  the  same  atmosphere  in  the  interstellar 
spaces;  that  its  rigidity  is  about  1/1000,000000  that  of  steel  — 
hence  that  it  is  easily  displaceable  by  a  moving  mass  ;  that  it  is 
not  discontinuous  or  granular ;  and  hence  that  as  a  whole  it  may 
be  compared  to  an  impalpable  and  all-pervading  jelly,  through 
which  Light  and  Heat  waves  are  constantly  throbbing,  which  is 
constantly  being  set  in  local  strains  and  released  from  them,  and 
being  whirled  in  local  vortices,  thus  producing  the  various  phe- 
nomena of  Electricity  and  Magnetism ;  and  through  which  the 
particles  of  ordinary  matter,  with  their  relatively  small  transla- 
tory  velocities,  move  freely,  like  bullets  through  cobbler's  wax, 
encountering  but  little  retardation  if  any,  for  the  elasticity 
of  the  Ether,  as  it  closes  up  behind  each  moving  particle,  is 
approximately  perfect. 

Nothing  of  the  nature  of  an  air-pump  can  remove  it  from 
any  given  space  ;  the  most  perfect  vacuum  conceivable  must  be 
denned  as  a  plenum,  a  space  fully  occupied,  but  occupied  by 
Ether  alone. 

Change  of  State.  —  Work  must  be  done  upon  a  solid  in 
order  to  convert  it  into  a  liquid :  energy  must  in  some  form  be 
imparted  to  it.  This  form  may  be  that  of  Heat,  directly  applied 
so  as  to  fuse  the  solid.  In  such  a  case  a  definite  amount  of  the 
energy  imparted  in  the  form  of  Heat  apparently  disappears  (see 
Latent  Heat,  p.  361),  for  it  does  the  mechanical  work  of  liquefy- 
ing the  solid.  If  the  liquid  again  assume  the  solid  form,  as  in 
freezing,  the  process  is  reversed:  the  energy  absorbed  during 
liquefaction  gradually  reappears  in  the  form  of  heat,  which  must 
be  dissipated  before  the  freezing  can  become  complete. 

If  a  solid  body  simply  assume  the  liquid  form  without  hav- 
ing external  heat  or  other  energy  applied  to  it,  the  absorption 
of  some  of  the  heat  of  the  body  itself  results  in  a  cooling  of  its 
substance,  as  in  the  case  of  a  freezing  mixture,  where,  on  certain 
chemical  salts  being  dissolved  in  cold  water,  the  resultant  solu- 
tion is  found  to  be  extremely  cold.  Solid  CO2  and  ether  sink 
to  -100°C. 


236  MATTER.  [CHAP. 

Again,  where  two  chemical  elements  combine,  their  combina- 
tion is  generally  attended  with  heat,  the  elements  losing  their 
potential  energy  of  separation.  The  supply  of  an  equivalent 
amount  of  energy  from  without  is  necessary  in  order  to  reverse 
the  process  of  combination  —  that  is,  to  effect  chemical  separa- 
tion or  decomposition.  When  the  processes  of  chemical  combi- 
nation and  liquefaction  go  on  together  —  the  product  of  the 
combination  of  elements  of  which  one  or  both  are  solid  being 
itself  liquid — the  result  may  be  that  the  cooling  effect  of  the 
latter  action  exceeds  the  heating  effect  of  the  former ;  thus,  in 
the  union  of  carbon  and  sulphur  to  form  carbon-disulphide, 
which  is  a  liquid,  the  absorption  of  heat  due  to  liquefaction  is 
greater  than  the  evolution  of  heat  due  to  combination,  and  the 
action  stops  unless  heat  be  supplied  from  without.  On  the  other 
hand,  in  the  combination  of  quicklime  with  water  to  form  slaked 
lime,  we  find  much  heat  evolved  —  partly  due  to  the  chemical 
combination,  partly  to  the  liquid  water  assuming  a  solid  form. 

The  transformation  of  a  solid  into  a  gas,  in  a  like  manner, 
involves  the  expenditure  of  heat  or  some  other  form  of  energy 
in  performing  the  mechanical  work  of  volatilisation.  Snow 
evaporates  in  a  cold  high  wind ;  arsenic  trioxide  under  ordinary 
atmospheric  pressures  is,  without  melting,  volatilised  by  heat, 
while,  if  a  sufficient  pressure  be  applied,  it  melts  before  volatilis- 
ing. Dr.  Carnelley  found  that  in  a  similar  way  ice,  if  heated 
under  an  exceedingly  small  pressure,  may  be  rendered  very  hot 
(180°  C.),  and  will  volatilise  freely,  yet  without  melting,  unless 
the  pressure  be  allowed  to  exceed  a  certain  low  maximum,  which 
he  called  the  Critical  Pressure,  this  being  very  low  for  water, 
very  high  for  arsenic  trioxide.  A  sheet  of  metal  may  be  dissipated 
in  vapour  by  an  electric  discharge,  part  of  the  energy  of  which 
becomes  spent  in  producing  this  mechanical  effect.  Again,  in 
chemical  combination  we  often  see  the  conversion  of  solids  into 
gases.  Carbon  and  oxygen  combine  to  form  carbon-monoxide ; 
of  the  heat  which  is  evolved  by  the  union  of  the  elements,  a 
large  part  is  absorbed  in  rendering  the  solid  carbon  gaseous.  If 
the  CO  produced  be  in  its  turn  burned  so  as  to  form  CO2,  none 
of  the  heat  of  combination  of  oxygen  and  carbon-monoxide  is 
absorbed  in  doing  mechanical  work  of  this  kind,  'and  the  amount 
of  heat  evolved  in  the  second  stage  of  the  production  of  CO2  is 
greater  than  that  evolved  in  the  first.  Conversely,  where  two 
gases  produce  a  solid,  as  chlorine  and  sulphuretted  hydrogen  do, 
the  amount  of  heat  liberated  is  determined  not  only  by  the 


ix.]  CHANGE  OF  STATE.  237 

amount  of  energy  absorbed  in  decomposing  H2S,  and  by  the 
amount  liberated  by  the  union  of  H2  and  C12,  but  also  by 
the  fact  that  the  sulphur  is  "deposited  in  the  solid  form. 

If  a  liquid  were  exposed  to  an  indefinite  and  perfect  vacuum, 
it  would  evaporate  at  once  at  any  temperature  above  the  absolute 
zero.  If  it  be  exposed  to  an  imperfect  vacuum,  it  will  still 
evaporate  readily,  but  not  so  readily  as  before,  for  its  vapour 
must  be  able  to  force  its  way  from  the  liquid  and  against  the 
superincumbent  pressure.  If  the  pressure  be  great,  the  amount 
of  heat  which  must  be  supplied  to  the  liquid  in  order  to  enable 
it  to  overcome  this  resistance  and  to  enter  into  ebullition  is  also 
greater,  and  the  Boiling-point  of  a  liquid  increases  with  the 
pressure. 

Let  a  liquid  be  supposed  heated  in  a  vessel  provided  with  a  piston,  by 
means  of  which  pressure  can  be  exercised  on  the  contents  of  the  vessel ;  the 
vessel  being  supposed  of  any  sufficient  length.  The  liquid  is  heated  and 
converted  into  vapour ;  the  vapour  forces  out  the  piston,  and  the  external 
air  pushes  it  in ;  the  piston  rests  when  the  external  and  internal  pressures 
are  equal.  If  we  press  home  the  piston,  the  vapour  is  partly  condensed  :  to 
retain  it  in  the  gaseous  form  we  must  simultaneously  apply  a  stronger  heat. 
This  process  may  be  supposed  continued  until,  at  a  certain  high  temperature 
(the  "  critical  temperature  ")  and  great  pressure,  we  have  the  whole  of  the 
liquid  evaporated,  and  its  vapour  compressed  into  the  same  space  as  the 
original  liquid.  If  expansion  be  altogether  prevented,  this  process  is  con- 
tinuous, and  the  temperature  at  which  water  can  be  wholly  converted  into 
vapour  under  such  circumstances  is  720  5>6  C. 

Liquids  are,  as  a  rule,  more  bulky  than  the  corresponding  solids ;  hence 
fusion,  which  involves  expansion,  obeys  the  same  law  as  evaporation,  which 
also  involves  expansion ;  it  is  hindered  by  pressure,  and  the  fusing-point, 
like  the  boiling-point,  is  raised  by  pressure.  A  few  liquids  —  water,  cast-iron 
—  are  denser  than  their  solids.  In  such  a  case,  an  increase  of  pressure  may 
be  said  to  predispose  the  particles  to  set  into  the  more  compact  and  denser 
form,  the  liquid  form,  and  fusion  is  facilitated  by  pressure.  Thus  the  melt- 
ing-point of  ice  is  lowered,  that  of  most  other  solids  raised,  by  pressure. 

Change  of  state  involves,  then,  either  an  absorption  or  a 
liberation  of  energy ;  and  the  amount  of  energy  which  must  be 
supplied  to  a  body  in  order  to  enable  it  to  undergo  a  change  of 
state  depends  on  the-  pressure  which  tends  to  resist  or  to  favour 
such  a  change,  as  well  as  on  the  intrinsic  energy  which  it  already 
possesses. 

There  is  no  known  means  of  effecting  any  transformation 
of  matter  in  any  of  its  ordinary  forms  into  the  Ether,  or  vice 
versd. 


238  MATTER.  [CHAP. 


THE  CONSTITUTION  OF  MATTER. 

The  question  as  to  whether  Matter  is  or  is  not  infinitely 
divisible  has  been  made  the  basis  of  much  acute  speculation  ;  but 
it  is  only  within  this  century  that  any  serious  proof  has  been 
adduced  in  favour  of  an  Atomic  Theory,  or  theory  according  to 
which  matter  is  considered  as  made  up  of  indivisible  particles. 
According  to  this  view,  matter  consists  of  particles  or  atoms, 
each  of  which  it  is  impossible  with  our  present  appliances  to 
divide,  and  the  division  of  which,  if  it  were  possible,  would 
probably  result  in  the  subversion  of  our  ideas  as  to  the  appar- 
ently fundamental  nature  of  some  of  the  properties  of  matter. 

Chemical  Views.  —  The  probability  of  this  atomistic 
view  was  raised  almost  to  the  rank  of  certainty  by  the  researches 
of  successive  chemical  investigators.  It  was  first  found  that 
every  definite  chemical  substance  in  a  state  of  purity  has  always 
the  same  constitution;  that  an  analysis  effected  for  one  pure 
sample  of,  say,  oxide  of  lead,  is  applicable  to  every  other 
pure  sample  of  the  same  substance.  Hence  the  law  of  Fixity 
of  Proportions  in  chemical  compounds. 

But  it  was  remarked  that  the  same  elements  often  unite  in 
different  proportions  to  form  compounds  possessed  of  essentially 
different  properties.  Carbon  and  oxygen  thus  unite  to  form  two 
well-known  compounds,  of  which  the  percentage  compositions 
by  weight  are  respectively  :  — 

Carbon    .     .     .     42-85  Carbon    .     .     .     27-27 

Oxygen  .     .     .     57-14  Oxygen  .     .     .     72-72  * 

Analytical  results  tabulated  in  this  way  are  not  very  instructive; 
but  if  the  second  example  be  multiplied  by  42-85/27'27  we  find 
the  respective  ratios  to  become 


Carbon    .     .     .     42-85 
Oxygen  .     .     .     57-14 


and 


Carbon   .     .     .     42-85 
Oxygen  .     .     .  114-28 


or  in  round  numbers, 

Carbon    ....       3  Carbon   ....       3 

Oxygen  ....       4  Oxygen  ....       8 ' 

Here  we  find  that  the  same  quantity  of  carbon,  united  in  the  one 
compound  (carbon-monoxide)  with  a  certain  quantity  of  oxygen, 
is  in  the  other  (carbonic  anhydride)  united  with  twice  the  quan- 
tity of  that  element.  From  a  large  number  of  instances  of  this 


ix.]  CONSTITUTION  OF  MATTER.  239 

kind  was  evolved  the  Law  of  Multiple  Proportions  —  that  the 
same  elements  may  form  a  series  of  different  compounds  by 
uniting  in  several  fixed  proportions  which  bear  a  whole-number 
ratio  to  each  other.  Nitrogen  and  oxygen  thus  form  five  com- 
pounds, in  which  the  nitrogen  and  oxygen  are  present  in  the 
respective  ratios  of  14  :  8,  14  : 16,  14  :  24,  14 :  32,  14  :  40  ;  and 
in  this  case  the  quantities  of  oxygen,  united  with  a  fixed  quan- 
tity (14)  of  nitrogen,  bear  to  one  another  the  relative  ratios  of 
1:2:3:4:5.  Iron  has  two  oxides,  in  which  the  iron  and  the 
oxygen  bear  to  one  another  the  respective  ratios  of  28  : 16  and 
28 :  24 ;  here  the  quantities  of  oxygen,  united  with  the  same 
quantity  of  iron,  bear  to  one  another  the  ratio  2  :  3. 

Then,  further,  the  law  of  Chemical  Equivalence  was  formu- 
lated: chemical  quantities,  which  are  equivalent  to  the  same  thing 
as  regards  power  of  doing  chemical  work  or  forming  chemical 
compounds,  are  equivalent  to  one  another.  One  part  by  weight  of 
hydrogen  will  combine  with  eight  of  oxygen  (7-98165  ±-00175); 
so  will  108  parts  of  silver  (107-896).  108  pts.  of  silver  and  1  of 
hydrogen  are  mutually  equivalent,  for  they  can  both  do  the  same 
chemical  work  —  they  can  enter  into  combination  with  8  pts.  of 
oxygen ;  and  they  are  both  equivalent  to  the  8  pts.  of  oxygen 
with  which  they  can  combine.  If  now  it  be  found  that  1  pt.  by 
wt.  of  hydrogen  can  combine  with  35-5  (35*478)  pts.  by  wt.  of 
chlorine,  then-  the  law  asserts  that  the  equivalent  quantity,  108 
pts.,  of  silver  should  also,  in  its  turn,  be  able  to  combine  with 
an  equal  quantity,  35-5  pts.,  of  chlorine.  This  law  is  a  general- 
isation, based  upon  facts  determined  by  the  aid  of  the  balance 
and  independent  of  theory ;  and  this  law  of  equivalence,  so 
based,  though  it  be  too  sweeping  a  generalisation  to  be  now 
accepted  in  its  full  sense,  yet  did  useful  service  in  its  day  in 
enabling  tables  of  Equivalent  Numbers  or  of  Combining  Propor- 
tions to  be  drawn  up,  and  a  system  of  Chemical  Formulae  to  be 
devised,  based  upon  these  equivalents.  According  to  this  sys- 
tem, the  composition  of  water  was  symbolised  as  HO ;  this  sym- 
bol might  be  read  in  words  as  —  one  equivalent  of  hydrogen  and 
one  of  oxygen,  united  to  form  one  equivalent  of  water.  The 
symbol  of  hydrogen  peroxide  was  HO2;  one  equivalent  (1  pt. 
by  wt.)  of  hydrogen  combined  with  two  equivalents  (2  x  8  =  16 
pts.  by  wt.)  of  oxygen. 

When  such  facts  as  these  were  known,  a  reasoned  explana- 
tion of  them  was  sought.  None  that  offered  was 'so  plausible  as 
Dalton's  atomic  theory,  a  revival  of  the  old  hypothesis  of  Leu- 


240  MATTER.  [CHAP. 

cippus,  Democritus,  and  Lucretius,  that  matter  consists  of  atoms, 
coupled  with  the  proposition  that  the  atoms  of  the  different 
elements  have  different  relative  weights.  According  to  this  view 
the  smallest  mass  of  water  must  consist  of  an  atom  of  hydrogen 
and  another  of  oxygen,  their  relative  atomic  weights  being  1  and 
8 ;  and  these  were  connected  as  one  might  couple  together  a  ball 
of  wood  and  one  of  lead.  More  complex  substances  were  pro- 
duced by  the  union  of  a  greater  number  of  such  atoms  —  as,  for 
instance,  HO2,  NO5,  (KO,  HO),  etc. ;  and  the  symbolic  formulae 
were  then  used  to  denote  the  relative  number  of  such  atoms 
entering  into  the  formation  of  compound  substances. 

But  it  was  found  that  the  system  of  formulas  based  upon  the 
facts  of  equivalence  did  not  work  well  when  made  to  signify 
the  relative  numbers  of  atoms  united  to  form  a  compound.  The 
equivalent  number  for  carbon  was  6,  because  that  quantity  of 
carbon  was  equivalent  (in  carbonic  oxide)  to  8  of  oxygen,  which 
quantity  was  in  its  turn  equivalent  to  the  standard  unity  of 
hydrogen.  In  marsh  gas  6  pts.  by  wt.  of  carbon  are  found  to  be 
combined  with  2  of  hydrogen  —  i.e.,  with  two  equivalents ;  hence 
the  formula,  according  to  this  system,  must  be  CH2.  It  is  known, 
however,  that  one-fourth  of  the  hydrogen  can  be  replaced  by  half 
an  equivalent  (17|-  pts.  by  wt.)  of  chlorine,  forming  CH^Cl^ : 
an  expression  intelligible  though  cumbrous  when  read  in  the 
language  of  equivalents,  but  absurd  when  read  in  terms  of  the 
atomic  theory.  This  last  formula  had  accordingly  to  be  modified 
to  C2H3C1 ;  and  then  the  original  marsh  gas  had  to  be  supposed 
to  enter  invariably  into  reactions  as  2CH2,  or  else  its  formula 
must  be  modified  to  C2H4.  The  latter  is  the  more  natural  sup- 
position. It  was  pointed  out  (Ge*rhardt)  that  throughout  the 
whole  of  the  chemistry  of  the  carbon  compounds,  similar  reason- 
ing shows  that  if  the  atomic  weight  of  carbon  be  6,  the  atoms 
always  appear  in  reactions  in  even  numbers ;  whence  the  infer- 
ence is  obvious  that  the  atomic  weight  of  carbon  must  be  12,  and 
the  proper  formula  of  marsh  gas  is  CH4.  In  a  similar  way  it  was 
shown  that  the  assumption  that  the  atomic  weight  of  oxygen,  as 
well  as  its  equivalent  number,  is  8,  leads  to  the  invariable  appear- 
ance of  O2,  or  of  an  even  number  of  oxygen  atoms,  in  every  equa- 
tion ;  whence  the  atomic  weight  of  oxygen  must  be  16 ;  and  the 
Atomistic  formula  for  water,  as  distinguished  from  the  Equiva- 
lence-formula, must  be  H2O.  The  Atomistic  Formulae  now  in  use 
do  not  directly  make  use  of  the  idea  of  equivalence  :  they  denote 
the  number  of  atoms  of  which  the  Molecule  —  a  fruitful  idea, 


ix.}  CONSTITUTION  OF  MATTER.  241 

due  to  Avvogadro  —  is  made  up.  The  symbol  H2O,  for  instance, 
signifies  a  molecule  of  water,  made  up  of  two  atoms  of  hydrogen 
(at.  wt.  =  1)  and  one  of  oxygen  (at.  wt.  =16).  When  attention 
was  first  directed  to  this  mode  of  representation,  it  was  found  to 
be  entirely  in  accord  with  the  half-forgotten  researches  of  Gay 
Lussac  on  the  relative  volumes  of  gases  which  enter  into  com- 
bination. He  had  found  that  one  volume  (say  1  cub.  cm.)  of 
oxygen  and  two  (2  cub.  cm.)  of  hydrogen  unite  to  form  two  vol- 
umes (2  cub.  cm.)  of  water-vapour.  The  atomistic  equation,  on 
the  other  hand,  is  O  +  2  H  =  H2O ;  that  is,  one  atom  of  oxygen 
unites  with  two  atoms  of  hydrogen  to  form  a  molecule  of  water. 
These  two  statements  are  closely  parallel ;  and  the  molecule 
H2O  formed  occupies,  in  water-vapour,  the  same  space  as  the 
original  two  atoms  of  hydrogen. 

Similarly,  it  had  been  found  that  the  electric  spark  decom- 
posed 2  cub.  cm.  NH3  into  1  cub.  cm.  N  and  3  cub.  cm.  H.  The 
equation  was 

NH3  N          +  H3 

One  molec.       One  atom.        Three  atoms. 

2  cub.  cm.        1  cub.  cm.          3  cub.  cm. 

Here  again  the  molecule  of  the  compound,  NH3,  occupies,  in 
gaseous  ammonia,  the  same  space  as  two  atoms  of  hydrogen. 

So  forth ;  the  general  rule  is  that  the  molecule  of  any  com- 
pound in  the  gaseous  state  occupies  the  same  space  as  two  atoms 
of  free  hydrogen. 

Hence  we  may  provisionally  establish  a  general  rule,  subject  to  excep- 
tions farther  to  appear:  —  If  in  a  chemical  equation  relating  to  gases  we 
write  "2  vols."  under  every  complete  molecule,  and  "1  vol."  under  every 
atom  of  any  element  entering  into  or  resulting  from  the  reaction  in  the  free 
state,  we  learn  the  relative  volumes  of  the  gases  concerned  in  the  reaction. 
Thus,  if  alcohol-vapour  be  burned  in  oxygen, 

Alcohol-vapour.  Oxygen.  Carbonic  anhydride.      Water-vapour. 

C2H60          +  60  2C02  +  3H20. 

One  molecule.  Six  atoms.          Two  molecules.       Three  molecules. 

2  vols.  6  vols.  4  vols.  6  vols. 

Thus  a  system  of  equations  based  on  the  atomic  theory  is  found 
readily  to  give  important  information  beyond  what  it  was 
designed  to  give.  This  lends  probability  to  the  system. 

It  is  not  proved,  however,  that  the  combining  weights  of  the  elements 
exactly  correspond  to  the  relative  masses  of  single  atoms  or  molecules.  If 
they  do,  then  the  number  of  atoms  of  each  kind,  in  a  giveiv  quantity  of  a 
binary  compound,  is  precisely  the  same ;  and  similarly  for  ternary  com- 
pounds and  so  on.  But  there  are  actual  instances  which  at  least  point  in  the 


242  MATTER.  [CHAP. 

contrary  direction.  Pentachloride  of  phosphorus  is  decomposed  by  heat 
into  chlorine  and  terchloride  of  phosphorus ;  but  in  presence  of  an  excess 
of  chlorine,  the  dissociation  is  balanced  by  recombination.  In  the  oxyhy- 
drogen  blowpipe,  the  highest  temperature  is  attained  not  by  means  of  2  vols. 
of  hydrogen  and  1  of  oxygen,  but  by  2  vols.  of  hydrogen  and  about  1]  of 
oxygen.  There  is  thus  an  excess  of  oxygen  required  in  order  to  keep  down 
dissociation.  For  all  that  appears,  it  might  have  been  the  hydrogen  that 
would  have  had  to  be  supplied  in  excess.  If  we  assume  that  there  is  a  small 
action  of  this  kind  at  ordinary  temperatures,  the  combining  weights  will 
give  numbers  only  approximately  proportional  to  the  atomic  masses,  or  to 
multiples  or  sub-multiples  of  these. 

The  Molecule  of  a  compound  substance  is  the  smallest  mass 
that  can  exist  in  the  free  state.  If  we  could  break  up  a  molecule 
we  would  sever  it  into  its  constituent  atoms  — as  HC1  into  H  and 
Cl  —  but  we  would  destroy  the  substance  on  which  we  operated, 
as  such.  A  molecule  of  gaseous  hydrochloric  acid  contains  2 
atoms ;  the  various  hydrates  of  CaCl2  contain  from  21  to  4500 
atoms ;  a  molecule  of  caoutchouc,  of  gum  arabic,  or  of  aluminic 
hydrate  contains  about  6000  atoms ;  while  one  of  egg-albumin 
has  something  short  of  30,000,  and  most  typical  protoplasmic 
colloids  have  (Sabanejew)  more  than  30,000. 

What  is  the  condition  of  elementary  substances  in  the  free 
state  ?  Here  such  equations  as  the  following  come  to  our  aid : 

CuH   +  HC1   =  CuCl  +  HH, 
Ag20  +  H202  =  Ag2  +  00  +  H20 ; 

and  we  learn  that  the  molecule,  even  of  an  elementary  substance, 
consists  of  two  atoms,  and  we  find  by  experiment  that  it  occupies, 
like  the  compound  molecules  already  discussed,  the  same  space 
as  two  atoms  —  i.e.,  one  molecule  —  of  hydrogen.  All  molecules, 
simple  as  well  as  compound,  are  thus  seen  each  to  occupy  the 
same  space ;  and  conversely,  the  same  space  must  be  occupied  by 
an  equal  number  of  molecules  of  whatever  kind  they  be.  This 
is  the  extremely  important  law  known  by  the  name  of  Awo- 
gadro's  Law.  All  gases  (at  the  same  temperature  and 
pressure)  consist,  within  equal  volumes,  of  equal  num- 
bers of  molecules. 

This  is  a  general  law,  and  its  direct  consequence  is  that  the 
specific  gravity  of  every  gas,  at  a  given  temperature  and  pres- 
sure, as  compared  with  that  of  hydrogen  under  the  same  condi- 
tions, is  the  relative  weight  of  a  molecule  of  the  gas  as  compared 
with  the  molecular  weight  (  =  2)  of  hydrogen.  Thus  the  molec- 
ular weight  of  alcohol,  C2H6O,  is  24  +  6  +  16  =  46;  that  of 
hydrogen  =  2 ;  the  single  molecule  of  alcohol  is  twenty-three 


IX.]. 


AVVOGADRO'S  LAW. 


243 


times  as  heavy  as  that  of  hydrogen,  and  accordingly  the  density 
of  alcohol-vapour  is  twenty-three  times  that  of  hydrogen,  other 
things  —  temperature  and  pressure  —  being  equal. 

There  are  some  apparent  exceptions.  Mercury-vapour  which,  if  two  atoms 
formed  its  molecule,  would  have  a  molecular  weight  of  400  and  a  sp.  gr.  of 
200,  has  a  sp.  gr.  of  only  100 ;  hence  its  molec.  wt.  (twice  its  sp.  gr.)  is  only 
200,  and  its  molecule  contains  only  one  atom.  Cadmium,  zinc,  potassium, 
sodium,  and  bismuth  have  monatomic  molecules  when  in  the  state  of  vapour  ; 
so  has  iodine  above  500°  C.  and  under  a  low  pressure.  Phosphorus  and 
arsenic  vapours  have,  on  the  other  hand,  an  excessive  sp.  gr. ;  that  of  phos- 
phorus is  62,  and  its  molec.  wt.  must  be  124 ;  but  its  at.  wt.  is  only  31 ; 
hence  its  molecule  musb  contain  four  atoms ;  at  1600°  C.  it  breaks  up,  how- 
ever, into  diatomic  molecules.  The  molecule  of  arsenic  is  also  tetratomic, 
while  that  of  ozone  is  triatomic.  Sulphur  at  500°  C.  is  hexatomic ;  at 
800°  C.  it  is  diatomic.  Chlorine  and  bromine  vapours  partially  break  up 
into  single  atoms  at  high  temperatures. 

Hence,  to  provide  for  these  exceptional  instances,  we  must  revise  the 
rule  provisionally  laid  down,  and  adjust  it  as  follows :  To  find  the  relative 
volumes  of  gases  entering  or  leaving  a  reaction,  modify  the  equation,  so 
that  it  represents  no  free  gaseous  atoms,  but  only  complete  gaseous 
molecules ;  then  under  every  complete  gaseous  molecule  write  "  1  vol." 
Thus  — 

+  O  H0 


H2 

becomes  2H2 

Two  molecs. 

2  vols. 
The  equation 

C2H60  + 

becomes  C2H6O  -f 

One  molec. 

1  vol.  + 

Again,  the  equation 


02 

One  molec. 
1vol. 


6O 

3O2 

Three  molecs. 
3  vols. 


3CaP2Oe 
becomes 
3CaP2O€ 


+         IOC      =       CaqP9Oe 


IOC      =       Ca3P2O8 


2H20. 

Two  molecs. 
2  vols. 

2CO2          + 
2CO2          + 
TWTO  molecs. 
2  vols. 

+        10CO 

+        10CO 
Ten  molecs. 
10  vols. 


3H20 
3H2O. 

Three  molecs. 
3  vols. 

-       4P 


One  molec. 
1vol. 


Another  order  of  exceptions  is  presented  in  cases  of  Dissociation  or 
Thermolysis.  When  NH4C1  is  volatilised,  its  vapour  has  half  the  sp.  gr. 
indicated  by  the  above  theory ;  in  other  words,  it  occupies  twice  the  theo- 
retical volume.  This  is  because  a  molecule  of  NH4C1  is  in  reality  split  up 
into  separate  molecules  of  NH3  and  HC1  (which  may  be  partly  separated  by 
diffusion),  each  of  which  occupies  the  whole  space  that  the  original  single 
molecule  would  have  been  able  to  occupy  had  it  not  been  decomposed  by  the 
heat  applied.  Similarly,  a  molecule  of  calomel  volatilised  occupies  twice  its 
normal  volume ;  for  instead  of  a  single  molecule  of  Hg2Cl2  we  have,  as  the 
result  of  dissociation,  a  molecule  of  HgCl2,  and  another  molecule,  complete 
though  monatomic,  of  mercury,  each  of  these  molecules  independently 
taking  up  as  much  space  as  the  original  Hg2Cl2  would  have  occupied  if  it 


244  MATTER.  [CHAP. 

had  not  been  decomposed ;  and  the  result  is  that  calomel  becomes,  on  sub- 
limation, contaminated  with  bichloride  of  mercury  (corrosive  sublimate)  and 
darkened  by  mercury.  Sulphuric  acid  vapour  has  twice  the  theoretical 
volume ;  H3SO4  =  H2O  +  SO3.  Colourless  N2O4  dissociates  more  and  more 
completely  into  dark  NO2  as  the  temperature  rises.  By  such  apparent 
exceptions  Avvogadro's  Law  is  thus  confirmed. 

Molecules  appear  in  many  instances  to  be  able  to  combine  with  one 
another.  We  thus  have  water  of  crystallisation  in  crystallised  salts,  coales- 
cence of  molecules  in  acetic  acid  to  form  double  molecules  which  are  torn 
asunder  by  water  or  by  heat,  and  probably  the  various  allotropic  conditions 
in  which  many  solids,  such  as  sulphur,  carbon,  phosphorus,  present  them- 
selves. 

Upon  this  basis  has  been  erected  the  modern  science  of 
Chemistry,  one  of  the  leading  auxiliary  ideas  in  which  is  that 
of  the  Atomicity  of  an  atom  —  the  number  of  atoms  of  hydrogen 
which  an  atom  of  any  element  in  question  can  combine  with 
or  replace.  Whether  the  special  manner  of  thought  and  expres- 
sion of  particular  chemists  be  or  be  not  adopted,  the  theoretical 
chemist  can  hardly  express  himself  without  making  some  use  of 
the  well-known  Graphic  Formulae,  by  means  of  which  the  rela- 
tions of  the  atoms  in  a  molecule  may  be  indicated  or  suggested. 
Yet  this  mode  of  representation  is  sadly  deficient,  although 
exceedingly  useful  and  suggestive.  It  gives  a  factitious  repre- 
sentation, in  one  plane,  of  a  statical  condition  of  the  molecule : 
it  does  not  account  for  the  energy  possessed  by  a  molecule  in 
virtue  of  any  one  arrangement  of  its  atoms,  as  compared  with 
that  possessed  by  a  molecule  of  an  isomeric  compound  in  vir- 
tue of  another  disposition  of  atoms  of  the  same  kind  and  num- 
ber; and,  indeed,  it  scarcely  touches  as  yet  at  any  point  the 
physical  molecule  or  atom  with  which  perfect  knowledge  would 
presumably  show  it  to  be  identical.  Still,  the  attempt  is  being 
made  to  bridge  over  the  gap  —  as,  for  example,  by  the  researches 
of  Le  Bel  and  Van't  Hoff,  who  trace  out  such  relations  as  those 
between  symmetry  of  the  molecule,  as  in  the  case  of  propi- 
CH3 

onic  acid,  H— C— H,  and  the  absence  of  rotary  power  as  affecting 

COOH 

the  plane  of  polarisation,  on  the  one  hand,  and  on  the  other 
between  graphic  asymmetry  of  the  molecule,  as  in  the  case  of 
CH» 

lactic  acid,  HO— C— H,  and  the  possession  of  this  rotary  power; 
COOH 


ix.  J  MOLECULES.  245 

and  by  those  of  Wislicenus,  who  has  done  much  good  work  in 
showing  how  the  arrangement  of  the  atoms  in  the  molecule  may 
be  more  comprehensively  represented  by  tridimensional  dia- 
grams. But,  on  the  whole,  Chemistry  and  Physics,  which  should 
be  parts  of  one  dynamical  Science  of  Matter  and  Energy,  are  still 
separated  by  a  wide  gap,  and  one  great  stride  which  the  Science 
of  the  future  has  to  take  is  that  of  assimilating  the  theories  of 
the  physical  and  the  chemical  molecules,  and  thereby  stepping 
over  this  gulf. 

Physical  Views. —  Physicists  have  been  obliged,  inde- 
pendently of  chemists,  to  develope  mechanical  theories  of  the 
Molecule  or  the  Atom,  as  they  have  indifferently  termed  it. 
That  such  a  thing  does  exist  is  manifest  to  them  on  several 
grounds.  Not  to  speak  of  -compressibility  and  porosity  of  matter 
as  showing  that  it  does  not  entirely  till  space,  we  learn  from 
Cauchy's  investigations  that  if  light  be  a  wave-motion,  there 
would  be  no  dispersion,  no  prismatic  colours  of  the  spectrum,  if 
the  glass  of  the  dispersing  prism  were  continuous  or  were  of  a 
granular  structure  with  indefinitely  small  grains.  According  to 
him,  matter  must  be  distinctly  granular,  whether  it  be  discontinu- 
ous or  not,  and  its  granulations  must  not  be  greatly  less  in 
diameter  than  about  YOWO  °^  the  wave-length  of  the  shortest 
wave  of  light  —  i.e.,  about  20.000000  mm->  or  about  sooToWooo" 
inch.  Lord  Kelvin  finds  that  there  must  be  from  200  to  600 
molecules  in  one  wave-length.  He  also  finds  by  his  Electro- 
meter that  plates  of  copper  and  zinc  exert  a  certain  measurable 
attractive  force  upon  one  another.  An  indefinite  number  of 
plates  would  multiply  this  attraction  to  an  indefinite  amount; 
and  if  such  plates  were  allowed  to  come  together,  the  heat  given 
out  and  representing  their  potential  energy  of  separation  would 
be  indefinite,  and  they  would  combine  after  the  manner  of  gun- 
powder. The  energy  observed  to  be  given  out  in  the  form  of 
heat  during  the  formation  of  brass  by  the  fusion  together  of 
copper  and  zinc  is  not  indefinite  :  it  corresponds  to  the  mutual 
attraction  of  a  number  of  plates  not  more  numerous  than 
100,000000  to  the  millimetre.  Hence  copper  and  zinc  could  not 
be  made  into  plates  thinner  than  this,  and  plates  of  this  tenuity 
would  be  only  one  molecule  thick.  A  soap  film  could  not  be 
stretched  beyond  a  certain  thickness  without  volatilising,  if  it  be 
maintained  at  the  same  temperature,  unless  it  become  materi- 
ally weakened  when  a  certain  limit  is  attained:  for  the  heat 
which  would  have  to  be  supplied  in  order  to  prevent  it  from 


246  MATTER.  [CHAP. 

cooling  upon  stretching  would  be  more  than  sufficient  to  volatil- 
ise it.  This  limit  appears  to  be  reached  when  a  thickness  of 
1/100,000000  mm.  has  been  attained.  Further,  considerations 
derived  from  the  kinetic  theory  of  gases  lead  to  the  conclusion 
that  a  cubic  cm.  of  solid  or  liquid  contains  a  number  of  mole- 
cules which,  though  exceedingly  large,  is  limited ;  and  the  dis- 
tance between  these  is  a  quantity  of  the  same  order  as  those 
above  mentioned.  From  these  considerations  Lord  Kelvin  con- 
cludes —  Thomson  and  Tait,  Natural  Philosophy,  vol.  i.  part  2, 
A  pp.  F,  1883,  and  Nature,  July  1883  (which  see  specially)  — 
that  if  a  globe  of  water  the  size  of  a  football  (16  cm.  diar.) 
were  magnified  to  the  size  of  the  earth,  the  molecules  or  gran- 
ules would  each  occupy  spaces  greater  than  those  filled  by  small 
shot,  less  than  those  occupied  by  footballs. 

But  this  tells  us  nothing  about  the  nature  of  the  atoms  or 
molecules.  It  would  at  first  sight  be  natural  to  conceive  them 
as  hard  balls,  but  this  would  not  explain  their  elasticity  and 
mutual  action ;  Faraday  regarded  them  as  "  centres  of  force  ;  " 
Macquorn  Rankine  as  nuclei,  each  surrounded  by  an  atmosphere 
in  which  there  are  whorls  and  currents  of  a  complicated 
character. 

The  most  interesting  hypothesis  is  that  of  Lord  Kelvin, 
who  supposes  each  Atom  of  matter  to  be  a  Vortex-ring  in  the 
universal  Ether.  The  Ether  itself  we  do  not  directly  perceive ; 
but  this  hypothesis  would  render  our  perception  of  matter  a 
phenomenon  of  exactly  the  same  order  as  that  of  light  or  radi- 
ant heat,  viz.,  a  perception  of  Matter  as  a  Mode  of  Motion  of 
the  Ether. 

If  one  look  at  a  smoke-ring  blown  from  a  cannon,  from  a 
locomotive-engine  chimney,  from  a  tobacco-pipe,  the  lips  of  a 
smoker,  or  from  an  exploded  bubble  of  phosphuretted  hydro- 
gen, it  will  be  seen  that  the  whole  of  the  matter  of  the  ring  is  in 
a  state  of  rotation  round  an  axis  disposed  in  a  circular  form,  and 
having  no  free  ends.  This  is  a  Vortex-ring;  and  such  is  that 
motion  in  the  Ether  which  is  supposed  to  constitute  a  vortex- 
atom.  A  rotating  ring  of  this  kind  in  an  imperfect  fluid,  such 
as  air,  must  be  the  result  of  friction ;  but  in  a  perfect  fluid  it 
could  only  originate  by  a  special  creation  of  some  kind.  Such 
a  vortex-atom  in  a  perfect  fluid  would  have  the  following  proper- 
ties :  it  could  move  about  in  the  fluid ;  its  volume  would  be 
invariable ;  it  would  be  indestructible ;  if  struck  by  another  it 
would  be  indivisible,  but  would  present  perfect  elasticity,  due 


ix.]  MOLECULES.  247 

to  its  motion,  and  though  for  the  moment  distorted,  it  would 
recoil  and  oscillate  through  its  mean  form :  it  would  thus  be 
capable  of  harmonic  vibration,  as  the  spectroscope  shows  the 
particles  of  matter  to  be  ;  it  would  be  capable  of  changes  of  form, 
becoming  narrow  and  thick,  or  wide  and  thin ;  and  it  is  practi- 
cally the  only  form  of  Motion  in  the  Ether  which  could  remain  in 
or  near  the  same  mean  position,  and  at  the  same  time  be  capable 
of  being  compounded  with  movements  of  translation.  This  kind 
of  atomic  structure  would  also  account  for  what  Tolver  Preston 
calls  the  open  structure  of  matter,  which  allows  light,  electric 
and  magnetic  stresses,  and  the  action  of  gravity,  to  be  trans- 
mitted through  it.  These  properties  of  the  vortex-ring  explain 
well  many  of  the  observed  properties  of  matter ;  but  knowledge 
falls  short,  for  we  have  not  only  the  chemical  atom  and  atomi- 
city, but  also  physical  mass  and  gravitation  to  explain  before  we 
can  form  any  full  theory  of  the  inner  structure  of  the  Molecule. 

In  this  view,  the  atom  would  consist  of  a  certain  quantity  of  the  Ether, 
possessed  of  a  certain  amount  of  energy.  If  that  be  so,  it  is  then  conceivable 
that  if  we  were  able  to  arrest  the  vortical  motion,  and  thus  to  destroy  the 
atom,  the  corresponding  energy  might  be  liberated. 

The  Kinetic  Theory.  —  The  next  question  is,  Do  these 
molecules  remain  at  the  same  spot,  rotating  round  it,  or  oscillat- 
ing in  its  vicinity  ?  or  have  they,  in  addition  to  whatever  intrin- 
sic motion  they  may  be  possessed  of,  a  motion  of  Translation  ? 
The  phenomena  of  Diffusion  help  us  to  arrive  at  a  conclusion 
on  this  subject.  If  a  solution  of  a  coloured  salt  be  placed  in 
a  vessel,  and  a  layer  of  a  colourless  solution  be  laid  upon  the 
coloured  stratum,  the  whole  being  left  at  rest  for  some  weeks 
and  protected  from  all  disturbance,  the  plane  of  demarcation 
between  the  strata  becomes  blurred,  the  strata  ultimately  mix, 
and  the  solution  becomes  uniform.  This  can  only  occur  through 
a  gradual  travelling  of  the  coloured  solution  into  the  colourless 
one,  and  vice  versd. 

Again,  if  a  jar  of  hydrogen  and  a  jar  of  oxygen  be  brought 
into  communication  with  one  another,  even  though  the  former 
be  uppermost,  the  gases  will  perfectly  mix  in  a  short  time.  This 
shows  that  the  hydrogen  rapidly  travels  into  the  oxygen,  and 
vice  versd.  The  particles  of  matter,  therefore,  cannot  be  at  rest, 
but  must  be  in  perpetual  relative  motion;  and  this  is  the 
Kinetic  Theory  of  Matter. 

Chemical  analogy  also  illustrates  this  position.  If  steam  be  passed  over 
red-hot  iron  filings,  the  iron  takes  the  oxygen,  and  hydrogen  passes  off ;  if, 


248  MATTER.  [CHAP. 

on  the  other  hand,  hydrogen  be  passed  over  oxide  of  iron,  it  forms  water- 
vapour,  and  reduced  metallic  iron  is  left  behind.  These  actions,  apparently 
so  contradictory,  are  explained  thus  :  There  is  a  molecular  agitation  and  a 
continued  process  of  decomposition  and  recomposition  of  chemical  molecules ; 
the  chemical  atoms  of  iron,  oxygen,  and  hydrogen  are  constantly  changing 
their  partners  and  forming  new  molecules ;  and  in  the  first  instance  any 
molecules  of  hydrogen,  in  the  second  any  molecules  of  steam,  that  happen  to 
be  formed  are  carried  off  in  the  current  of  gas  which  passes  through  the 
apparatus.  The  particles  even  of  one  and  the  same  substance  appear  to  be 
in  this  ceaselessly  restless  state  of  decomposition  and  recomposition  :  when 
the  substance  is  heated,  the  molecules  are  easily  broken  up,  but  are  not  so 
easily  formed  again,  whence  we  have  the  phenomena  of  Thermolysis  or  Dis- 
sociation ;  but  even  at  ordinary  temperatures  the  atoms  associated  within  the 
molecules  break  asunder,  and  must  but  seldom  happen  to  meet  each  other 
again.  Agitation  and  break-up  thus  occurring  within  the  molecules  are 
incompatible  with  rest,  and  must  necessarily  be  associated  with  violent 
translatory  movements  of  the  whole  molecules. 

Dissociation  also  takes  place  within  a  solution.  Sal  ammoniac  dissolved 
in  water  gives  up  ammonia  on  boiling. 

In  a  gas,  then,  we  must  figure  to  ourselves  a  very  large 
number  of  physical  atoms,  moving  about  with  great  velocity, 
striking  one  another  and  the  sides  of  the  containing  vessel. 
Then  the  energy  of  any  given  quantity  of  gas,  so  far  as  that  is 
due  to  movements  of  Translation,  will  depend  on  the  aggregate 
mass  m  and  on  the  mean  velocity  v  of  the  particles ;  and  it  will 
be  \m  -  v2. 

This  mean  velocity  is  the  geometrical  mean  of  all  the  individual 
velocities. 

If  we  consider  a  cube  of  unit-volume,  filled  with  any  gas,  and  take  any 
one  internal  face  of  it ;  that  face,  whose  area  must  be  unity,  is  struck  by 
particles  travelling  with  an  average  velocity  u  in  a  direction  at  right  angles 
to  that  face,  or  having  an  average  component  of  velocity  =  u  in  that  direc- 
tion, and  having  therefore  a  certain  aggregate  momentum.  This  momen- 
tum, with  which  the  particles  strike  the  wall  during  a  unit  of  time,  must  be 
equal  to  the  counter-pressure  p  exerted  by  the  wall  of  the  vessel  per  unit  of 
area ;  *  the  pressure  p  exerted  by  the  gas  on  unit-area  of  the  walls  of  the 
vessel  is  therefore  equal  to  the  wallward  momentum  of  the  particles  imping- 
ing on  a  unit-area  of  the  wall  in  the  course  of  a  unit  of  time.  But  what  is 
the  amount  of  this  momentum?  It  is  the  product  of  the  number  of  particles 
which  strike  the  unit-area  wall  in  a  unit  of  time,  into  the  average  momen- 
tum of  each. 

1.    The  number  of  the  striking  particles  — 

If  the  gas  contain  N  particles  per  unit  of  volume,  and  if  these 
move  towards  the  wall  struck  by  them  with  an  average  velocity  u 
per  second,  the  number  of  particles  which  must  strike  the  unit- 
area  wall  in  a  unit  of  time  is  Nu. 

*  This  momentum  would  be  lost  to  the  gas  within  the  cube  were  the  particles 
conveying  it  not  prevented  by  the  counter-pressure  of  the  wall  from  escaping :  this 
loss  -^  time  during  which  it  would  have  been  effected  is  the  Rate  of  Change  of 
Momentum  (see  p.  20)  prevented  during  that  time,  per  unit  of  area  of  the  wall. 


ix.]  KINETIC  THEORY.  249 

2.   The  average  momentum  of  each,  towards  the  wall  of  the  vessel  — 

The  mass  m  of  each  particle  is  the  same ;  the  average  velocity  is 
u;  the  average  wallward  momentum  of  each  particle  is  Jhu. 

The  momentum  with  which  the  wall  is  struck  is  thus  Nu-  mvL  per  unit  of 
area,  per  unit  of  time  :  and  this  =  p,  the  pressure  on  the  wall  per  unit  of  area. 
But  the  cube  is  one  of  unit  volume  ;  its  volume  b  =  1 ;  the  aggregate  mass 
of  the  gas  is  b  Nm=bp;  .-.  Nm  =  p ;  whence  p  =  pu2. 

Next,  what  is  the  average  velocity  u  in  any  one  direction  ?  The  average 
speed  v  is,  if  resolved  into  components  u,  v,  w,  at  right  angles  to  one 
another,  v  =  V_u2  +  v2  +  w2. 

But  u,  v,  w  are  equal  to  one  another,  for  there  is  no  difference  between 
one  direction  and  another  in  respect  of  velocity :  whence  v  =  -y/3  u2,  and 

u=  V^A 

Therefore  the  pressure  per  unit  of  area  on  the  bounding  surface,  and  at 
right  angles  thereto,  is  p  or,  generally,  in  any  direction,  the  hydrostatic  pres- 
sure per  unit  of  area  is  p  =  pu2  =  p  •  v  2/3  ;  and  consequently,  whatever  be 
the  volume  b,  the  product  pb  =  pb  •  v2/3  =  m~v2/3  —  #(£ mv2)  =  f  the  aggre- 
gate molecular-translational  kinetic  energy*  of  the  gas  whose  mass  m  is  con- 
fined within  volume  b. 

Since  the  molecular-translational  kinetic  energy  of  a  mass  m  of  any  gas 
is  equal  to  \m  -  v2  ergs,  and  this  is  equal  to  f  jt>b,f  where  b  is  the  volume  occu- 
pied by  the  mass  m,  it  follows  that  this  energy  is,  per  gramme,  fpb/m  =  f  jo/p ; 
and  also,  this  energy  is,  per  cub.  cm.,  fj0b/b  =  f  p  ergs.  Hence  this  energy 
is,  at  the  same  temperature,  equal  in  equal  volumes  of  all  gases. 

If  two  gases  have  the  same  Temperature,  the  particles  have  the  same 

mean  molecular  energy  (^mr2)  of  translation.     This  is  a  hypothesis;  but 

/if  it  were  otherwise,  two  gases  at  the  same  temperature  would  change  in 

y   temperature  when  mixed  ;  for  their  average  molecular  energy  would  become 

equalised  throughout. 

If  the  aggregate  kinetic  energy  of  translation  (Imv2}  be  equal  in  equal 
volumes  of  two  gases,  and  if  at  the  same  time  the  molecular  energy  of  the 
molecules  of  each  be  equal  (their  temperatures  being  equal),  it  follows  that 
the  number  of  molecules  must  be  equal  in  the  equal  volumes  of  the  two 
gases,  and  hence  Avvogadro's  Law  is  true  in  the  physical  as  well  as  in  the 
chemical  sense,  being  a  direct  deduction  from  the  kinetic  theory. 

If  there  be  two  gases  whose  respective  densities  at  equal  temperatures 
and  pressures  are  p  and  p,,  Avvogadro's  law  shows  that  their  unequal  masses 
are  divided  among  equal  numbers  of  molecules :  hence  the  mass  of  each 
single  molecule  must  be  proportional  to  the  density  of  its  gas ;  for  if  m  and 


*  Let,  for  example,  the  gas  be  one  gramme  of  hydrogen,  at  0°  C.  and  76  cm.  baro- 
metric pressure,  and  occupying  under  these  conditions  a  volume  fa  =  11,105  cub. 
cm.;  then  p  =  1,013663'4  dynes  per  sq.  cm.;  whence  pb  =  11,317,207000;  and  the 
Tran slational  Kinetic  Energy  of  one  gramme  of  hydrogen  at  0°  C.  and  76  cm. 
barometric  pressure  is  ipfa  =  16,975,810000  ergs  =  1697*681  Joules,  or  408*14  ca  (see 
p.  353),  or  1251-65  ft.-lbs. 

t  We  here  assume  the  absence  of  intermolecular  forces.  If  there  be  such,  inde- 
pendent of  collisions,  the  molecular  Kinetic  Energy  =  ipfa  +  5  S(Rr)  (Clausius),  where 
the  last  expression  (the  "  Virial ")  is  half  the  sum  —  a  sum  which  forgiven  values  of 
p  and  fa  retains  an  appreciably  constant  value  —  of  the  products  of  the  mutual 
distances  r  of  every  pair  of  particles  into  the  corresponding  mutual  attractive 
force  R. 


250  MATTER.  [CHAP. 

m,  be  the  respective  masses  of   single   molecules  of   the  respective   gases, 
m  =  piJ/Nij  =  p/N,  and  ml  =  py/N  ;  whence  m  :  mt  :  :  p  :p,  . 


Thus  the  Molecular  Kinetic  Theory  of  Gases  explains  the 
pressure  on  the  sides  of  the  vessel  containing  a  gas  :  it  explains 
the  tendency  of  gases  to  indefinite  expansion  :  it  explains  Heat 
as  the  energy  of  molecular  agitation  ;  equality  of  temperature 
as  equality  of  the  mean  energy  of  agitation  in  the  several  mole- 
cules. It  also  arrives  at  Avvogadro's  Law,  and  explains  the 
numerical  identity  of  ratio  existing  between  the  relative  weights 
of  the  several  kinds  of  molecules  and  the  specific  densities  of 
the  corresponding  aggregate  gases. 

The  equation  p  =  py2/3  given  above  yields  v  =  -\/3p/p,  ^J  means  of 
which  y,  the  mean  velocity  of  movement  of  the  particles  of  any  gas,  may 
be  found.  Thus  for  hydrogen  p,  the  pressure  per  sq.  cm.,  is  equal  to  the 
Weight  of  say  76  cm.  of  mercury  (density  =  13-596),  or  of  1033-296  grms.- 
mass  resting  on  every  square  cm.  But  the  weight  of  1033-296  grins,  is  mg  ; 
1033-296  grms.  x  981  =  1,013663-376  dynes.  Again,  p,  the  density  of  hydro- 
gen, is  -0000895682  grammes  per  cubic  cm.  Hence  V3.P/P  =  184260  cm.  per 
second,  the  average  velocity  of  the  particles  of  hydrogen. 

Hence  also  the  mean  velocities  of  gases  vary  inversely  as  Vpl  or,  which 
is  an  equivalent  statement,  the  mean  velocities  of  the  particles  of  gases  vary 
inversely  as  the  square  root  of  the  molecular  weight  :  whence  oxygen-atoms 
have  one-fourth  the  velocity  of  hydrogen-atoms,  because  they  are  sixteen 
times  as  massive.  This  is  the  law  governing  the  relative  speed  with  which 
the  different  components  of  a  gaseous  mixture  will  travel  through  a  mem- 
brane. 

The  kinetic  theory  also  informs  us  that  when  we  double  the 
number  of  molecules  which  move  in  a  given  space  with  a  given 
mean  velocity  we  double  the  number  of  molecules  which  strike 
the  walls,  and  accordingly  we  double  the  pressure  ;  or  in  other 
words,  the  pressure  varies  directly  as  the  density  of  a  given 
quantity  of  gas,  this  being  another  form  of  Boyle's  Law. 

It  also  tells  us  that  if  we  mix  a  particles  of  one  gas,  b  parti- 
cles of  another,  c  of  a  third,  and  so  on,  the  average  kinetic 
energy  of  all  the  particles  being  the  same,  or  soon  becoming 
equalised,  the  pressure  (per  sq.  cm.  of  bounding  surface)  pro- 
duced by  the  a  molecules  of  the  first  gas  is  proportional  to  their 
number,  the  pressure  produced  by  the  second  gas  is  proportional 
to  6,  and  so  forth  ;  or  in  other  words,  that  in  a  mixture  of  gases 
the  pressure  produced  by  each  component  of  the  mixture  is  inde- 
pendent of  the  rest,  and  depends  only  on  the  amount  of  such 
component  which  is  present  in  the  mixture  (Dalton's  Law). 

Again,  when  the  temperature  is  increased,  the  energy  of  the 


ix.]  KINETIC   THEORY.  251 

particles  is  increased;  each  particle  strikes  both  oftener  and 
harder;  the  pressure  experienced  by  the  walls  of  the  vessel 
therefore  varies  as  the  square  of  the  velocity,  and  is  proportional 
to  the  molecular  energy  of  the  particle  —  that  is,  to  the  abso- 
lute amount  of  heat-energy  possessed  by  it.  This  if  the  volume 
be  kept  constant ;  but  if  the  pressure  be  kept  constant  and  the 
volume  allowed  to  increase,  then  the  volume  varies  in  the 
same  proportion;  that  is,  as  the  "absolute  temperature"  (see 
p.  364).  (Charles's  Law,  often  attributed  to  Gay  Lussac.) 

The  kinetic  theory  of  gases  also  explains  how  it  is  that  when 
a  stream  of  gas  passes  through  air,  its  progress  is  retarded  by 
"viscosity;  "  rapidly-moving  particles  of  the  gas  travel  later- 
ally into  the  air ;  slowly-moving  particles  of  the  air  travel  into 
the  gas,  and  thus  its  progress  is  hampered.  Similarly,  the  vis- 
cosity of  a  gas  will  bring  to  rest  a  current  set  up  within  its  own 
substance.  This  viscosity  is  proportional  to  the  absolute  tem- 
perature, but  is  independent  of  the  density  in  any  given  gas. 

The  theory  also  explains  the  conduction  of  heat  in  gases  ; 
rapidly -moving  particles,  by  collision,  part  with  some  of  their 
energy  to  others,  which  in  their  turn  enter  into  collision  with 
those  beyond  them:  and  we  have  already  seen  it  explain  the 
diffusion  of  gases. 

The  mutual  impact  of  elastic  solid  particles  would  necessarily  result  in 
the  ultimate  transformation  of  the  whole  translational  energy  into  energy  of 
vibration ;  that  of  vortex-rings  seems  to  imply  no  such  result.  The  latter 
seems,  therefore,  the  preferable  form  of  the  kinetic  theory  of  matter,  although 
it  is  as  yet  far  from  complete. 

These  molecules,  thus  travelling  with  such  great  velocities 
and  entering  into  a  practically  infinite  number  of  collisions  with 
one  another  (in  hydrogen  17750  millions  per  second),  can  never 
travel  very  far  in  an  undisturbed  path.  At  the  ordinary  tem- 
perature and  pressure  the  mean  free  path  of  the  molecules  of 
hydrogen,  which  have  the  longest  trajectory,  seems  to  be  about 
2"olhro  mm-»  or  a  tenth  part  of  the  average  length  of  a  wave  of 
light  (Maxwell) ;  y-0-^o  o"  mni*  (Crookes).  The  diameter  of  mole- 
cules is  not  the  same  in  the  case  of  all  elements,  but  is  on  the 
average  perhaps  ^-^VoTo  mm*  Thus  the  smallest  visible  organic 
particle  (^oVo"  mm.  diar.)  may  contain  about  480,000000  atoms, 
which  may  be  arranged  in  as  few  as  16,000  molecules.  The  num- 
ber of  molecules  in  a  cubic  cm.  of  a  gas  at  the  ordinary  tempera- 
ture and  pressure  is  about  19,000000,000000,000000- (Maxwell), 
1000,000000,000000,000000  (Crookes),  not  more  than  6000,- 
000000,000000,000000  (Lord  Kelvin). 


252  MATTER.  [CHAP. 

These  numbers  are  arrived  at  by  using  a  proposition  formulated  by 
Clausius,  that 
8*  x  free  path  of  each  molecule  _  Whole  space  occupied  by  the  molecules 

Diameter  of  each  molecule  Their  real  aggregate  volume 

From  this  the  real  aggregate  volume,  which  does  not  differ  very  widely  from 
that  of  the  corresponding  liquid,  is  found,  and,  if  divided  by  the  cubic  space 
occupied  by  each  molecule,  gives  the  number  of  molecules. 

Ultragaseous  Matter.  —  When  gas  is  rarefied  the  number 
of  molecules  in  a  given  space  is  diminished.  Let  us  suppose 
that  the  rarefaction  is  carried  on  so  far  that  only  one  particle 
out  of  every  original  million  is  left  in  the  space  exhausted. 
The  pressure  is  one-millionth  of  its  original  amount;  but  any 
molecule  once  in  motion  has  one-millionth  its  former  chance  of 
encountering  any  other  molecule,  and  consequently  its  average 
free-path  is  magnified  a  millionfold.  The  mean  path  would 
then  be  (Crookes)  YOOTO  mm-  x  1*000,000  =  100  mm.,  or  about 
4  inches.  By  means  of  a  good  Sprengel  pump  exhaustion  to  a 
hundred-millionth  of  an  atmosphere  can  be  attained,  and  the 
mean  free-path  of  the  gas  so  rarefied  would  be  about  33  feet. 
In  our  atmosphere  at  a  height  of  210  miles  the  single  molecules 
are  relatively  so  few  (1000  to  the  cubic  cm.),  that  each  molecule 
might  travel  through  a  uniform  atmosphere  of  that  density  for 
60,000000  miles  without  entering  into  collision ;  beyond  a  height 
of  300  miles  the  atmosphere  is  so  rare  (less  than  one  molecule 
per  cubic  foot)  that  the  particles  might  freely  travel  through 
such  an  atmosphere  from  one  fixed  star  to  another;  while  in 
the  fields  of  space,  at  distances  practically  infinite  from  the 
earth  or  any  other  star,  the  number  of  cubic  miles  containing  a 
single  molecule  would  be  represented  by  the  figure  1  followed 
by  314  cyphers. 

This  opens  up  to  us  an  extraordinary  view  of  the  nature  of  our  atmos- 
phere. We  must,  —  though  the  process  cannot  be  rapid,  for  each  particle 
rising  from  the  earth  is  retarded  by  gravity  and  falls  back  towards  the 
earth,  —  constantly  be  losing  particles  of  nitrogen  and  oxygen  as  we  are 
dragged  through  space,  and  we  may  constantly  be  picking  up  new  ones.  If 
we  entered  regions  of  space  in  which  there  were  no  particles  fit  to  make  up 
our  losses,  it  would  be  an  interesting  question  how  short  a  time  would  suf- 
fice altogether  to  deprive  us  of  our  atmosphere. 

The  region  of  space  through  which  the  earth  is  at  present  travel- 
ling contains  much  benzene  vapour  with  ethyl-hydride  and  other  alcohol- 
derivatives  (Abney). 

Thus  our  ideas  on  the  subject  of  the  constitution  of  matter 
have  undergone  a  profound  modification.  Matter  is  discontinu- 

*  \/72  =  8.48,  Clerk  Maxwell;  8.86,  Tait. 


ix.]  CONSTITUTION   OF   MATTEK.  253 

ous  in  the  highest  degree,  for  it  consists  of  separate  particles  or 
molecules,  which  are  mutually  non-interpenetrable ;  the  special 
properties  of  the  different  states  of  matter  depend  on  the  num- 
ber of  molecules  which  are  contained  within  a  given  space,  as 
well  as  on  the  energy  of  movement  which  is  possessed  by  each ; 
and  each  particle  is  susceptible  not  only  of  translation  as  a 
whole,  but  also  of  vibration  or  rotation,  and  may  besides  be  in 
a  state  of  vortex-motion,  upon  the  continuance  of  which  its  con- 
tinued existence  may  depend. 

MOLECULAR  FORCES. 

Hitherto  we  have  conducted  our  reasoning  on  the  implied 
assumption  that  the  molecules  had  no  mutual  action,  and  we 
have  arrived  at  results  such  as  Boyle's  law,  Dalton's  law,  and 
others,  which  we  have  deduced  from  theory.  Now  we  must 
confirm  our  theory  by  reference  to  facts,  and  we  find  this  assump- 
tion overruled  by  such  material  discrepancies  as  the  following. 
Boyle's  law,  though  obeyed  on  the  whole,  is  disobeyed  by  every 
gas  when  the  pressure  is  so  high  or  the  temperature  so  low  that 
condensation  is  not  far  off :  this  departure,  though  not  exten- 
sive, is  significant.  All  gases  just  about  to  become  condensed 
are,  except  in  the  single  case  of  hydrogen,  more  easily  com- 
pressed than  the  law  would  indicate.  Dalton's  law  is  departed 
from  by  a  mixture  of  gases  condensible  with  difficulty :  such  a 
mixture  is  found  to  be  even  less  condensible  than  the  compo- 
nent gases,  and  the  critical  temperature  is  lowered.  Charles's 
law  is  not  obeyed  throughout  the  whole  range  of  experimental 
pressures  and  temperatures  ;  at  a  high  pressure  any  increment  of 
heat  produces  a  disproportionately  large  increment  of  pressure. 

In  fact,  Gases  obey  these  laws  only  when  their  pressure  is 
very  feeble  and  their  temperature  at  the  same  time  high  above 
the  critical  temperature  —  that  is,  when  the  molecules  are  com- 
paratively far  from  one  another.  At  ordinary  temperatures 
and  pressures  the  particles  do  affect  one  another  not  merely  by 
mutual  impact  or  mutual  gravitation,  but  also  by  mutual  actions 
or  molecular  forces,  effectively  coming  into  play  when  the  par- 
ticles are  at  exceedingly  small  distances  from  one  another. 
When  the  pressure  is  small,  the  free  path  is  comparatively  long, 
and  the  molecules  are  mutually  removed  from  each  other's  influ- 
ence :  and  the  higher  the  rate  at  which  the  particles  £re  moving, 
the  less  will  be  the  proportionate  effect  of  the  molecular  forces ; 


254  MATTER.  [CHAP. 

or  in  other  words,  the  higher  the  temperature  the  less  appreci- 
able will  be  the  effect  of  intermolecular  action. 

When  a  gas  is  being  compressed  into  a  liquid  we  know  that 
in  the  first  place,  in  all  gases  except  hydrogen,  the  particles  are 
attracted  slightly  towards  one  another,  and  also  that  there  is  on 
the  other  hand  a  practical  repulsion  from  one  another  caused 
by  their  energetic  movement.  We  further  find,  however,  that 
though  the  particles  become  approximated  with  relative  ease 
while  liquefaction  is  approaching,  yet  when  the  liquid  state  has 
been  attained,  and  even  before  it  has  been  attained,  repulsion 
takes  the  place  of  attraction ;  the  liquid  when  formed  offers  a 
relatively  enormous  resistance  to  compression.  This  is  well 
seen  in  the  case  of  carbonic  anhydride  merging  insensibly  from 
the  gaseous  into  the  liquid  state ;  just  before  ceasing  to  be  a 
gas  it  is  very  compressible ;  just  after  becoming  a  liquid  it  is 
relatively  very  slightly  so. 

Air  obeys  Boyle's  law  precisely,  and  the  air  manometer  is  therefore  cor- 
rect, at  a  pressure  of  200  atmospheres  ;  below  that  pressure  the  volume  is  in 
defect ;  above,  it  is  in  excess  (Andrews).  (See,  also,  p.  375.) 

In  Liquids  the  molecules  are  within  the  spheres  of  one 
another's  action.  This  accounts  for  the  viscosity  of  all,  even  of 
the  most  mobile  liquids :  the  particles  detain  one  another  by 
their  mutual  attraction,  and  a  flowing  liquid  is  thus  hindered  in 
its  flow  by  molecular  friction.  Molecular  action  also  accounts 
for  the  fact  that  a  stream  of  liquid  has  a  certain  tenacity  and 
will  not  readily  break :  such  is  the  condition  of  a  stream  of 
liquid  in  a  siphon.  Again,  it  explains  why,  under  ordinary  cir- 
cumstances, the  effects  of  molecular  attraction  are  strikingly 
manifest  in  liquids  only  at  the  surface,  and  in  the  form  of  Sur- 
face-Tension. In  the  interior  of  a  mass  of  fluid  each  particle  is 
rig  100  ^-^  — *^L  ^ree  t°  adjust  its  mean  position  under 

'  the  influence  of  the  surrounding  mole- 
cules ;  the  mean  position  which  it  as- 
sumes is  that  in  which  it  is  acted  on 
equally  on  all  sides,  and  there  is  then 
nothing  to  render  the  mutual  attrac- 
tions manifest.  At  the  surface  of  a 
liquid  mass,  however,  if  it  be  a  free 
surface,  the  particles  can  only  be  acted 
upon  by  others  lying  internal  to  them. 
The  result  is,  as  is  shown  in  Fig.  100,  a  system  of  forces  acting 
at  right  angles  to  the  free  surface  of  the  fluid,  and  tending  to 


ix.]  MOLECULAR   FORCES.  255 

reduce  that  free  surface  to  the  least  possible  area.  We  may, 
indeed,  consider  a  drop  of  water  as  consisting  of  a  quantity  of 
water  enclosed  in  a  superficial  skin  of  water  which  is  under  ten- 
sion, and  whose  particles  attract  one  another  into  the  least  pos- 
sible superficial  area;  and  since  of  all  surfaces  the  sphere  has 
the  greatest  content  for  the  least  area,  the  superficial  film  may 
be  said  to  mould  the  drop  to  the  spherical  form,  which  in  the 
case  of  falling  raindrops  is  approximately  perfect,  as  is  shown 
by  the  rainbow.  To  these  surface-tensions  are  also  due  the 
important  phenomena  of  Capillarity. 

Many  of  the  properties  of  Solids  are  also  due  to  molecular 
forces.  Such  are  toughness,  hardness,  and  the  like,  which  may 
be  grouped  under  the  generic  name  Strength  of  Materials; 
these  depending  probably,  in  part  at  least,  on  the  proximity  of 
the  particles  to  one  another,  and  perhaps  on  their  form.  The 
molecular  grouping  of  molecules  is  also  very  important,  though 
very  little  can  be  said  about  it ;  but  upon  it  depend  not  only 
the  crystalline  or  amorphous  condition  of  a  substance  and  in 
part  its  strength,  but  also  that  stable  or  unstable  equilibrium 
upon  which  the  phenomena  of  elasticity  or  the  properties  of 
such  things  as  Rupert's  drops  depend.  These  last  consist  of  lit- 
tle masses  of  fused  glass  dropped  into  cold  water ;  the  outside  is 
suddenly  chilled  and  solidified  while  the  interior  is  still  in  a 
state  of  fusion.  The  internal  mass  has  to  accommodate  itself  as 
it  best  can  to  the  dimensions  of  the  outer  skin ;  it  does  so  under 
tension,  but  the  moment  that  the  relations  are  disturbed  by 
breaking  off  the  narrow  end,  or  even  by  the  slightest  scratch, 
the  whole  flies  to  powder :  it  is  in  a  state  of  unstable  equilib- 
rium, and  the  slightest  displacement  precipitates  a  downfall  of 
the  whole  arrangement.  In  the  same  way  the  slightest  scratch 
in  the  interior  of  a  large  glass  tube,  especially  if  it  have  local 
variations  of  thickness,  —  even  such  a  scratch  as  is  produced 
by  the  fall  of  a  crystal  of  quartz  or  a  rub  with  the  end  of  an 
iron  wire,  —  will  often  shiver  the  tube ;  for  which  reason  no 
rough  treatment  should  be  internally  applied  to  such  tubes  with 
any  metal  harder  than  copper.  This  state  of  internal  tension 
accounts  for  the  danger  in  the  use  of  cast-iron  in  structures. 

Many  solid  masses  have,  however,  their  particles  so  arranged 
as  to  form  Conservative  Systems,  which  tend  to  restore  any  work 
done  on  them,  and  consequently  are  in  stable  molecular  equi- 
librium ;  the  details  of  the  molecular  grouping  are  unknown, 
but  in  a  perfectly  elastic  body,  or  practically  in  any  solid  body 


256  MATTER.  [CHAP,  ix.] 

within  its  Limits  of  Elasticity,  any  displacement  among  the 
molecules  produces  a  restitution-pressure  equal  and  opposite  to 
the  distorting  force  or  stress ;  and  it  is  observed  that,  as  a  gen- 
eral rule,  the  distortion  is  proportional  to  the  distorting  force 
(  Ut  tensio  sicut  vis  ;  "  Hooke's  Law  "),  and  hence  the  restitution- 
pressure  is  proportional  to  the  distortion.  This  elasticity  may 
in  solids  be  observed  more  or  less  perfectly  to  obtain,  whether 
the  distortion  be  that  of  form  or  of  volume ;  while  liquids  have 
elasticity  of  volume  alone,  never  of  form. 

To  the  same  order  of  Molecular  Forces  must  be  attributed 
the  effects  of  Cohesion  between  masses  or  particles  of  the  same 
substance,  and  of  Adhesion  between  those  of  different  sub- 
stances; and  also  the  phenomena  of  Chemical  Affinity,  the 
potential  energy  of  chemical  separation,  and  the  liberation  of 
energy  attendant  on  chemical  combination. 


CHAPTER  X. 

SOLIDS. 

THE  special  properties  of  solids  are  due  to  the  relative  con- 
tiguity of  their  molecules.  Their  definite  free  surface  is  due  to 
the  mutual  attraction  of  their  molecules,  and  is  retained  in  vir- 
tue of  the  same  forces  which  in  the  aggregate  manifest  them- 
selves as  causes  of  cohesion,  tenacity,  etc.,  and  the  result  of 
which  is  that  a  solid  can  persist  under  the  action  of  a  stress  not 
evenly  applied  —  that  is,  of  a  stress  which  is  not  hydrostatic. 

Still,  the  particles  appear  to  have  some  power  of  travelling  past  one 
another,  though  not  much ;  carbon  soaks  into  iron  in  the  old  "  cementa- 
tion "  steel-making  process ;  it  can  also  travel  through  porcelain  ;  in  crystals 
made  up  of  successive  layers  of  different  alums  the  different  layers  become 
more  or  less  blended ;  in  nickel-plating  steel  plates  for  printing  purposes, 
the  nickel  sinks  into  the  steel  to  some  depth ;  powders  of  potassium  nitrate 
and  sodium  acetate,  when  mixed  and  compressed,  form  hygroscopic  nitrate 
of  sodium  and  potassium  acetate ;  and  some  solids  are,  to  a  small  extent, 
affected  by  an  electric  current  as  if  they  were  liquids  containing  moveable 
molecules  (Electrolysis).  The  phenomena  of  Magnetism  point  towards  the 
molecules  of  iron  being  turned  round  when,  the  iron  is  magnetized. 

Cohesion  is  the  mutual  attraction  of  the  particles  of  a  solid 
for  one  another,  and  is  measured  by  the  amount  of  force  which 
must  be  applied  in  order  to  overcome  it.  The  term  cohesion  is 
generally  applied  to  the  mutual  attraction  of  particles  of  the 
same  substance,  adhesion  to  that  of  different  substances.  When 
two  pieces  of  white-hot  iron  or  platinum  are  brought  in  contact 
they  weld  by  cohering.  When  a  piece  of  silver  and  a  piece  of 
platinum  are  brought  in  contact  at  500°  C.  they  adhere.  If 
metals  in  the  state  of  dust  be  mixed  and  exposed  to  a  pressure 
of  7000  atmospheres  they  will  form  a  firm  metallic  mass,  and 
will  even  combine  and  form  an  alloy.  Even  sulphides  and 
arsenides  may  thus  be  formed ;  for  pressure  promotes  contact. 
Cohesion  is  manifested  by  two  surfaces  of  glass,  which,  if  ground 
exceedingly  smooth  and  placed  in  contact,  will  cohere  firmly ; 
and  the  well-known  Barton's  cubes  are  little  cubes  of  metal  pol- 

s  257 


258  SOLIDS.  [CHAP. 

ished  so  smoothly  that  mere  apposition  causes  them  to  cohere, 
the  force  of  cohesion  being  so  great  that  a  string  of  a  dozen  may 
be  supported  in  the  air  by  this  mutual  attraction  alone.  Com- 
mon graphite  is  ground  to  powder  and  punned  by  boiling  with 
nitric  acid  and  chlorate  of  potash :  it  is  then  washed  and  dried ; 
the  powder  is  placed  in  a  mould  and  exposed  to  extreme  pres- 
sure produced  by  a  hydraulic  press ;  after  compression  the  black 
powder  is  found  to  have  been  converted  into  a  solid  mass  of 
coherent  pencil-graphite,  which  may  be  sawn  into  strips  and 
used  for  pencils.  If  a  leaden  bullet  be  cut  into  two  with  a  sharp 
and  heavy  knife,  the  two  halves  will  cohere  firmly  if  pressed 
together  by  their  bright  surfaces. 

Hardness  —  Softness.  —  A  body  is  said  to  be  harder  than 
another  when  it  can  be  used  to  scratch  the  latter  but  cannot  be 
scratched  by  it.  In  this  sense  the  diamond  is  the  hardest  of  all 
solids.  The  scratching  body  must  not  have  too  sharp  a  point, 
for  this  would  prove  a  pin  to  be  harder  than  glass,  which  is  not 
the  case.  Hardness  is  a  property  that  cannot  be  measured.  All 
that  we  can  do  is  to  make  up  a  list  of  substances  in  their  rela- 
tive order  of  hardness,  and  to  express  the  hardness  of  any 
particular  substance  by  stating  its  place  in  that  series.  The 
standard  series,  due  to  Mohl,  is  the  following :  — 

1.  Green  laminated  Talc.  2.  Crystallised  Gypsum.  3.  Trans- 
parent Calcspar.  4.  Crystalline  Fluorspar.  5.  Transparent  Apa- 
tite. 6.  Pearly  cleavable  Felspar  (Adularia).  7.  Transparent 
Quartz.  8.  Transparent  Topaz.  9.  Cleavable  Sapphire.  10.  Dia- 
mond. Flint  scratches  quartz  with  difficulty,  but  is  easily 
scratched  by  topaz :  hence  its  hardness  is  set  down  as  7-25  on 
this  arbitrary  scale.  The  rapidity  of  movement  of  the  attacking 
substance  is  a  matter  of  practical  importance:  thus  the  sand- 
blast (a  stream  of  sand  rapidly  blown  from  a  tube)  is  capable  of 
cutting  through  rocks  and  even  through  steel  with  relatively 
great  rapidity;  and  the  same  result  is  seen  in  the  mechanical 
operation  of  filing. 

Mr.  Edison  finds  that  platinum  wire  may  be  rendered  as 
hard  as  steel  pianoforte  wire  by  heating  in  vacua,  keeping  up  the 
vacuum,  and  gradually  increasing  the  temperature.  The  par- 
ticles of  platinum  have  all  air  removed  from  their  interstices, 
they  cohere  very  firmly,  and  the  metal  welds  together. 

Hardness  —  Fragility. — This  is  a  distinct  use  of  the  word 
Hardness.  In  this  sense  the  diamond  possesses  little  hardness, 
for  if  struck  a  blow  with  a  hammer  it  flies  to  pieces. 


x.]  PROPERTIES   OF   SOLIDS.  259 

Malleability,  the  property  of  yielding  to  the  hammer  with- 
out breaking  at  the  edges.  —  Gold  can  be  hammered  out  into 
leaves  extremely  thin.  A  half  square-inch  of  gold  of  the  thick- 
ness of  letter  paper  is  hammered  out  to  81  square  inches ;  each 
square  inch  of  this  thin  sheet  is  again  hammered  out  into  81 
square  inches,  of  which  each  one  is  in  its  turn  again  hammered 
out  to  81  square  inches.  Antimony,  on  the  other  hand,  flies  to 
powder  at  the  first  blow  of  the  hammer. 

Plasticity.  —  Some  solids  can  be  moulded,  as  lead  in  a 
bullet-mould,  coins  at  the  Mint,  bars  in  a  rolling-mill.  Any 
force  above  a  certain  limit  produces  a  permanent  set.  Plastic 
solids,  under  pressure,  follow  the  laws  of  the  motion  of  liquids. 

Ductility.  —  Some  metals  can  be  drawn  through  fine  aper- 
tures in  a  draw-plate,  and  wires  can  thus  be  formed :  other 
metals  are  incapable  of  this,  for  they  snap.  The  order  of  duc- 
tility is  —  Gold,  Silver,  Platinum,  Iron,  Copper,  Palladium,  Alu- 
minium, Zinc,  Tin,  Lead.  Platinum  wires  of  exceeding  tenuity, 
such  as  are  adapted  to  the  eye-pieces  of  microscopes  for  micro- 
metric  work,  are  made  by  constructing  a  thick  silver  bar  with  a 
thin  platinum  core,  drawing  this  out  to  an  extreme  fineness,  and 
dissolving  off  the  silver  by  steeping  the  drawn  wire  in  nitric  acid. 

Resistance  to  Deformation.  —  When  a  solid  undergoes 
deformation  under  the  influence  of  an  applied  force,  a  condition 
of  equilibrium  is  ultimately  reached,  and  is  thereafter  main- 
tained, except  in  so  far  as  excessive  forces  or  protracted  duration 
of  the  experiment  may  bring  to  view  the  fact  that  there  is  always 
a  slow  yielding,  even  of  the  most  rigid  solids,  under  continu- 
ously applied  forces.  This  equilibrium  is  one  between  the  Force 
acting  and  an  equal  and  opposite  reaction  or  counter-force  or 
Resistance  developed  by  the  body  during  the  deformation ;  and 
this  resistance  is  so  developed,  whether  the  force  applied  be 
tensile,  compressive,  shearing,  or  torsive. 

Cubical  Compressibility.  —  Most  solids  are  only  slightly 
compressible  under  the  action  of  a  pressure  equably  applied  to 
their  whole  surface.  •  Their  Resistance  to  Compression  is  called 
their  Elasticity  of  Volume. 

When  a  uniform  pressure,  p  dynes  per  sq.  cm.,  applied  to  the  whole  sur- 
face of  a  solid,  reduces  its  volume  to  to  (b  —  Sto),  the  Compressibility  is  meas- 
ured by  the  proportionate  change  of  volume,  8to/to,  produced  per  dyne  of 
applied  pressure  per  sq.  cm.  It  is  therefore  equal  to  8ij/b  -4-  p.  The 
Resistance  to  Compression,  or  the  Elasticity  of  Volume,  is^the  reciprocal 
of  this,  and  is  p  -=-  8ij/b  =  &•  The  work  done  on  compression  is  \p-fo  = 


Of 

V 


260  SOLIDS.  [CHAP. 

Shearability  varies  greatly  in  solids ;  compare  steel  and  india- 
rubber  ;  the  latter  is  much  more  readily  pressed  out  of  shape. 

Shearability  is  measured  by  the  Shear  ( =  tan  0,  Fig.  25)  produced,  per 
unit  of  Tangential  Force  applied  per  unit  of  area,  when  AB  and  CD  (Fig.  25) 
are  one  unit  distance  apart ;  that  is,  it  is  tan#./f.  The  reciprocal  of  this  is 
the  Rigidity,  the  Resistance  to  Transverse  Distortion,  n,  =  f /tan  0 :  a  shear- 
ing force  f  will  produce  a  shear  tan  6  =  f /n ;  and  to  produce  a  given  shear, 
tan  0,  the  shearing  force  applied  per  sq.  cm.  must  be  f  =  n  •  tan  0,  propor- 
tional to  the  rigidity  n- 

The  amount  of  Work  done  in  producing  a  given  shear,  tan  0,  in  a  layer 
whose  thickness  is  d,  is  equal  to  the  product  of  the  average  Resistance,  f /2, 
overcome  by  the  shearing  force  f  (=  n  •  tan  0),  into  the  space  (=  d  •  tan  6) 
through  which  the  moving  plane  AB  (Fig.  25),  or  CD  (Fig.  99),  is  displaced 
parallel  to  itself  ;  that  is,  it  is  \n-d-  tan'20  =  Jf  •  d  -  tan  0,  for  each  sq.  cm. 
of  the  moving  plane  AB  or  CD  so  displaced;  or  £n  •  tan2 0  =  £f  •  tan  0  for 
each  cub.  cm.  of  volume  so  sheared. 

Extensibility  —  Inextensibility.  —  Some  substances  can, 
like  indiarubber,  be  extended  greatly  by  the  application  of  a 
stretching  force  or  Traction  :  others,  like  baked  clay,  very  little. 
When  bodies  are  so  treated,  they  mostly  become  thinner  at  the 
same  time.  The  ratio  of  the  Elongation  produced  to  the  Trac- 
tion t  is  the  "  extensibility  " ;  this  Elongation  being  measured 
by  the  ratio  of  the  increase  in  length  to  the  original  length. 

Under  a  given  longitudinal  traction,  t  dynes  per  sq.  cm.  of  cross-section, 
E,  the  actual  lengthening,  is  equal  to  I- 1  -  (l/3n  4-  l/9ft),  where  I  is  the 
length  of  the  rod  or  wire  in  centimetres;  the  Elongation  E/L  is  therefore 
t  •  (l/3n  +  l/9fc)=  A  •  t,  where  A  is  the  Coefficient  of  Extensibility;  or  it 
is  AT/o,  where  T  is  the  Total  Tension  applied,  in  dynes,  and  o  is  the  cross- 
section  of  the  rod  or  wire  employed. 

At  the  same  time,  the  rod  thins  out,  unless  it  be  like  cork,  exception- 
ally compressible ;  its  transverse  measurements  are  all  diminished  by 
t  •  (l/6n  —  l/9ft)  cm.  per  linear  centimetre.  The  ratio  of  this  proportionate 
Contraction  per  cm.  to  the  simultaneous  proportionate  Elongation  is 
(3ft  —  2n)  -«- (6ft  +  2n),  which  is  called  Poisson's  Ratio. 

When  t  =  1,  E/l  =  A  ;  A  therefore  measures  the  Elongation  produced 
by  a  Traction  of  one  dyne  per  sq.  cm.  cross-section.  If,  on  the  other  hand, 
E  =  Z,  t  =  I/A;  and  I/A  then  measures  the  Traction  per  sq.  cm.  which 
would  be  necessary  in  order  to  double  the  length  of  the  rod  or  wire,  if  that 
strain  could  be  effected  without  rupture. 

In  Cast  Steel,  tempered,      .     .     .    A  =  {1  -=-  (2520,000000  x  981)} 

Wrought  Iron »»{!•*•  (2000,000000  x  981)} 

Copper „   „  (1  -f-  (1050,000000  x  981)} 

Wood „   „  (1  -4-      (10,000000  x  981)} 

Leather „   „  {1  -=-       "    (175000  x  981)} 

Fresh  Bone „„{!-*-    (230,466000  x  981)} 

Tendon »»{!•*•      (16,341000  x  981)} 

Nerves „   „  {1  -        (1,890000  x  981)} 

Living  Muscle  at  Rest    .     .     .    „   „  (1  -         '    (95000  x  981)} 
Arteries „„{!-=-  (5200  x  981)} 


x.]  EXTENSIBILITY".  261 


Problem. 

How  many  grammes'  weight  would  be  necessary  in  order  to  double  the 
length  of  a  piece  of  steel  wire  1  sq.  mm.  in  cross-section,  if  that  were  possi- 
ble? Here  t  =  E/IK  =  I/ A,  for  E  =  1;  therefore  t  =  (2520,000000  x  981) 
dynes :  and  t  =  T/o ;  whence  the  Total  Tension  required  is  T  =  t  •  o  = 
0-Olt  =  0-01  x  (2520,000000  x  981)  dynes  =  the  weight  of  T  /  g  = 
25,200000  grammes. 

French  engineers  are  in  the  habit  of  reducing  these  inconveniently  large 
physical  constants  by  expressing  extensibility  in  terms  of  the  number  of 
kilos.'  weight  which  would  be  required  to  double  the  length  of  a  bar  whose 
sectional  area  is  one  square  millimetre :  the  resultant  numbers  are  -nnnnnr 
of  those  obtained  when  the  extensibility  is  measured  in  terms  of  the  number 
of  grammes'  weight  which  would  be  required  to  double  the  length  of  a  bar 
whose  sectional  area  is  one  square  centimetre. 

Muscles  are  more  extensible  when  they  are  in  a  state  of  contraction  than 
when  they  are  at  rest ;  and  if  a  muscle  when  loaded  by  a  certain  weight  be 
stimulated  to  contraction,  the  mere  effort  to  contract  may  so  diminish  the 
resistance  to  extension  or  increase  the  extensibility  that  the  contracting 
effort  may  be  more  than  counterbalanced  by  the  mechanical  stretching  of 
the  muscle  produced  by  the  weight  hanging  upon  it,  and  the  overloaded 
muscle  may  actually  stretch  when  stimulated  to  contract.  Muscles  also 
become  a  little  less  resistant  or  more  extensible,  under  a  given  load,  shortly 
after  death. 

•There  is  no  substance  of  which  wires  or  rods  could  be  loaded 
with  indefinite  weights,  or  even  with  such  weights  as  would 
double  the  length :  there  is  for  each  substance  a  special  limit  of 
tenacity  or  cohesion,  when  extension  can  go  no  farther,  and  the 
rod  is  ruptured.  This  breaking  weight,  per  sq.  cm.,  measures 
the  cohesion. 

According  to  Wertheim,  bone  ruptures  when  800,000  grammes  are  sus- 
pended on  it,  per  sq.  cm.  of  its  cross-section ;  tendon,  625,000  ;  nerve,  135,100 ; 
veins,  18,500 ;  arteries,  13,700 ;  muscle,  4500.  Thus  a  nerve  whose  section 
is  i  sq.  cm.  could  bear  a  stretching  force  equal  to  the  weight  of  33-7  kilo- 
grammes or  over  5  stone ;  but  the  danger  of  stretching  an  artery  or  a  vein 
by  mistake  is  obvious.  There  is  a  great  difference  in  the  breaking  weight 
of  the  same  tissue  in  persons  of  different  age  and  habit.  Wertheim  found 
that  the  fibula  of  a  young  man  of  thirty  had  a  breaking  weight  of  1,503,000 
grammes  per  centimetre,  while  that  of  the  same  bone  in  an  old  man  of 
seventy-four  was  reduced  to  432,500. 

One  of  the  highest  breaking-weights  is  that  of  steel  pianoforte  wire. 
Wire  1  mm.  in  diameter  may  sustain  a  pull  equal  to  142  tons. per  sq.  inch, 
or  22,120000  grammes'  weight  per  sq.  cm. 

The  reciprocal  of  the  Coefficient  of  Extensibility,  A,  is  I/A, 
the  Coefficient  of  Resistance  to  Extension,  or  Young's  Modu- 
lus, 2 ;  it  is  the  fraction  (Longitudinal  Traction  per  sq.  cm.  of 
cross-section) -f- (the  Elongation  produced). 


262  SOLIDS.  [CHAP. 

This  is  (l/3n  +  l/9k)-]  =  9nfc/(3ft  +  n)  =  g;  and  in  order  to  produce  a 
given  lengthening  or  extension  E  in  a  rod  of  a  given  original  length  I,  we  must 
apply  a  longitudinal  traction  t  =  g  •  £//  dynes  per  sq.  cm.  of  cross-section. 

In  steel,  for  example,  g  =  1/A  =(2520,000000  x  981),  in  dynes  per  sq. 
cm. ;  and  this  is  equal  to  the  Weight  of  g/#  =  2520,000000  grammes'  mass 
suspended  per  sq.  cm.  cross-section  ;  or  to  the  weight  of  g/#p  =  (2520,000000 
-T-  7-8)  linear  cm.  of  the  wire  which  is  being  experimented  upon,  whatever 
be  its  cross-section. 

The  Work  done  in  producing  extension  is  the  average 
Resistance  x  the  space  through  which  it  is  overcome ;  and  this 
is  equal  to  half  the  product  of  the  Total  Tension  into  the 
Extension. 

If  a  rod  be  exposed  to  a  traction  t  dynes  per  sq.  cm.  of  its  cross-section, 
stretching  will  go  on  until  the  ultimate  resistance  arrived  at  is  in  equilib- 
rium with  the  traction.  When  this  is  the  case,  g  •  E/l  =  t.  The  average 
resistance  encountered  by  the  traction  is  half  this,  or  ^g  •  E/l  =  £t,  per  sq. 
cm.  of  cross-section  of  the  rod.  The  space  through  which  the  resistance  is 
overcome  is  s  —  E.  The  work  done  =  average  resistance  x  space  =  -?g-  Ez/l 
=  |t  •  E,  per  sq.  cm.  of  cross-section  of  the  rod ;  or  ^g  •  o  •  E2/l  =  £T .  E  for 
the  whole  rod,  of  cross-section  o. 

Linear  Compressibility  follows  the  same  laws  as  extensi- 
bility. Within  narrow  limits  the  coefficients  of  compressibility 
and  of  extensibility  have  the  same  value,  A.  Excessive  com- 
pression leads  to  crushing,  by  lateral  dilatation  ;  and  each  sub- 
stance has  its  own  Crushing  Weight,  found  by  experiment  on 
masses  of  determinate  size. 

Flexibility. — In  every  rod  undergoing  flexion,  if  this  be 
due  to  the  weight  of  a  mass  suspended  from  a  free  end,  there 
must  be  a  certain  extension  of  the  upper  aspect  of  the  rod,  a 
compression  of  the  lower,  and  a  Neutral  Line  between,  which 
retains  its  original  length.  If  the  flexure  be  due  to  weight 
pressing  down  the  middle  of  the  rod  which  is  supported  at  its 
extremities,  the  extension  is  in  the  lower  aspect  of  the  rod,  the 
compression  in  the  upper.  In  the  former  case  a  cut  in  the  upper 
aspect  Avould  weaken  the  rod ;  in  the  latter  the  same  effect  would 
only  be  produced  by  a  cut  on  the  lower  aspect.  Flexion  may 
bring  about  compression  and  extension  beyond  the  range  of  the 
breaking  or  crushing  strengths,  and  the  body  may  thus  be  broken. 
If  this  occur  before  there  has  been  any  perceptible  flexion,  the 
body  is  said  to  be  brittle :  if  it  allow  a  considerable  range  of 
flexion  it  is  said  to  be  tough  —  it  bends  much  before  breaking. 
The  crystalline  or  granular  or  fibrous  structure  of  a  substance 
has  much  to  do  with  its  brittleness  or  toughness.  For  example, 
tin,  which  is  very  crystalline,  is  very  brittle ;  wrought-iron  axles, 


x.]  FLEXIBILITY.  263 

which  are  at  first  fibrous  and  very  tough,  are  subject  to  a  molec- 
ular rearrangement  facilitated  by  vibration,  and  become  crystal- 
line and  brittle. 

The  amount  of  Flexure  of  a  rod  depends  upon  Young's  Modulus. 
Thus,  if  a  beam,  supposed  weightless,  be  fixed  at  one  end,  and  if  its  free 
end  be  loaded  with  a  mass  m ;  then,  if  its  length  be  /,  its  horizontal  breadth 
6,  and  the  vertical  depth  of  its  rectangular  section  be  d,  the  free  end  will 
descend  through  a  height  h  =  4:ingl3/bdsg. 

Torsibility  of  a  solid  may  be  measured  in  the  simplest  case — that  of  a  rod 
or  wire  —  by  specifying  the  angle  through  which  a  unit  force,  applied  at  a 
distance  of  1  cm.  from  the  axis  of  the  wire,  can  twist  it.  This  angle  is  1/t ; 
and  it  is  inversely  proportional  to  the  fourth  power  of  the  radius  of  the 
wire.  Its  reciprocal,  t,  is  the  resistance  to  torsion. 

This  angle,  1/t  =  2//rt7rrt4,  where  /  is  the  length  of  the  twisted  wire,  a 
its  half  diameter,  and  tt  its  coefficient  of  rigidity  to  transverse  distortion. 

If  a  force  F  be  applied  at  the  end  of  a  lever  r,  the  torque  or  twisting 
moment  is  Fr,  and  the  angle  of  twist  becomes  0  =  Fr  •  2//rt7ra4  =  Fr/t. 
Hence,  Fr  —  1 0.  Conversely,  if  such  a  wire  is  to  be  twisted  through  an  angle 
6,  the  torque  to  be  applied  must  be  equal  to  t0,  whatever  the  distance  of  the 
point  of  application  of  the  twisting  force  F  may  be. 

The  work  done  in  producing  torsion  0  is  \i&  •  0  =  liB2  =  %  Fr  •  9. 

If  a  bar,  loaded  so  that  its  Moment  of  Inertia  is  N,  be  suspended  hori- 
zontally at  its  midpoint  by  a  wire,  and  if  it  be  turned  round  its  point  of 
support  through  a  horizontal  angle  6,  it  will  oscillate  with  a  period 
T  =27r-\/N/t  =  27rV-N//n7ra4,  where  t,  /,  n,  and  a  are  data  pertaining  to 
the  suspending  wire,  twisted  by  the  oscillation. 

A  bar  suspended  at  its  midpoint  by  a  wire  capable  of  twist,  and  acted 
upon  by  a  torque  or  twisting  moment  Fr,  will  rotate  and  cause  the  lower  end 
of  the  wire  to  rotate  with  it ;  if,  however,  the  upper  end  of  the  wire  be  at 
the  same  time  twisted  in  an  opposite  sense,  to  so  great  an  extent  that  the 
reverse  twisting  moment  due  to  the  torsion  0  of  the  wire  itself  becomes 
equal  to  Fr,  there  is  then  no  change  in  the  position  of  the  suspended  bar  at 
the  lower  end.  The  counter-force  at  the  point  of  application  of  the  force  F 
to  the  bar  is  t#/r;  this  is  numerically  equal  to  F,  which  it  holds  in  check. 
Any  other  force  F',  similarly  applied,  would  be  equal  to  t@'/r;  whence 
F  :  F' : :  0 :  6' ;  and  Forces  may  be  compared  by  observing  the  ratio  between 
the  angles  of  opposite  twist  that  must  be  given  to  the  one  end  of  a  wire  or 
fibre  in  order  to  prevent  those  forces,  similarly  and  successively  applied, 
from  causing  twist  at  the  other  end. 

By  means  of  fibres  of  quartz  ^-$  cm.  thick,  obtained  by  suddenly  draw- 
ing asunder  a  drop  of  melted  quartz,  Professor  Vernon  Boys  has  repeated 
Cavendish's  experiment  (p.  202),  and  measured  the  attraction  between 
masses  of  800  grammes  and  1  gramme,  at  such  a  distance  that  this  amounts 
to  about  T^oVoo  dyne>  or  the  weight  of  ^Vooooo  gramme.  With  fibres 
-^~  times  this  diameter,  it  would  be  easy  to  measure  the  attraction  between 
two  No.  5  shot,  or  awo  °f  ^ne  former  amount.  These  fibres  are  very  per- 
fectly elastic,  are  unaffected  by  moisture,  and  have  extreme  tensile  strength. 

The  Dimensions  of  fc,  n,  and  g  are  [M/LT2]  ;  those  of  t  are  [ML2/T2]. 

Elasticity.  —  "  Elasticity  is  the  property  in  virtue  of  which 
a  body  requires  force  to  change  its  bulk  or  shape,  and  requires  a 


264  SOLIDS.  [CHAP. 

continued  application  of  the  force  to  maintain  the  change,  and 
springs  back  when  the  force  is  removed ;  and  if  left  at  rest  with- 
out the  force,  does  not  remain  at  rest  except  in  its  previous 
bulk  and  shape  "(Lord  Kelvin). 

Power  of  Restitution.  —  There  are  two  properties,  Resis- 
tance and  Restitution,  which  must  concur  in  any  given  body 
before  it  can  be  said  to  be  elastic.  Resistance  is  not  the  only 
criterion  of  Elasticity.  A  body  may  resist  extension,  compres- 
sion, torsion,  shear,  and  yet  not  be  elastic.  In  order  that  it  may 
be  perfectly  elastic,  it  must  have  all  the  following  properties:  — 

(1.)  It  must  offer  a  definite  resistance  to  distortion. 

(2.)  The  distortion  is  not  permanent,  and  if  the  deforming 
force  be  removed,  the  distorted  body  springs  back  to  its  orig- 
inal form  or  bulk. 

(3.)  The  distorting  force  must  be  continuously  maintained 
in  order  to  keep  up  the  distortion. 

(4.)  So  long  as  the  distorting  force  is  kept  up,  there  is  a 
counter-pressure  or  restitution-pressure  (P)  developed  and 
sustained  in  the  elastic  substance.  As  this  holds  the  deforming 
force  (F)  in  check,  and  is  in  equilibrium  with  it,  thus  setting 
up  a  condition  of  stress  in  the  substance,  it  must  be  numerically 
equal  and  opposite  to  it ;  P  +  F  =  0. 

(5.)  The  restitution-pressure  does  not  become  diminished 
by  lapse  of  time. 

In  both  extension,  compression,  shear,  and  twist,  the  Restitution- 
Force  is  opposite  and  equal  to  the  Displacing  Force.  Thus,  in 
Extension,  P  +  T  =  0,  or  P  =  -  T  ;  i.e.,  P  =  -  g  •  E  •  o/l,  or  P  oc  -  E.  If  the 
distortion  (E  •  o/l)  =  1,  P  =  —  g,  and  the  restitution-force  is  represented  by  a 
number,  the  Coefficient  of  Restitution  (or  "  coefficient  of  elasticity  "),  which 
is  equal  to  the  Coefficient  of  Resistance  to  Extension.  In  general,  when  the 
Deformation  (tan  0,  Sij/ij,  E/l,  or  the  angle  of  torsion  0)  is  unity,  the  resti- 
tution-force per  unit  of  area  (or,  in  the  last  case,  the  restitution-torque)  is 
represented  by  the  coefficient  of  resistance  to  deformation,  n,  k,  g,  or  t,  as  the 
case  may  be  ;  and  when  the  Deformation  has  any  other  value,  the  restitution- 
force  per  unit  of  area  (or,  in  the  case  of  torsion,  the  restitution-torque)  is 
equal  to  the  product  of  that  value  into  the  corresponding  Coefficient  of 
Resistance.  Hence  the  restitution-pressure  on  any  particle  is  proportional 
to  the  Displacement  of  that  particle,  and  is  oppositely  directed. 

Under  any  given  distortion  within  the  limits  of  restitutive 
power,  the  restitution-pressure  is  equal  to  the  product  of  the 
Coefficient  of  Restitution  into  the  distortion;  the  coeffi- 
cient of  restitution  being  numerically  identical  with  the  recipro- 
cal of  the  deformability.  It  is  usual  to  profess  to  measure  the 
elasticity  of  a  solid  by  a  "  coefficient  of  elasticity"  which  is  stated 


x.]  ELASTICITY.  265 

to  be  equal  to  the  resistance  to  distortion.  There  is  an  equality, 
a  numerical  identity,  between  the  Resistance  to  distortion  and 
the  Coefficient  of  Restitution  (upon  which  the  amount  of  resti- 
tution-pressure depends),  provided  that  any  one  system  of  units 
be  strictly  adhered  to,  that  the  body  be  perfectly  elastic,  and  that 
the  distortion  be  unity.  It  seems,  however,  strange  to  set  up  a 
method  of  measuring  elasticity  based  on  a  tacit  fundamental 
assumption  that  the  bodies  dealt  with  are  perfectly  elastic." 

If  there  be  two  bodies,  of  which  one  has  a  low,  the  other  a 
high  coefficient  of  restitution,  and  if  the  same  displacement  be 
effected  in  both,  the  restitution-pressures  in  the  two  substances 
differ  in  the  same  ratio  as  their  respective  coefficients :  and  in 
these  two  bodies  the  relative  amounts  of  work  stored  up  in  the 
form  of  tensional  or  potential  energy  also  differ  in  the  same 
ratio.  Elasticity  thus  presents  two  aspects,  the  Statical  and 
the  D}^namical.  On  liberation  of  a  strained  body,  the  whole  of 
the  energy  stored  up  in  it  may  be  restored  in  the  kinetic  form. 

This  restitution  may  be  due  to  the  solid  body  being  a  con- 
servative system  of  particles,  a  small  displacement  amongst 
which  acts  as  a  disturbance  of  masses  in  stable  equilibrium :  by 
such  a  displacement  an  aggregate  force  is  called  into  action 
which  tends  to  produce  restoration  to  the  original  form  or  bulk. 
In  an  elastic  body  the  greater  the  displacement  or  distortion  the 
greater  the  restitution-pressure,  and  that  in  direct  proportion 
(Hooke's  Law). 

Perfect  and  Imperfect  Elasticity.  —  A  body  is  perfectly 
elastic  when  any  given  stress  produces  no  permanent  set  or 
deformation,  restitution  being  always  complete.  It  is  imper- 
fectly elastic  when  it  does  permanently  retain  such  a  set.  It  is 
said  to  be  strained  beyond  its  Limits  of  Elasticity  when  it  is 
so  far  strained  that  it  retains  such  a  set :  it  is  said  not  to  be 
strained  beyond  its  limits  of  elasticity  when  it  is  not  deformed 
so  far  that  it  cannot  exactly  return  to  its  original  form  or  bulk. 
When  the  limits  of  Elasticity  are  narrow,  as  in  the  case  of  lead 
(which,  though  exceedingly  easily  bent  so  as  to  take  a  permanent 
set,  can  yet  be  induced  to  enter  into  vibration,  and  must  there- 
fore be  elastic  within  narrow  limits),  the  body  is  again  said  to 
be  "imperfectly  elastic,"  or  to  possess  little  elastic  toughness. 
When  it  can  suffer  distortion  within  wide  limits  without  taking 
up  a  permanent  set,  it  is  said  to  have  great  elastic  tough- 
ness; and  a  body  which  has  infinitely  wide  limits^'of  elasticity 
is  said  to  be  perfectly  elastic.  There  is  no  body  perfectly 


266  SOLIDS.  [CHAP. 

elastic,  but  any  body  may  within  the  limits  of  its  elasticity  be 
considered  as  a  perfectly  elastic  body. 

In  popular  language  a  body  is  said  to  be  very  elastic  when 
it  has,  like  indiarubber,  great  elastic  toughness  —  i.e.,  when  it 
can  be  distorted  through  wide  ranges  without  taking  up  a  per- 
manent set;  but  this  use  of  the  word  should  be  discouraged 
in  favour  of  that  use  in  which  it  is  made  to  signify  conjoined 
powers  of  Resistance  to  deformation,  and  of  Restitution  of 
shape,  of  bulk,  and  of  work  done  upon  the  elastic  object. 

The  elastic  toughness  exemplified  in  a  Toledo  sword-blade 
must  be  distinguished  from  the  ordinary  ultimate  toughness  or 
breaking  toughness ;  the  former  may  be  much  less  than  the  latter. 

Residual  Restitution  with  Deferred  Restitution-pressure.  —  When 
a  body  which  has  been  distorted  is  left  to  itself  without  vibration,  it  may, 
when  it  has  come  to  rest,  be  fixed  between  supports ;  it  then  exerts  no  elastic 
pressure ;  but  in  the  course  of  a  little  time  it  will  be  found  to  be  exerting 
an  elastic  pressure  which  has  been  in  the  meantime  slowly  developed,  and 
which  tends  to  restore  the  body  more  nearly  to  its  normal  condition. 
Mechanical  disturbances  —  such  as  vibration,  shaking,  jarring,  etc.,  — which 
allow  the  molecules  to  glide  past  one  another,  facilitate  the  development  of 
this  deferred  restitution.  Boltzmann  found  that  successive  torsions,  differ- 
ing in  amount  and  in  sense,  caused  the  subsequent  successive  emergence  of 
deferred  restitution-pressures  whose  order  of  succession  was  the  inverse  of 
that  of  the  torsions  which  had  given  rise  to  them. 

Vibrations  due  to  Elasticity.  —  When  a  body  is  distorted, 
not  beyond  the  limits  of  elasticity,  and  liberated,  the  work  done 
upon  it  is  restored.  The  body  exactly  regains  its  original  form 
or  bulk,  but  at  the  moment  of  complete  restitution  the  energy 
possessed  by  the  body  (if  perfectly  elastic)  has  wholly  assumed 
the  kinetic  form,  and  the  body  passes  rapidly,  if  it  be  free  to  do 
so,  through  its  mean  form  or  bulk,  and  suffers  an  equal  dis- 
tortion or  alteration  of  volume  in  the  opposite  sense.  Again  a 
restitution-pressure  is  developed,  and  the  body  swings  back 
through  its  mean  position.  This  is  repeated,  and  thus  we  have 
vibrations  produced  as  the  result  of  elasticity.  The  force  bring- 
ing back  every  particle  towards  the  mean  position  is  proportional 
to  the  displacement  from  that  mean  position,  and  this  is  the  cri- 
terion of  harmonic  motions.  Hence  in  a  solid  body,  which  is  a 
system  of  particles,  any  displacement  sets  up  an-  intermolecular 
restitution-pressure,  which  results  in  harmonic  motion  (Fourier- 
motion)  of  the  separate  particles,  and  the  particles  of  a  disturbed 
tuning-fork  or  stretched  string  may  execute  harmonic  vibrations, 
particles  equidistant  from  one  another  generally  assuming  equal 


x.]  ELASTICITY.  26  T 

differences  of  phase  in  their  respective  S.H.M.'s.  The  whole 
body  executes,  like  a  pendulum,  isochronous  vibrations;  as,  for 
example,  the  vibrating  mainspring  of  a  watch. 

Viscosity  of  Elastic  Solids.  —  When  an  elastic  body  has 
entered  into  vibration  it  appears  more  or  less  rapidly  to  lose  its 
energy;  its  vibrations  wane  away.  This  waning  away  is  due  to 
the  "  viscosity  "  of  the  solid :  the  energy  of  vibration  becomes 
converted  into  heat.  The  amplitude  of  each  successive  oscilla- 
tion bears  to  that  of  the  one  immediately  preceding  a  constant 
ratio.  If  the  amplitudes  of  the  first  and  second  oscillations  be 
1 :  a,  the  third  will  be  a2,  the  nth  will  be  an~l.  On  account  of  this 
viscosity  a  tuning-fork  cannot  be  made  of  lead  or  zinc,  the  vibra- 
tions of  which  too  rapidly  die  away :  but  even  pipe-clay  can 
slightly  vibrate  in  this  manner.  This  "  Viscosity  "  is  what  is 
frequently  understood  by  the  term  imperfect  elasticity :  the 
restitution  of  form  or  bulk  may  be  perfect,  but  that  of  energy  is 
not,  for  some  of  it  is  dissipated  in  the  form  of  Heat. 

Fatigue  of  Elasticity.  —  When  a  tuning-fork  is  kept  (as 
by  an  electromagnetic  arrangement,  p.  735)  continuously  vibrat- 
ing for  a  long  time,  it  stops  almost  instantaneously  when  the 
exciting  cause  is  removed.  The  steel  requires  periods  of  rest : 
if  it  be  kept  continuously  vibrating  it  has  a  tendency  to  become 
viscous,  and  to  return  comparatively  slowly  to  its  mean  form 
after  each  displacement. 

Effect  of  repeated  variations  of  Stress.  —  Metal  requires 
intervals  of  rest  in  order  to  enable  it  to  recover  from  fatigue ; 
and  if  these  be  not  allowed,  it  will  break  down  and  fracture 
under  the  repeated  application  of  forces  far  less  than  the  break- 
ing weight.  The  greater  the  variations  of  stress,  and  the  more 
frequent  their  recurrence,  the  sooner  does  the  metal  collapse. 

Velocity  of  propagation  of  a  compressional  wave-motion.  —  The 

elastic-restitution-pressure  developed  in  consequence  of  a  Compression  varies 
as  fc,  the  coefficient  of  restitution ;  the  acceleration  produced  by  the  restitu- 
tion-pressure varies  as  the  restitution-pressure ;  the  velocity  in  the  circle  of 
reference  (in  S.H.M.)  varies  as  the  square  root  of  the  acceleration;  the 
velocity  of  propagation  varies  as  the  velocity  in  the  circle  of  reference  :  there- 
fore the  velocity  of  propagation  varies  as  the  square  root  of  fe,  the  coefficient 
of  restitution,  or  of  resistance  to  compression. 

Given  the  same  elastic  pressures  and  the  same  work  done  upon  two  bodies 
whose  respective  densities  are  p  and  p,,  the  energy  being  equal,  the  respec- 
tive velocities  produced  must  vary  inversely  as  the  square  roots  of  p  and  pr 
Hence  v  varies  as  Vfe/p ;  and  it  can  be  shown  that  no  multiplier  intervenes, 
and  that  v  is  equal  to  Vfc/p. 

In  this  it  is  assumed,  however,  that  in  a  compressional  wave  the  rigidity 


268  SOLIDS.  [CHAP. 

n  may  be  neglected.  This  is  practically  the  case  in  gases,  to  which  the 
formula  v  —  "v/ft/p  is  applicable,  subject  to  further  discussion  (p.  324)  as  to 
what  the  true  value  of  ft  may  be  ;  but  in  solids  the  rigidity  does  come  into 
play,  even  in  compressional  waves  ;  and  for  such  waves,  in  a  tridimensional 
elastic  solid,  v  =  V(ft  +  tn)  •*•  P- 

The  velocity  of  propagation  of  a  transversely  distortional  vibration  is 
Vn/p  in  a  tridimensional  medium,  and  Vt/p  along  a  uniform  and  perfectly 
flexible  stretched  string  ;  that  of  a  longitudinal  vibration  is  Vg/p  along  a 
wire  or  rod,  stretched  or  unstretched  ;  that  of  a  torsional  vibration  is 
Vn/p,  along  a  wire  or  rod. 

Hence  along  a  steel  wire  (p  =  7-85,  g  =  981  x  2520,000000,  ft  =  1-84  x  1012, 
fl  =  0'95  x  1012),  a  longitudinal  compressional  wave,  such  as  a  sound-wave, 
will  travel  with  a  velocity  v  =  ^U/p  =  V981  x  2520,000000  -5-  7-85  = 
V314,919,745,223  =  561177  cm.  per  second  =  5611-77  metres  per  second; 
whereas  in  an  extended  mass  of  steel  the  rate  of  propagation"  of  a  compres- 
sional wave  will  be  V(ft  +  fn)  -*•  p  =  655000  cm.  per  second,  and  that  of 
a  pure  transverse-distortional  wave  (without  change  of  volume)  would  be 
=  348000  cm.  per  second. 


The  property  of  Elasticity  is  not  inconsistent  with  brittle- 
ness  :  glass  has  very  narrow  limits  of  pliability,  and  is  accord- 
ingly brittle,  but  within  these  limits  it  is  eminently  elastic. 

Physiological  Examples  of  Elasticity.  —  The  whole  ligamentous 
system  affords  examples,  and  many  of  the  bones  also  possess  this  property. 
The  ligaments  of  the  elastic  arch  of  the  foot,  the  vertebral  ligaments,  and 
the  intervertebral  discs  acting  against  the  down-dragging  weight  of  the  vis- 
cera; those  ligaments  which  by  their  very  molecular  constitution  (however 
this  may  be  accounted  for)  are  always  on  the  stretch,  such  as  the  elastic  liga- 
ment of  the  eye,  the  filled  arteries,  the  ligaments  of  the  symphysis  pubis  ;  the 
combined  flexion  and  twist  of  the  ribs  in  inspiration  and  their  elastic  resti- 
tution in  expiration  ;  the  ligaments  of  the  lamellibranch  shell,  the  tracheae 
of  insects,  —  all  furnish  examples  of  Elasticity. 

The  Mechanical  Advantages  of  Elasticity.  —  These  can 

be  studied  in  a  well-hung  vehicle  with  light  springs.  Any  sud- 
den jolt  or  jar  is  not  communicated  to  the  body  of  the  vehicle 
with  its  original  abruptness,  but  gives  rise  to  a  wave-motion, 
which  lifts  the  body  of  the  carriage  and  allows  it  to  oscillate 
until  it  returns  to  relative  rest.  If  a  person  jump,  landing  on 
his  feet,  the  shock  is  partly  broken  by  the  elastic  arches  of  the 
feet;  but  further,  before  it  reaches  the  brain  it  has  to  pass 
through  a  succession  of  elastic  discs,  the  ultimate  effect  of 
whose  intervention  is  a  gradual  and  not  an  abrupt  arrest  of 
the  downward  movement  of  the  head.  Were  it  not  for  this 
the  brain  would  be  ruptured  by  exceedingly  small  leaps. 

Work  is  directed  by  elastic  intermediaries  so  that  it  may 
become  useful  and  not  harmful.  Jolts  and  jars  —  which,  as 


X.] 


ELASTICITY. 


269 


we  have  seen  under  Momentum,  involve  the  disappearance  of 
Energy  in  doing  harmful  mechanical  work  —  are  converted 
into  smooth  wave-motions.  Thus  energy  is  saved,  mischief  pre- 
vented, and  the  mechanism  rendered  more  durable.  If  a  person 
run  over  a  gravelly  road  with  a  heavy  vehicle  attached  to  his 
person  by  a  non-elastic  cord,  he  will  feel  greatly  bruised.  If 
he  interpose  a  steel  spring  or  a  thick  piece  of  indiarubber 
between  himself  and  the  vehicle,  the  pain  is  infinitely  lessened 
,and  the  actual  energy  expended  is  about  25  per  cent  less  (Prof. 
Marey  and  M.  Him) ;  work  has  not  been  spent  in  jolting  and 
jarring  himself  and  the  vehicle. 

The  use  of  elastic  intermediaries  suggests  itself  in  all  cases  where  jolts 
of  any  kind  would  be  injurious.  Compare  the  effects  of  an  involuntary 
movement  made  by  a  patient,  whose  limbs  are  under  extension  by  a  weight 
and  non-elastic  cord,  with  the  effect  of  the  same  movement  when  a  light 
spring  intervenes  between  the  limb  affected  and  the  extending  weight.  The 
spring  persists  and  keeps  up  the  tension,  but  it  yields  to  the  momentary 
twitch ;  the  weight  rises  a  little  and  sinks  back  to  its  former  position. 

If  a  piece  of  thin  cord,  tied  round  a  somewhat  heavy  mass  of  iron,  be 
pulled  with  a  jerk,  it  may  snap  without  lifting  the  heavy  mass ;  whereas,  if 
an  indiarubber  band  be  interposed  somewhere  between  the  hand  and  the 
iron,  the  same  jerk  may  first  extend  the  indiarubber  band  which,  in  its 
turn,  may  then  lift  the  heavy  mass. 

Strength  of  structures  as  depending  on  their  form.  — 

This  is  the  special  study  of  the  Engineer.  Here  we  can  only 
state  a  few  principles. 

Galileo  found  that  a  given  weight  of  material  if  disposed  in 
solid  bars  presents  less  resistance  to  crushing  or  bending  than 
the  same  material  arranged  in  the  form  of  tubes,  provided  that 
the  walls  of  these  tubes  be  not  excessively  thin.  The  following 
table  is  from  Weisbach's  Engineering  Mechanics :  — 


Resistance  to  Breaking. 

Resistance  to  Crushing. 

Massive  cylinder,  rad.  =  4. 
Mass  =  7rr2/p  =  167r/p. 

1 

1000 

1000 

Hollow;  radii  5  and  3. 

) 

Mass  =  2oirlp  —  9?rA.p  = 

1700 

2125 

Ifcr/p. 

) 

Massive  cylinder,  rad.  =  5. 

1000 

1000 

Hollow,  radii  5  and  4. 

87040 

870-40 

5  and  3. 

590-40 

>69040 

270  SOLIDS.  [CHAP.  x. 

Hence,  mass  for  mass,  the  hollow  tube  is  stronger :  diameter  for 
diameter,  the  solid  is  the  stronger.  The  strongest  tube  for  all 
purposes  has  the  relative  radii  11  and  5. 

Examples  of  this  kind  of  structure  we  find  in  the  hollow  stems  of  plants, 
in  the  feathers  of  birds,  in  the  long  bones  of  the  human  body. 

Economy  of  material  may  be  carried  still  farther  by  the 
adoption  of  the  lamellar  or  trabeculated  structure.  We  see  in 
lattice-girders  how  the  judicious  arrangement  of  struts  which 
support  each  other  makes  a  structure  strong  enough  for  all 
practical  purposes,  though  very  light ;  often  much  stronger  than 
if  it  were  burdened  with  the  excessive  weight  of  its  own  sub- 
stance which,  if  it  were  solid,  it  would  have  to  support. 

In  the  spongy  structure  of  bones  we  find  a  similar  arrangement.  In  the 
upper  end  of  the  femur  we  find  a  disposition  of  horizontal,  vertical,  and 
oblique  trabeculae,  which  together  form  a  rigid  triangular  framework  sup- 
porting the  weight  of  the  body.  In  the  astragalus  we  have  a  comparatively 
light  and  porous  structure,  but  the  trabeculae  are  so  arranged  as  to  resist 
and  to  distribute  the  downward  pressure  of  the  body  and  the  compressing 
pressure  exerted  by  those  bones,  the  os  calcis  and  the  scaphoid,  which  abut 
against  it  in  the  arch  of  the  foot. 


CHAPTER  XL 

OF  LIQUIDS. 

THIS  chapter  may  be  divided  into  three  parts,  treating  of  (1) 
Molecular  Actions,  (2)  the  Statics  of  Liquid  Masses, 
(3)  the  Kinetics  of  Liquid  Masses. 

1.  MOLECULAR  ACTIONS. 

Cohesion.  —  If  a  ring  of  iron  wire  be  dipped  into  a  solution 
of  soap,  it  will  be  seen  on  taking  it  out  that  the  cohesion  of  the 
liquid  causes  a  film  to  be  formed,  which  remains  stretched  across 
the  ring.  The  force  of  cohesion  is  also  manifested  by  a  drop  of 
water  hanging  from  a  glass  rod.  Such  a  drop  may  be  gradually 
increased  in  size  until,  at  a  certain  maximum,  its  weight  over- 
comes its  cohesion,  the  water  breaks  asunder,  and  the  drop  falls. 
A  thin  rod  of  glass  or  wire  of  metal  may  similarly  be  fused  at  the 
end,  and  the  fused  drop  may  be  increased  in  size  by  continued 
fusion  until  the  molecular  forces  can  no  longer  counteract  its 
weight.  A  little  water  on  the  end  of  a  thick  glass  rod  will 
retain  a  piece  of  paper  placed  in  contact  with  it,  even  though 
some  grains'  weight  be  suspended  from  the  paper. 

The  above  examples  furnish  us  with  indications  merely,  and  do  not 
enable  us  directly  to  measure  the  attractions  inside  a  liquid.  These  cannot 
be  directly  measured,  because  no  apparatus  can  be  applied  to  the  interior  of 
a  liquid  without  creating  a  new  surface  at  its  own  boundary.  But  we  can 
infer  their  amount.  Referring  to  Fig.  100,  we  see  that  one  of  the  particles 
is  at  the  surface,  and  that  the  molecular  forces  acting  upon  it  are  only  half 
those  acting  upon  an  interior  particle.  To  move  a  particle  from  the  interior 
to  the  surface  would  consume  a  certain  amount  of  work ;  to  remove  it  from 
the  interior  through  the  surface,  as  on  boiling,  would  require  twice  as  much. 
If  we  suppose  some  water  to  be  boiling  in  a  tube  whose  cross-sectional  area 
is  1  sq.  cm.,  and  the  level  in  which  is  maintained  constant,  then,  as  it  is 
known  that  the  Energy  which  must  be  supplied  in  the  form  of  Heat  in 
order  to  boil  away  one  gramme  of  water  is  (p.  390)  equal  to  (536  x  41,593000) 
ergs,  half  this  amount,  or  (536  x  41,593000  -r-  2)  ergs,  would  bring,  molecule 
by  molecule,  one  gramme  of  water  from  the  interior  to  the  surface.  But  as 

271 


272  OF  LIQUIDS.  [CHAP. 

1  gramme  of  water  =  1  cub.  cm.,  and  as  the  area  of  evaporating  surface  is 
1  sq.  cm.,  the  path  of  the  molecules  is  on  the  average  $  cm. ;  and  the  inter- 
nal pressure  overcome  is  {(536  x  41,593000  -^  2)  -H  -£-}  dynes  per  sq.  cm.,  or 
22000  atmospheres.  The  internal  forces  are  thus  seen  to  be  enormous. 

Now  let  the  average  diameter  of  the  molecules  be  taken  as  1/x  cm. : 
then  there  will  be  x3  molecules  in  one  cub.  cm.,  and  a  layer  one  molecule  thick, 
made  up  out  of  one  gramme  or  one  cub.  cm.  of  water,  will  have  x2  mole- 
cules per  sq.  cm.,  and  will  cover  x  sq.  cm.  To  make  such  a  layer  or  film 
out  of  1  cub.  cm.  of  water  would  be  the  same  thing  as  to  bring  1  cub.  cm. 
of  water,  molecule  by  molecule,  from  interior  to  surface  without  evaporation. 
This,  as  we  have  seen,  would  require  (536  x  41,593000  -=-  2)  or  11146,900000 
ergs.  To  produce  in  a  mass  of  water  1  sq.  cm.  of  additional  free  surface 
would  therefore  require  (11146,900000/x)  ergs.  The  numerical  value  of 
this  would  of  course  depend  on  the  diameter  of  the  molecules  of  water;  and 
as  the  diameter  \/x  is  evidently  very  small,  the  divisor  x  is  very  great,  and 
only  a  very  small  part  of  the  internal  attraction  can  make  itself  obvious  at 
the  surface  by  resisting  stretching  or  causing  contraction  of  the  free  surface. 
Still,  a  measurable  proportion  of  it  does  so,  and  gives  rise  to  the  phenomena 
described  in  the  succeeding  paragraphs. 

Surface-Tension.  —  It  has  already  been  remarked  that  the 
molecular  forces  are  most  strikingly  manifest  at  the  surface  of 
a  liquid,  and  that  every  liquid  may  be  regarded  as  bounded  by 
a  superficial  skin  or  film,  which  behaves  like  a  stretched  mem- 
brane, and  which  in  time  reduces  the  contained  liquid  to  that 
form  which  gives  to  the  greatest  cubical  content  the  least  super- 
ficial area.  The  tension  of  this  superficial  film  is  the  Surface- 
Tension  of  the  liquid.  A  raindrop,  a  shot  falling  from  a 
shot-tower,  assumes  the  globular  form  because  the  sphere  is  the 
simplest  geometrical  form  which  fulfils  these  conditions.  A  bub- 
ble of  air  in  water  assumes  a  spherical  form  —  not  perfectly  so 
on  account  of  the  resistance  to  its  ascent.  The  most  convenient 
way  of  studying  the  various  forms  assumed  by  masses  of  liquid 
under  the  influence  of  surface-tension  is  to  relieve  them  of  the 
effect  of  gravity  by  floating  them  in  liquids  of  their  own  specific 
density,  with  which  they  will  not  readily  mix. 

A  mixture  of  alcohol  and  water  is  made,  of  the  same  specific  density  as 
olive  oil.  Masses  of  olive  oil  placed  in  this  fluid  will  neither  rise  nor  sink, 
but  will  assume  the  globular  form.  If  they  be  not  free  to  assume  the  globu- 
lar form,  but  have  limiting  conditions  imposed  upon  them,*  they  may  assume 
geometrical  forms  of  great  interest,  all  having  the  smallest  superficial  area 
possible  under  the  given  conditions.  If,  for  example,  into  such  a  globular 
mass  of  oil,  an  oiled  circular  disc  of  iron  be  suspended,  having  a  diameter 
greater  than  that  of  the  mass,  the  mass  of  oil  will  spread  over  each  face  of  the 
disc,  and  will  form  on  each  side  of  it  a  segment  of  a  larger  sphere.  If  such 

*  Refer  to  an  exceedingly  charming  work  by  M.  Plateau,  Statique  des  Liquides 
soumis  aux  seules  Forces  moUculaires,  a  treasure-house  of  experiments  devised  by 
a  savant  afflicted  with  total  blindness. 


XL]  SURFACE-TENSION.  273 

a  disc  be  brought  up  to  the  globular  mass  by  one  face  only,  the  oil  will  not 
pass  round  the  edge  of  the  disc.  The  curvature  of  the  segments  of  spheres 
produced  may  be  modified  by  adding  or  removing  oil  by  means  of  a  pipette. 
If  a  short  oiled  cylinder  with  open  ends  be  put  into  the  dilute  alcohol,  and  a 
mass  of  oil  inserted  by  means  of  a  pipette,  the  oil  will  fill  the  cylinder  and 
form  a  kind  of  biconvex  lens  of  oil ;  by  means  of  a  pipette,  oil  may  be  taken 
from  the  mass  until  it  becomes  a  biconcave  lenticular  mass ;  or,  if  the  opera- 
tion be  stopped  at  the  right  instant,  a  plane-ended  cylindrical  mass  of  oil 
will  be  obtained.  If  a  circular  ring  be  immersed  in  a  large  mass  of  oil,  and 
some  of  the  oil  be  then  removed,  the  mass  will  assume  a  lenticular  form. 
If  a  little  iron  framework  be  constructed,  representing  in  outline  the  sides 
(one  inch)  of  a  cube,  and  hung  into  a  mass  of  oil  which  is  then  gradually 
removed,  the  mass  of  oil  will  have  part  of  its  periphery  determined  by  the 
iron  framework,  and  will  assume  the  appearance  successively  of  a  cube  with 
convex  sides,  of  a  cube  with  plane  sides,  of  a  cube  with  concave  sides. 

But  we  can  study  the  effect  of  surface-tension  to  even 
greater  advantage  when  we  diminish  the  mass  of  a  liquid  with- 
out decreasing  the  area  of  the  superficial  film.  This  we  can  do 
by  using  soap  films  or  collodion  films. 

Soap  Films.  —  Plateau's  method.  1  part  fresh  moist  Marseilles  soap 
(much  better,  pure  oleate  of  soda)  cut  into  very  small  pieces ;  dissolve  with 
moderate  heat  in  40  parts  by  weight  of  distilled  water.  Filter  through 
moderately-thin  filter  paper.  Mix  thoroughly  15  volumes  of  this  solution 
with  11  volumes  of  Price's  glycerine ;  let  the  mixture  stand  for  seven  days. 
On  the  eighth  day  cool  down  to  3°  C.  for  six  hours  ;  a  considerable  deposit  is 
formed.  Filter  through  porous  paper,  but  take  the  precaution  of  placing  in 
each  filter  along  with  the  solution  a  little  closed  stoppered  bottle  full  of  ice. 
This  will  prevent  the  re-solution  of  the  precipitate  formed  by  cold.  The 
first  filtrate  is  turbid,  but  this  must  be  refiltered  till  it  is  limpid.  Films  and 
bubbles  made  with  this  solution  last  for  eighteen  hours  if  kept  under  a  glass 
shade  in  very  slightly  moist  air. 

Collodion  Films.  —  (Gernez.}  Ether  89  parts  by  weight,  absolute 
alcohol  5 1,  photographic  gun-cotton  5| ;  dissolve.  Decant.  To  100  parts 
by  volume  of  the  clear  solution  add  70  to  100  parts  of  pure  castor  oil.  This 
mixture  is  tenacious  enough  to  permit  the  use  of  frameworks  8  cm.  in 
diameter. 

Prof.  B.  P.  Thompson's  Films.  —  Rosin  46  by  weight;  Canada  balsam 
53 ;  melt ;  add  1  of  turpentine.  For  use,  heat  to  a  little  above  100°  C. 

If  a  roughened  iron  ring  be  dipped  into  any  of  these  mix- 
tures, and  taken  out,  a  plane  film  will  be  found  stretched  across  it. 
A  pipette  (whose  point  has  been  dipped  into  the  same  mixture) 
may  be  employed  to  blow  bubbles  and  place  them  on  this  film, 
and  then  to  enlarge  or  diminish  these  bubbles.  Such  films  and 
bubbles  stretch  themselves  into  the  most  singularly  beautiful 
forms  when  iron  frameworks  forming  the  complete  angular 
periphery  of  cubes,  pyramids,  cylinders,  and  so  fortfc  are  sub- 
stituted for  the  roughened  ring  above  described;  and  these 


274  OF  LIQUIDS.  [CHAP. 

forms  may  be  infinitely  varied  by  modifying  the  size  of  the  bub- 
bles placed  on  the  films,  or  by  breaking  with  a  hot  needle  the 
connection  of  the  film  with  one  or  more  of  the  peripherical 
bounding  lines ;  in  the  latter  case  the  most  beautiful  skew-sur- 
faces are  formed,  all  presenting  the  least  area  possible  under 
the  limiting  conditions. 

If  on  a  simple  film  stretched  over  a  ring,  a  piece  of  silk 
thread  (moistened  beforehand  with  the  same  solution)  be  laid 
in  such  a  way  that  the  thread  crosses  itself  at  some  one  point  on 
the  film,  and  if  the  film  be  pierced  inside  the  loop  of  thread,  the 
loop  will  fly  open  and  form  a  perfect  circle :  for  the  rest  of  the 
film  tends  to  occupy  as  small  an  area  as  possible.  If  a  drop  of 
alcohol  be  laid  within  the  loop,  the  loop  flies  open  in  the  same 
way ;  although  the  film  is  not  broken,  yet  its  surface-tension  is 
diminished. 

If  a  shallow  dish  containing  mercury  be  tilted  over,  so  that 
the  mercury  is  on  the  point  of  pouring  out ;  if  then  some  of  the 
mercury  be  drawn  over  so  as  to  start  a  flow,  the  stream  will 
drag  the  mercury  out  of  the  dish. 

Mercury  can  even  be  blown  or  shaken  into  bubbles  by  means  of  water 
in  place  of  air,  and  a  film  of  it  can  be  produced  on  a  small  amalgamated 
copper  ring. 

If  a  piece  of  camphor  be  placed  on  clean  water  it  partly 
dissolves  in  the  water.  A  strong  solution  of  camphor  has  less 
superficial  tension  than  a  weak  solution,  which  in  its  turn  has 
less  tension  than  pure  water.  At  any  part  of  the  surface  where 
the  solution  happens  to  be  more  dilute,  there  the  weaker  solu- 
tion, having  the  greater  tension,  pulls  the  camphor  towards 
itself.  The  camphor  accordingly  flies  about  the  surface  of  the 
water.  This  is  generally  prevented  by  allowing  the  finger  to 
touch  the  water,  unless  the  finger  be  beforehand  specially  puri- 
fied; so  slight  a  trace  of  fatty  matter  communicated  to  the 
water  prejudices  its  surface-tension  to  so  great  an  extent.  If 
a  drop  of  ether  be  suspended  by  a  glass  rod  close  to  a  thin  layer 
of  water,  the  rest  of  the  water  is  observed  to  flee  from  the  spot; 
the  surface-tension  of  the  rest  of  the  water  is  unchanged,  but 
just  under  the  drop  of  ether  the  tension  is  diminished  by  absorp- 
tion of  the  ether-vapour.  A  thin  layer  of  water,  into  the  centre 
of  which  a  drop  of  alcohol  is  thrown,  leaves  the  alcohol  for  a 
similar  reason.  If  a  glass  of  strongly  alcoholic  wine  be  tilted 
so  as  to  moisten  the  side  of  the  glass,  the  film  of  wine  left  will 
gradually  lose  some  alcohol,  and  becoming  more  aqueous  it  will 


XL] 


SURFACE-TENSION. 


275 


Fig.ioi. 


have  a  greater  superficial  tension  than  the  wine  ;  it  will  pull 
itself  up  the  sides  of  the  glass  and  gather  into  drops.  A  thin 
layer  of  water  on  a  metallic  plate,  the  midpoint  of  which  is 
heated,  withdraws  to  the  edges. 

Measurement  of  Surface-Tension.  —  A  soap  or  collodion  film  has 
two  surfaces,  and  if  the  film  be  not  too  thin,  these  are  independent  of  one 
another.  Consequently  the  tension  of  a  film  is  double  that  of  a  single  sur- 
face of  the  same  liquid  and  of  the  same  area.  The  tension  of  a  film  can  be 
measured  directly.  A  little  framework,  consisting  of  a  transverse  bar  AB, 
and  two  grooved  slips  CD  and  EF, 
allows  the  piece  of  wire  GHIJ  to  A  c 
slip  freely  up  and  down  the  grooves.  - 
The  wire  is  pushed  home,  and  a 
quantity  of  the  liquid  is  applied 
between  HI  and  CE.  The  little 
pan  X  may  be  loaded  with  sand 
until  the  wire  HI  is  pulled  out  to 
a  certain  distance  Cp  from  AB. 
When  it  is  in  that  position,  the 
film  has  an  area  CE  •  Cp.  The 
Weight  mg  of  the  total  mass  m  sus-  . 
pended  on  the  film,  and  the  Tension 
over  the  area  CE  •  C/>,  are  equal  to  one  another.  If  the  total  weight  applied 
be  increased,  the  area  assumed  by  the  film  is  increased  in  direct  proportion. 
The  total  weight  mg  is  distributed  over  the  breadth  CE  ;  whence,  if  T  repre- 
sent the  superficial  tension  across  unit  of  length  of  CE,  then  mg  =  T-  CE. 
The  energy  of  the  film  is  the  work  done  upon  it  ;  weight  or  force  mg  has 
pulled  the  film  through  a  space  Cp:  the  energy  is  mg  x  Cp  =  T  -  CE  •  Cp. 
The  energy  may  also  be  represented  as  the  product  of  S  (the  energy  per 
unit  of  area)  x  CE  •  Cp  (the  area)  =  S  •  CE  •  Cp.  Hence 

T.CE.Cp  =  S-CE-Cp 
T=  S. 

The  energy  per  unit  of  area  (which  in  the  case  of  a  soap  film  is  54-936  ergs 
per  sq.  cm.)  is  numerically  equal  to  the  surface-tension  across  each  unit  of 
length  (which  in  the  same  case  is  54-936  dynes  per  cm.).  These  are  inde- 
pendent of  the  form  of  the  film,  and  depend  only  on  its  actual  area,  not  on 
its  curvature.  For  each  single  surface  T  and  S,  as  found  by  experiment  on 
films,  must  be  halved. 

T  may  also  be  measured  by  observing  the  height  to  which  the  tension 
of  the  curved  surface  will  raise  the  level  of  liquid  in  a  clean  open  capillary 
tube,  wetted  by  the  liquid  and  dipped  into  the  liquid.  The  liquid  wetting 
the  tube,  and  the  superficial  layer  of  the  liquid  in  the  tube,  contract  towards 
one  another:  the  result  is  a  curved  surface  whose  outer  boundary  is  2?rr, 
where  r  is  the  radius  of  the  tube.  If  the  superficial  tension  across  unit  of 
length  of  this  boundary  be  T,  the  total  superficial  tension  will  be  2irrT. 
The  Weight  of  the  liquid  lifted  by  this  tension  is  its  volume  x  pg  ;  that  is, 
Trr2hpg,  where  h  is  the  height  of  the  column  supported.  Hence  in  proper 
units  T  =  rhpg/2  ;  and  the  height  of  the  column  is  inversely  proportional  to 
the  diameter  of  the  tube.  Again,  if  m  be  the  maximum  mass"  of  a  large 
hanging  drop  depending  from  a  small  wetted  circular  disc  of  radius  r,  the 


276  OF  LIQUIDS.  [CHAP. 

boundary  is  2irr ;  mg,  the  .Weight  of  the  hanging  drop,  =  27rr-  T;  and  T  = 
mg/27rr  dynes  per  cm. 

This  Modulus  of  Superficial  Tension,  T,  is,  in  all  the  instances 
which  we  have  considered,  that  of  a  surface  between  the  liquid  and  the  sur- 
rounding air.  In  the  case  of  pure  and  perfectly  clean  water  and  air,  this 
modulus  is  equal  for  each  surface  to  81-96173  dynes  per  cm.,  very  nearly 
three  times  the  superficial  tension  of  a  single  surface  of  soap  solution  in  con- 
tact with  air.  The  tension  of  the  bounding  surface  separating  olive  oil  and 
air  is  36-8856  dynes;  that  of  the  surface  between  olive  oil  and  water  is 
20-56176  dynes,  per  cm.  At  the  meeting-place  of  oil,  water,  and  air,  these 
three  surfaces  meet ;  the  tension  of  the  water-air  surface  decidedly  prepon- 
derates, and  the  edge  of  an  oil-drop  floating  on  water  is  drawn  out  indefi- 
nitely. If  a  drop  of  water  be  placed  on  chloroform  —  the  respective  tensions 
being  water-air  81-96173,  chloroform-air  30-6072,  and  chloroform-water 
29-5281  —  its  surface-tension  (water-air)  at  first  preponderates  and  pulls  it 
into  a  drop.  When  water,  air,  and  clean  glass  are  placed  in  contact  there 
is  again  a  triplet  of  tensions,  the  resultant  of  which  pulls  the  water  over  the 
glass,  which  is  thus  wetted  by  the  water.  The  water  tends  to  stand  so  that 
its  surface  makes  a  certain  angle  with  the  glass ;  this  is  the  angle  of  capil- 
larity, 180°  between  water  and  wet  glass,  45°-30  between  mercury  and  glass. 

In  the  case  of  water,  this  angle  is  such  that  the  upper  surface  of  water 
in  contact  with  glass  is  concave ;  in  the  case  of  mercury,  the  upper  surface 
is  convex.  Water  will,  however,  spread  over  perfectly  clean  mercury. 

On  the  curvature  of  the  upper  surface,  thus  determined,  depends  the 
direction  in  which  the  contractile  tension  of  the  superficial  film  acts.  The 
concave  surface  of  water  tends  to  contract  and  become  flat,  and  it  does  so 
in  proportion  to  the  curvature  imposed  on  it. 

The  narrower  a  capillary  tube  is,  the  greater  is  the  curva- 
ture of  the  surface  of  any  liquid  standing  in  it,  and  therefore 
the  greater  is  the  contractile  tendency  of  that  surface.  The 
effect  of  this  tendency  is,  in  the  case  of  water,  to  exert  an 
upward  pull,  to  neutralise  to  some  extent  the  downward  effect 
of  gravity,  and  to  support  a  column  of  water  in  the  tube. 
Hence  water  stands  at  a  higher  level  in  a  narrow  tube  whose 
lower  open  end  is  dipped  in  water  than  it  does  in  a  wider  one ; 
and  the  height  of  the  column  supported  is  inversely  proportional 
to  the  radius  of  the  tube  at  the  spot  where  the  curved  surface 
of  the  liquid  is  situated.  The  height  at  which  a  solution  stands 
depends  on  its  strength  and  on  the  salt  dissolved.  Mercury 
under  similar  circumstances  stands  at  a  lower  level. 

This  tendency  of  a  curved  surface  to  exert  traction  or  pres- 
sure on  a  .fluid  may  be  seen  in  a  conical  capillary  tube ;  if  a 
drop  of  water  be  introduced,  the  smaller  concave  surface  will 
pull  the  drop  towards  the  apex,  if  a  drop  of  mercury,  the 
smaller  convex  surface  will  push  the  mercury  from  the  apex. 

Capillary  phenomena  are  thus  phenomena  of  surface-tension  ;  and  it  is 
futile  to  explain  the  rise  of  sap  in  plants  or  the  passage  of  fluids  through 


xi.]  SURFACE-TENSION.  277 

minute  vessels  by  "  capillary  attraction  "  when  there  is  no  free  surface.  An 
experiment  which  may,  on  the  other  hand,  illustrate  these  movements,  con- 
sists in  oiling  the  interior  of  an  open  capillary  tube,  filling  it  with  water, 
and  dipping  the  end  of  the  tube  in  oil ;  the  attraction  of  the  sides  of  the 
tube  for  oil  will  cause  the  oil  to  run  along  the  tube  and  to  drive  out  the 
water;  this,  how-'  er,  is  not  an  exclusively  capillary  phenomenon. 

If  two  plates  of  clean  glass  be  set  to  stand  vertically  and 
parallel  to  one  another  in  a  shallow  dish  of  water,  the  water 
will  rise  up  the  sides  of  each  to  a  height  half  that  which  it 
would  attain  in  a  tube  whose  diameter  is  equal  to  the  distance 
between  the  plates.  If  the  two  plates  have  two  vertical  edges 
in  contact,  the  liquid  will  rise  indefinitely  where  they  are  in 
contact ;  at  other  parts  it  rises  to  a  height  inversely  proportional 
to  the  local  distance  between  the  plates,  and  it  thus  presents  the 
outline  of  an  equilateral  hyperbola. 

Just  as  the  surface  of  a  liquid  is  raised  against  a  fixed  plane 
of  clean  glass,  so  a  floating  vessel  of  clean  glass  may  by  surface- 
tension  be  pulled  down,  so  as  to  lie  more  deeply  in  the  liquid 
than  its  specific  gravity  would  lead  us  to  expect.  A  floating 
hydrometer  mostly  gives  an  abnormally  low  reading  on  this 
account;  it  is  pulled  into  the  liquid,  so  that  the  liquid  appears 
to  be  lighter  than  it  really  is.  If  a  little  vapour  of  ether  be 
poured  on  the  surface  of  the  liquid  so  as  to  diminish  the  surface- 
tension,  the  hydrometer  rises.  If  the  water  have  any  grease  on 
its  surface,  the  same  effect  follows.  If  the  hydrometer  be  greasy, 
it  is  repelled  and  stands  abnormally  high  in  the  liquid.  Hence 
great  confusion  and  inaccuracy  may  result  from  films  of  grease 
on  the  glass  or  on  the  fingers  of  the  manipulator. 

Objects  which  are  wetted  by  the  liquid  in  which  they  float 
are  thus  apparently  attracted  by  it ;  those  which  are  not  so  are 
apparently  repelled.  Two  wetted  objects  floating  on  water  seem 
to  attract  one  another;  two  objects  floating  on  a  liquid  which 
does  not  wet  them  seem  also  to  attract  one  another.  This 
may  be  seen  by  throwing  upon  the  surface  of  water  a  number  of 
wooden  balls,  of  which  some  are  smoked  with  lampblack,  while 
others  are  purified  firs,t  with  soap  and  water,  then  with  pure 
water;  the  smoked  balls  approach  each  other,  the  clean  ones 
approach  each  other,  but  the  clean  balls  appear  to  avoid  the 
smoked  ones. 

We  may  mention  another  consequence  of  surface-tension. 
A  jet  of  water  issuing  from  a  rectangular  orifice  is  most  acted 
upon  by  surface-tension  at  its  narrow  edges.  These  ^ire  pressed 
together  ;  they  meet,  and  when  they  do  so,  spread  out  laterally ; 


278  OF  LIQUIDS.  [CHAP. 

the  same  action  is  repeated,  and  the  whole  jet  is  a  succession  of 
flat  segments  at  right  angles  to  one  another.  At  first  sight  such 
a  jet  seems  to  have  a  screw  form. 

The  distances  at  which  molecular  forces  act  are  not  immeas- 
urably small.  Quincke  found  that  while  water  stands  against 
glass  at  one  angle,  against  silver  at  another  angle  of  capillarity, 
yet  against  glass  coated  with  silver  it  stands  at  such  an  angle  as 
to  show  that  the  influence  of  the  glass  is  felt  through  the  silver 
when  the  layer  of  silver  is  less  than  -000,005  cm.  thick;  this 
thickness  being  one-tenth  of  the  average  length  of  a  wave  of 
light,  and  being  further  (Meyer)  very  much  the  same  thing  as 
the  mean  free  path. 

Superficial  Viscosity.  —  This  is  a  property  of  the  superfi- 
cial film  of  liquids  after  exposure  to  the  air  for  some  time  :  and 
it  is  independent  of  the  surface-tension.  If  a  magnetic  needle 
be  so  adjusted  as  to  have  its  lower  surface  in  contact  with  the 
surface  of  a  solution  of  saponine,  it  will  remain  in  any  position, 
in  defiance  of  the  directive  force  of  the  earth's  magnetism.  On 
the  surface  of  most  other  liquids  it  will  move  into  the  magnetic 
meridian,  but  the  whole  superficial  film  of  the  liquid  will  move 
with  it,  as  may  be  shown  by  strewing  lycopodium  over  the  sur- 
face. The  superficial  film  is,  as  a  rule,  exceedingly  viscous  as 
compared  with  the  interior  mass;  it  is  consequently  hard  to 
move  or  to  break.  If  a  liquid  have  great  superficial  viscosity 
and  small  surface-tension  (as  in  the  case  of  soap-and-water), 
a  bubble  rising  through  the  liquid  may  raise  the  surface  film, 
which  the  tension  is  not  able  to  break :  the  bubble  may  therefore 
persist.  If  a  wire  ring,  bearing  a  soap  film,  be  :;wept  rapidly 
through  the  air,  the  air  may  fill  and  stretch  the  film,  and  sepa- 
rate part  of  it  in  the  form  of  a  complete  bubble.  A  bubble  ris- 
ing with  very  great  rapidity  through  a  liquid  may  tear  off  some 
of  the  viscous  superficial  film  and  form  a  complete  bubble :  this 
is  seen  when  a  mixture  of  olive  oil  and  strong  sulphuric  acid  is 
vigorously  stirred. 

Pure  water  has  great  surface-tension,  which  is  able  to  over- 
come the  superficial  viscosity.  Perfectly  clean  water  has  no 
superficial  viscosity.  Thus  pure  water  will  not  froth.  Some 
liquids,  such  as  a  solution  of  gum  arabic  or  of  albumen,  will 
froth  when  shaken,  but  their  superficial  viscosity  is  not  suffi- 
ciently great  to  enable  bubbles  to  be  blown  with  them.  Alcohol, 
sulphuric  ether,  bisulphide  of  carbon,  and  some  other  liquids, 
have  a  superficial  viscosity  less  than  their  internal  viscosity,  and 


XL]  SUPERFICIAL  VISCOSITY.  279 

consequently,  when  alcohol  is  mixed  with  a  superficially  viscous 
liquid,  it  neutralises  its  relative  superficial  viscosity,  and  froth- 
ing is  rendered  impossible.  Hence  the  practice  of  adding  a  few 
drops  of  spirit  in  order  to  check  frothing  in  pharmaceutical 
operations. 

To  this  toughness  of  the  superficial  film,  the  floating  of  an 
oiled  needle  or  the  walking  of  an  insect  on  water  must  be  in 
part  ascribed.  The  depth  of  the  dimple  produced  by  the  needle 
is  not  sufficient  to  account,  by  displacement,  for  the  support 
afforded  to  so  heavy  a  body:  the  superficial  tension  is  dimin- 
ished by  the  oil :  the  tenacity  of  the  surface  film  plays  its  part 
in  supporting  the  needle.  To  the  same  cause  we  may  attribute 
the  smoothing  of  the  surface  of  a  rough  sea  when  oil  is  poured 
upon  it:  the  new  surface  has  great  superficial  tenacity  and 
small  superficial  tension,  and  is  not  readily  broken  up  into  surf. 
The  new  surface  of  the  sea  is  relatively  rigid ;  waves  press 
against  it  from  beneath,  but  their  energy  is  spent  in  producing, 
not  ripples,  but  eddies  below. 

The  superficial  film  of  a  liquid  is  thus  seen  to  be  a  seat  of 
energy  and  to  be  physically  different  from  the  interior. 

A  bubble  in  bursting  does  so  with  an  audible  sound :  it  scatters  parti- 
cles of  its  substance  and  of  the  contained  gas  to  a  height  of  three  or  four 
feet ;  this  happens  during  the  effervescence  of  sewage  which  is  undergoing 
fermentation. 

Cohesion-Figures.  —  If  the  surface  of  a  tumblerful  of  salt 
water  (J  teaspoonful  to  the  tumbler)  be  touched  with  a  pen  not 
too  full  of  ink,  the  ink  will,  in  falling  through  the  liquid,  assume 
very  remarkable  vortical  movements.  A  shower  of  rain  falling 
on  a  troubled  sea  produces  similar  vortex-rings,  which  are  carried 
down  into  regions  of  comparative  stillness,  and  moderate  the  tur- 
bulence of  the  water  by  equalising  its  distribution  of  momentum. 
The  forms  assumed  by  drops  of  water  or  of  mercury  falling  on  a 
flat  surface,  at  the  instant  when  they  spread  out  and  break,  are 
very  remarkable,  and  may  be  seen  when  the  spreading  drops  are 
momentarily  illuminated  by  the  electric  spark.  The  edge  of  the 
spreading  drop  breaks  up  into  thinner  and  thicker  nodes  and 
loops  which  vibrate :  very  roughly  the  result  may  be  seen  in  a 
cooled  splash  of  candle-wax. 

Solubility  of  Solids  in  Liquids. —  When  a  solid  is  dissolved 
in  a  liquid,  work  is  done  in  overcoming  its  cohesion.  This 
cohesion  is  overcome  by  the  adhesion  between  the  solid  and  the 
liquid.  Ice  put  into  sulphuric  acid  has  its  superficial  particles 


280  OF  LIQUIDS.  [CHAP. 

successively  torn  off,  and  a  mass  of  dilute  sulphuric  acid  (which 
on  account  of  liquefaction  assumes  a  low  temperature  unless  heat 
be  supplied)  is  produced.  Such  union  may  or  may  not  be  asso- 
ciated with  a  play  of  chemical  affinities ;  in  the  case  of  ice  and 
sulphuric  acid  there  is  a  tendency  to  the  production  of  definite 
hydrates  of  sulphuric  acid,  the  formation  of  which  is  accom- 
panied by  the  evolution  of  a  certain  amount  of  heat.  If  sul- 
phate of  magnesia  be  placed  in  water  it  will  be  dissolved  to  a 
certain  limited  extent ;  if  the  salt  be  added  in  excess  above  this 
limit,  no  more  will  be  dissolved;  when  this  limit  has  been  reached 
the  solution  is  a  saturated  solution.  This  limit  is  expressed  by 
the  coefficient  of  solubility,  a  number  indicating  the  quantity 
of  solid  which  can  be  dissolved  and  remain  in  solution  in  unit- 
mass  of  the  liquid  at  the  particular  temperature  for  which  the 
coefficient  is  or  ought  to  be  specified.  A  saturated  solution  can 
dissolve  no  further  quantity  of  the  same  salt  at  the  same  tem- 
perature, for  the  adhesion  of  such  a  solution  to  the  salt  is  no 
longer  greater  than  the  cohesion  of  the  salt  itself :  or,  in  other 
words,  just  as  many  particles  then  leave  the  liquid  for  the  salt 
as  leave  the  salt  for  the  liquid.  If  the  cohesion  of  the  salt  be 
lessened  by  heat,  more  may  be  dissolvedj  and  as  a  general  rule, 
with  but  few  exceptions  —  hydrate  of  lime,  sulphate  of  soda, 
phosphate  of  magnesia  —  salts  are  more  soluble  in  hot  than  in 
cold  water.  The  adhesion  of  a  liquid  to  the  solid  which  it 
holds  in  solution  may  be  relatively  great  or  feeble ;  and  its 
relative  amount  may  be  indicated,  though  not  measured  quanti- 
tatively —  (1)  by  a  high  or  low  coefficient  of  solubility ,  (2)  by 
the  amount  of  energy  which  must  be  imparted  to  the  molecules 
in  order,  by  boiling,  to  tear  the  water  away  from  the  salt,  or,  in 
other  words,  by  the  high  or  low  boiling-point  of  a  saline  solu- 
tion ;  (3)  by  the  relative  effect  of  charcoal  filters  in  retaining 
the  salts  of  a  saline  solution  while  allowing  the  water  to  pass,  a 
property  made  use  of  in  the  analysis  of  poisons ;  and  sometimes 
(4)  by  the  detachment  of  the  liquid  from  the  solid  by  a  stronger 
molecular  attraction,  as  in  the  case  of  iodide  of  starch,  a  solu- 
tion of  which  is  precipitated  by  acetate  of  potash,  the  water 
leaving  the  iodide  of  starch  and  adhering  to  the  salt. 

'  There  is  a  general  relation  of  concurrence  between  the  solubility  and 
the  fusibility  of  a  salt ;  but  there  are  important  exceptions,  e.g.,  chloride  of 
silver,  which  is  fusible,  but  not  soluble  in  water. 

Dissociated  Molecules  in  Solutions.  —  When  such  a  chem- 
ically inert  substance  as  sugar  is  dissolved  in  water,  its  mole- 


XL]  DISSOCIATION   IN   SOLUTIONS.  281 

cules  seem  to  remain  undecomposed  ;  but  in  an  aqueous  solution 
of  a  salt,  of  a  chemically  strong  acid  or  base,  or  generally  of  any 
substance  which  presents  in  solution  a  marked  chemical  activity 
or  susceptibility  to  chemical  reaction,  there  is,  somehow,  more 
or  less  Dissociation  of  the  molecules  of  the  dissolved  sub- 
stance into  sub-molecules  or  free  Ions.  For  example,  NaCl 
splits  up,  on  solution  in  water,  into  Na  and  Cl  atoms;  H9SO4 
into  H,  H,  and  (SO4),  A12(SO4)3  into  Al,  Al,  (SO4),  (SO4), 
and  (SO4).  This  seems  quite  contrary  to  experience;  but  it 
is  clear  that  the  physical  properties  of  the  solution  can  only  be 
explained  by  assuming  that  there  is  within  it  a  Number  of  mole- 
cules or  sub-molecules,  which  cannot  be  accounted  for  on  any 
other  hypothesis ;  and  then,  as  the  dilution  or  the  temperature 
of  the  solution  increases,  the  more  nearly  is  the  increase  in  the 
number  of  molecules  such  as  to  correspond  exactly  with  their 
derivation  in  the  above  manner.  An  extremely  dilute  solution  of 
a  salt  thus  does  not  contain  the  salt,  as  such ;  it  only  contains 
ions.  If  an  aqueous  solution  of  hydrochloric  acid  (in  which 
there  is  almost  complete  dissociation)  be  mixed  with  one  of 
potash,  in  which  the  condition  is  the  same,  the  reaction  on 
neutralisation  is  (H  +  Cl)  +  (K  +  HO)  =  K  +  Cl  +  H2O. 
Water  is  formed  on  neutralisation,  but  the  ions  K  and  Cl 
remain,  for  the  most  part,  separate  until  crystallisation  takes 
place. 

The  physical  properties,  the  peculiarities  in  which  have  led  to  the  fore- 
going conclusion,  are  the  Osmotic  Pressure,  the  Freezing-Point,  the  Vapour- 
Pressure,  the  Density,  the  Colour,  and  the  Electric  Conductivity  of  aqueous 
solutions  of  those  substances  which  are  chemically  most  active  or  undergo 
chemical  reactions  in  the  shortest  time. 

When  a  saturated  solution  is  cooled,  the  coefficient  of  solu- 
bility diminishes,  and  the  solid  segregates  in  a  separate  form : 
thus  hot  saturated  solutions  may  be  set  aside  to  cool,  and  on 
cooling  they  crystallise,  the  materials  dividing  into  crystals  of 
the  salt  and  an  ordinary  cold  saturated  solution  of  the  same. 
Sometimes,  as  in  the  case  of  sulphate  of  soda,  such  a  solution 
(though  cooled  down  to  a  temperature  at  which  it  cannot  per- 
manently retain  all  the  salt  which  it  holds  in  solution)  does  not 
crystallise,  but  forms  a  supersaturated  solution.  Such  a  solu- 
tion is  in  a  state  of  unstable  molecular  equilibrium,  and  the 
instant  it  is  touched  with  a  crystal  of  the  same  salt  or,  with  less 
certainty,  by  a  crystal  of  an  isomorphous  substance^  or  by  the 
dust  of  the  air  containing  the  same  substance,  or  by  an  oil 


282  OF  LIQUIDS.  [CHAP. 

(especially  if  somewhat  oxidised),  or  by  a  bubble  of  gas  solu- 
ble in  the  liquid,  or  when  it  is  exposed  to  the  least  vibration, 
the  whole  molecular  arrangement  topples  over,  and  the  excess 
of  salt  assumes  the  solid  form.  It  does  so  with  evolution  of 
heat,  if  the  act  of  solution  had  been  accompanied  by  cooling. 

A  similar  delay  in  solidification  occurs  in  the  case  of  melted  phosphorus, 
which  can  be  kept  fluid  at  10°  C.  (its  solidification  point  being  44°-2  C.)  for 
weeks,  especially  if  the  water  lying  above  it  contain  a  trace  of  potash 
hydrate  or  of  nitric  acid.  The  slightest  shake  or  contact  with  a  piece  of 
phosphorus  determines  solidification. 

Miscibility  of  Liquids.  —  If  a  bottle  be  filled  with  oil  and 
water,  and  shaken,  the  layers  separate  as  soon  as  the  disturbance 
ceases,  though  there  is,  in  such  cases,  always  a  certain  small 
amount  of  evaporation  of  the  one  liquid  into  the  other.  Alco- 
hol and  water  treated  in  the  same  way  mutually  dissolve  each 
other,  and  mix  perfectly  in  any  proportions.  Ether  and  water 
will  each  take  up  a  certain  proportion  of  the  other,  which  pro- 
portion depends  upon  the  temperature,  and  when  shaken  together 
they  separate  into  two  layers,  the  one  a  solution  of  ether  in 
water,  and  the  other  a  solution  of  water  in  ether.  These  two 
liquids  are  miscible  only  in  certain  proportions,  which  depend 
upon  the  temperature ;  in  some  cases  a  sufficiently  high,  in 
others  a  sufficiently  low  temperature  brings  about  complete 
miscibility.  Very  often,  as  in  the  case  of  alcohol  and  water, 
there  is  a  contraction  of  volume  and  evolution  of  heat,  there 
being  some  potential  energy  of  separation  somehow  liberated  by 
the  approximation  of  mutually  attracting  molecules  of  the 
different  substances ,  or  there  may  be  expansion  and  cooling,  as 
in  the  case  of  alcohol  and  carbon  bisulphide. 

Imbibition.  —  Porous  objects,  such  as  a  lump  of  sugar,  blot- 
ting paper,  a  heap  of  sand,  a  sponge,  a  lamp-wick,  absorb  liquids 
with  a  rapidity  which  depends  on  the  nature  of  the  porous  sub- 
stance itself  and  on  that  of  the  liquid  absorbed,  and  which  is 
greater  if  the  materials  be  warm.  This  takes  place  by  reason  of 
an  attraction  between  the  solid  and  the  liquid  (which  Chevreul 
called  affinitS  capillaire),  and  heat  is  evolved  when  this  attrac- 
tion is  satisfied,  as  in  the  case  of  a  wetted  rope,  which  rises  in 
temperature  from  2  to  10°  C.  (part  of  this  effect  being  due  to  the 
concurrent  shrinkage  of  the  rope).  When  a  porous  body  which 
has  thus  taken  up  a  quantity  of  liquid  is  subjected  to  pres- 
sure, the  whole  of  the  liquid  can  by  no  means  be  squeezed  out ; 
some  water  still  remains,  which  can  be  evaporated  away.  Imbi- 


XL]  MISCIBILITY   OF  LIQUIDS.  283 

bition  will  fill  the  pores  of  a  solid  with  a  liquid,  but  will  not  set 
up  a  permanent  current  in  those  pores  unless,  as  in  the  case  of  a 
lamp-wick,  there  be  constant  removal  of  the  liquid  at  one  extrem- 
ity of  the  porous  object  while  imbibition  goes  on  at  the  other. 

Diffusion  —  Jar-diffusion.*  —  If  a  phial,  filled  to  within 
say  half-an-inch  of  the  top  with  a  saline  solution,  be  placed  in  a 
jar ;  if  water  be  poured  into  the  jar  so  as  to  surround  the  phial, 
and  if  more  water  be  cautiously  added  until  the  phial  is  covered 
with  a  layer  of  water  of  about  half-an-inch  in  depth,  the  whole 
being  set  aside  in  a  quiet  place,  the  solution  in  the  phial  will 
diffuse  into  the  surrounding  water.  The  quantity  of  substance 
diffused  into  the  water  in  a  given  time  depends  (1)  on  the 
length  of  that  time ;  (2)  on  the  quantity  of  substance  in  the 
phial  solution,  being  (within  narrow  limits)  proportional  to  its 
strength ;  (3)  on  the  temperature,  being,  for  dilute  solutions, 
nearly  proportional  to  the  absolute  temperature ;  (4)  on  a 
Coefficient  of  Diffusibility  special  to  each  substance.  Other 
things  being  equal,  urea  and  salt  diffuse  twice  as  fast  as  sugar, 
sugar  twice  as  fast  as  gum  arabic,  gum  arabic  more  than  four 
times  as  fast  as  egg-albumen. 

Sugar  travels  as  far  in  a  column  of  water  in  two  days  as  albumen  in 
fourteen.  The  following  numbers  indicate  the  relative  times  necessary  for 
the  process  of  diffusion  to  convey  in  water  through  equal  distances  equal 
amounts  of  the  several  substances  :  —  Hydrochloric  acid,  1 ;  chloride  of 
sodium,  2-33 ;  sugar,  7 ;  sulphate  of  magnesia,  7  ;  albumen,  49  ;  caramel,  98 
(Graham). 

The  rate  of  diffusion  of  all  substances  is  increased  by  moderate  heat, 
but  in  those  substances  whose  coefficient  of  diffusibility  is  small,  it  is  more 
increased  by  heat  than  it  is  in  those  substances  which  are  already  very  dif- 
fusible. Hence  the  greatest  proportionate  differences  in  diffusion-rates  are 
found  in  the  coldest  solutions. 

Some  liquids,  such  as  water  and  sulphuric  acid,  ether  and  chloroform, 
mercury  or  molten  gold  or  silver  and  molten  lead,  diffuse  into  one  another 
with  considerable  rapidity. 

If  a  mixture  be  placed  in  the  diffusion-phial,  the  approxi- 
mate rule  is  that  each  component  of  the  mixture  is  diffused  out 
at  its  own  rate,  and  independently  of  the  others.  There  is, 
however,  a  departure  from  strict  adherence  to  this  rule,  in  the 
sense  that  the  ordinary  differences  of  diffusibility  are  exagger- 
ated in  such  a  mixture.  If  the  phial  contain  a  double  salt,  such 
as  alum,  diffusion  may  effect  chemical  decomposition:  sulphate 
of  potash  and  sulphate  of  alumina  are  separated,  the  former 
being  diffused  more  rapidly. 

*  See  Graham's  Chemical  and  Physical  Researches. 


284  OF  LIQUIDS.  [CHAP. 

A  high  boiling-point  of  any  solution  (which  indicates  adhe- 
sion of  water  to  the  salt  dissolved)  is  associated  with  rapid 
diffusibility  of  the  same  salt ;  but  there  is  no  close  relation 
between  the  rapidity  of  diffusion  of  a  salt  and  its  solubility. 

Colloids  and  Crystalloids.  —  On  surveying  a  number  of 
objects  which  have  a  wide  range  of  relative  diffusibilities,  we 
see  that  at  one  end  of  the  scale  we  have  such  things  as  urea 
and  chloride  of  sodium,  and  at  the  other  such  things  as  starch, 
gum,  gelatine,  albumen.  The  former  are  bodies  of  rapid  dif- 
fusibility, have  generally  a  certain  chemical  stability  and  a 
crystalline  form,  and  are  called  Crystalloids.  The  latter  are 
bodies  of  slow  diffusibility,  have  large  molecules  very  probably 
compounded  of  groups  of  their  simplest  molecules,  have  in  gen- 
eral (with  rare  exceptions,  such  as  the  blood-crystals)  the  non- 
crystalline  amorphous  glue-like  character  which  gives  them, 
the  name  of  Colloids,  and  are  for  the  most  part  in  a  state  of 
unstable  equilibrium  when  in  the  moist  condition.  Colloids 
have  great  cohesion,  and  adhere  firmly  to  other  colloids :  thus 
isinglass  heated  with  acetic  acid  forms  a  cement  which  adheres 
firmly  to  glass ;  and  when  they  dry  they  tend  to  contract  firmly, 
so  that  a  strong  solution  of  gum  arabic,  drying  in  a  test  tube, 
will  sometimes  break  the  tube.  They  often,  when  drying  up, 
extrude  their  contained  water,  and  form  clots,  on  the  surface  of 
which  the  water  presents  itself  in  drops.  Colloids  also  in  many 
instances  possess  the  power  of  taking  up  alcohol  or  olein  in  the 
room  of  their  water  of  constitution.  This  property  is  possessed 
even  by  such  a  substance  as  colloid  silicic  acid. 

An  animal  tissue,  which  is  in  great  part  composed  of  colloids,  may  have 
its  water  expelled  and  replaced  by  alcohol,  by  dint  of  repeated  washing  in 
that  liquid. 

Colloids  being  very  slightly  diffusible  are  tasteless ;  they  do  not  reach 
the  nerve-ends.  For  the  same  reason  they  are  very  indigestible  —  e.g.,  gela- 
tine—  unless  peptonised;  peptones  being,  by  exception,  diffusible  though 
otherwise  colloidal. 

If  a  layer  of  pure  jelly  be  laid  on  a  layer  of  jelly  charged 
with  soluble  salts  and  also  with  caramel,  the  salts  will  diffuse 
into  the  upper  layer  of  solid  jelly  nearly  as  fast  as  if  it  were 
pure  water ;  the  caramel  will  not  travel  at  all:  If  a  film  of  col- 
loid matter  (starched  paper)  be  placed  between  a  mass  of  pure 
water  and  a  saline  solution  containing  colloid  matters,  the  colloid 
septum  will  offer  little  obstruction  to  the  passage  of  the  salts 
into  the  water,  but  will  prevent  the  colloid  matter  from  passing 


XL]  COLLOIDS   AND   CRYSTALLOIDS.  285 

through.  Colloid  matter  is  thus  impervious  to  the  diffusion  of 
other  colloids,  but  does  not  hinder  the  diffusion  of  crystalloids. 

Diffusion  through  Membranes  —  Osmosis. — If  three  lay- 
ers of  liquid,  chloroform,  water,  ether,  be  placed  in  a  closed 
bottle  and  set  aside,  it  will  be  found  that  in  course  of  time  the 
ether  travels  into  the  chloroform,  but  that  the  water  does  not 
to  any  appreciable  extent  allow  the  chloroform  to  pass  into  the 
ether.  The  ether  dissolves  to  some  extent  in  the  water  and 
diffuses  through  it:  it  is  removed  from  the  water  by  the  chloro- 
form :  step  by  step  the  upper  layer  of  ether  may  wholly  travel  into 
or  through  the  water.  A  thin  caoutchouc  membrane  lying 
between  alcohol  and  water  allows  the  alcohol  to  pass  through 
it  into  the  water;  but  the  reverse  passage  of  water  into  the 
alcohol  is  barred.  If  an  organic  septum  be  used  it  is  wetted, 
and  the  water  passes  into  the  alcohol.  If  hydrochloric  acid 
and  water  be  separated  by  an  animal  membrane,  the  hydro- 
chloric acid  passes  through  in  greater  quantitj^ :  both  fluids  wet 
the  membrane  ;  the  hydrochloric  acid  is  most  attracted.  Hence 
molecules  may  travel  through  a  septum  devoid  of  perceptible 
pores  as  well  as  through  one  in  which  pores  exist. 

If  the  membrane  employed  be  porous,  we  have  the  process 
of  Osmosis.  The  membrane  becomes  penetrated  by  that  one 
of  the  two  liquids  ("  liquid  A  ")  for  which  the  walls  of  its  pores 
have  the  greater  attraction  or  affinity.  When  each  small  capil- 
lary column  of  the  liquid  A  comes  at  the  farther  surface  of  the 
membrane  into  contact  with  the  liquid  B,  the  molecules  of  liquid 
B  diffuse  into  it.  Thus  the  end  of  the  column  of  liquid  A 
comes  more  to  resemble  that  liquid  B  which  is  less  attracted 
by  the  walls  of  the  pore,  and  it  is  extruded  from  the  pore  and 
pushed  into  liquid  B.  This  process  is  continuous,  and  a  stream 
of  liquid  A  is  drawn  through  each  pore  of  the  membrane  into 
liquid  B.  Down  the  centre  of  the  stream  there  is,  however,  a 
backward  diffusion-current  of  molecules  passing  from  the  liquid 
B.  This  happens  if  the  pores  be  wide  enough  to  allow  the  cen- 
tre of  the  stream  to  be  .comparatively  out  of  reach  of  the  imme- 
diate influence  of  the  walls  of  the  channels,  an  influence  which 
we  have  seen  to  extend  to  a  distance  of  2~oTo~o  mm-  or  5  o  o^  o  o 
inch.  If  the  liquid  stream  be  not  too  rapid,  these  molecules 
will  make  their  way  against  it  into  liquid  A.  If  the  channels 
be  very  narrow  the  liquid  stream  is  relatively  accelerated  ;  thus 
the  ratio  between  the  amount  of  water  that  passes^  through  a 
porous  membrane  into  a  saline  solution  and  the  amount  of  salt 


286  OF  LIQUIDS.  [CHAP. 

that  passes  in  the  opposite  direction  is  increased  by  diminution 
of  the  pores.  This  ratio  is  called  the  Endosmotic  Equivalent. 
It  is  not  a  constant,  but  depends  on  the  original  concentration 
of  the  solution  and  on  the  nature  of  the  membrane ;  and  even 
with  the  same  membrane  it  differs  according  to  its  thickness  or 
state  of  freshness,  and  may  be  increased  by  tanning  with  tannin 
or  chromic  acid,  which  diminish  the  size  of  the  pores. 

Thus  for  a  membrane  on  one  side  of  which  is  dry  common  salt,  on  the 
other  side  water,  if  the  membrane  be  a  piece  of  cow's  pericardium,  for  every 
grain  of  salt  which  passes  into  the  water,  4  grains  of  water  pass  into  the  salt ; 
with  a  piece  of  cow's  bladder,  the  endosmotic  equivalent  is  6.  If  on  one  side 
of  an  animal  membrane  there  be  placed  a  strong  solution  of  sulphate  of 
magnesia  and  on  the  other  a  quantity  of  blood  serum,  the  fluid  of  the  blood 
serum  will  pass  into  the  saline  solution,  taking  some  albumen  with  it,  and 
some  sulphate  of  magnesia  will  pass  into  the  blood  serum  (Milne-Edwards). 

The  mechanical  structure  of  the  membrane  has  a  marked 
influence  on  the  process ;  thus  water  will  pass  more  readily  in- 
wards through  frogskin,  more  readily  outwards  through  eelskin. 

The  matters  already  moistening  the  membrane  also  affect  the 
rate  of  transmission  ;  thus  albumen  more  readily  passes  through 
a  membrane  previously  moistened  with  alkalies.  If  between 
alcohol  and  water  there  be  arranged  a  membrane  previouslv 
soaked  in  oil,  the  membrane  cannot  be  wetted,  and  the  alcohol 
now  passes  into  the  water. 

If  the  saline  solution  be  in  a  state  of  movement  relative  to 
the  membrane,  the  particles  are  drawn  away  from  the  membrane, 
and  the  diffusion-stream  is  hindered;  if  the  water  into  which 
the  salts  are  passing  be  constantly  renewed,  the  molecular  dif- 
fusion is  accelerated. 

Heat  increases  the  rapidity  of  Osmosis.  An  electric  current 
(the  "electrodes  "  being  on  opposite  sides  of  the  membrane)  has 
the  singular  effect  of,  as  it  were,  pushing  the  liquid  bodily 
through  the  membrane  towards  the  negative  electrode.  Even 
gelatine  and  the  fatty  matters  of  milk  can  be  thus  driven 
through  a  membrane. 

If  a  mixture  of  different  substances  be  exposed  to  osmosis 
through  a  porous  membrane,  the  colloids  will  remain  or  will 
pass  through  in  very  small  quantities,  the  crystalloids  pass 
through  freely.  This  is  the  basis  of  the  process  of  Dialysis. 
Various  mechanical  arrangements  for  carrying  out  dialysis  sug- 
gest themselves :  a  phial  with  the  bottom  cut  off,  or  a  wide 
glass  tube,  over  the  lower  end  of  which  a  piece  of  membrane  is 
stretched ;  the  material  to  be  dialysed  being  placed  in  this,  and 


XL]  OSMOSIS.  287 

the  whole  suspended  in  water.  The  most  convenient  arrange- 
ment in  many  respects  is  a  piece  of  parchment  paper  (the  leaks 
in  which  are  stopped  with  albumen  coagulated  by  heat)  or, 
better,  gold-beaters'  skin,  laid  upon  a  wooden  ring,  into  which 
a  smaller  ring  is  thrust  so  as  to  form  a  dish  with  a  membranous 
bottom ;  this  is  floated  on  a  mass  of  water,  and  the  substance  to 
be  dialysed  is  placed  in  a  thin  layer  on  the  dish.  The  crystal- 
loids (strychnine,  etc.)  pass  into  the  water,  the  colloids  (mucus, 
etc.)  remain  in  the  dish.  This  method  is  peculiarly  applicable 
to  the  separation  of  poisons  from  animal  matters. 

If  the  mixed  solution  exert  pressure  upon  the  membrane, 
colloids  as  well  as  crystalloids  may  be  found  to  pass  in  consider- 
able quantities  through  that  membrane,  along  with  the  fluid 
forced  through  by  the  pressure. 

If  peroxide  of  iron  be  dissolved  in  a  solution  of  perchloride  of  iron,  and 
the  whole  be  then  dialysed,  the  chloride  of  iron  will  pass  through  the  mem- 
brane, leaving  the  colloid  oxide  of  iron  behind  in  solution.  Neutral  Prus- 
sian blue  (as  used  in  microscopical  work)  is  also  a  colloid,  and  may  be  puri- 
fied in  the  same  way :  so  is  sucrate  of  copper,  a  soluble  compound  of  copper 
oxide  with  sugar,  which  is  reduced  on  heating.  Albumen  may  also  be 
obtained  in  a  relatively  pure  form  by  separating  it  by  dialysis  from  the 
greater  part  of  the  salts  that  it  may  contain. 

If  the  membrane  used  be  the  gastric  or  intestinal  membrane,  taken 
after  death,  it  is  found  that  curare  or  snake  poison  will  not  pass  through  it, 
while  they  are  absorbed  readily  by  the  dermis  or  by  serous  membranes. 
They  seem  not  to  wet  the  former ;  hence  the  selective  absorption  of  poisons 
has  a  certain  physical  basis. 

Absorption  by  the  dermis  is  seen  to  be  a  physical  process ;  the  walls  of 
the  vessels,  both  lymphatic  and  venous,  are  known  to  be  physically  perme- 
able to  osmose,  and  the  salt,  if  it  be  placed  on  a  vascular  region,  is  quickly 
absorbed,  the  osmose  being  accelerated  by  the  flow  of  liquid  in  the  vessels. 
Substances  brought  in  contact  with  the  pulmonary  epithelium  are  also  very 
rapidly  absorbed.  Lymph  acts  towards  blood  as  water  does  towards  a  saline 
solution,  and  the  tendency  of  osmotic  action  is  to  carry  the  fluids  of  the 
body  into  the  blood-stream.  Repletion  of  the  vessels  checks  this  tendency. 
Adhesion  between  water  and  oil  is  greatly  increased  if  a  little  alkali  be 
dissolved  in  the  water.  When  the  mucous  membrane  is  covered  with  bile 
it  has  much  more  affinity  for  oil  globules,  which  are,  besides,  each  endowed 
by  emulsionising  with  an  aqueous  or  soapy  covering,  which  makes  them  act 
like  minute  masses  of  water,  and  enables  them  not  to  experience  any  rela- 
tive repulsion  when  carried  with  the  rest  of  the  aqueous  stream. 

Osmose  through  porous  membranes  is  thus  related  to  capil- 
lary affinity  and  to  diffusion,  but  it  bears  no  exact  numerical 
relation  to  either  of  these,  for  it  depends  on  the  relation  between 
the  pores  and  the  solid  parts  of  the  membrane,  uponrthe  nature 
of  the  material  (colloidal  or  otherwise^)  of  the  membrane,  upon 


288  OF  LIQUIDS.  [CHAP. 

the  width  of  the  pores,  upon  the  temperature  and  electrical  con- 
dition, upon  the  mutual  action  of  the  fluids,  and  in  physio- 
logical cases  (Milne-Edwards,  Physiologie,  tome  V)  it  seems  to 
depend  on  the  influence  of  the  nervous  system. 

Solution-Pressure  or  Osmotic  Pressure.  —  The  above  phenomena  are 
explained  as  follows  :  —  The  molecules  of  any  chemically  inactive  substance, 
such  as  sugar,  when  dissolved  in  water,  act  precisely  as  if  they  were  mole- 
cules of  an  independent  Gas,  which  exerts  its  own  Pressure.  Accordingly, 
where  there  are  differences  of  concentration  within  a  solution,  there  are  dif- 
ferences in  the  pressure  exerted  by  these  molecules ;  and  the  molecules  of 
the  dissolved  substance  tend  proportionately  to  travel  towards  the  region 
of  less  concentration,  thus  giving  rise  to  the  phenomena  of  Diffusion; 
and  equilibrium  is  not  attained  until  their  quasi-gaseous  pressure,  the  Solu- 
tion-Pressure or  Osmotic  Pressure,  and  along  with  it  the  concentration  of 
the  solution,  have  become  equalised  throughout  the  mass.  This  equalisa- 
tion is  slow,  because  the  liquid  obstructs  the  transference  of  the  molecules. 
If  the  liquid  be  contained  in  a  vessel  terminated  above  by  a  long  tube,  and 
if  it  be  separated  from  pure  water  by  a  membrane  or  pellicle  (such  as  that 
formed  by  precipitation  through  the  contact  of  a  solution  of  copper  with  one 
of  a  ferrocyanide),  which  is  permeable  by  water  but  not  by  the  dissolved 
substance,  water  will  enter  through  the  membrane  until  the  liquid  stands  in 
the  tube  at  a  height  which  measures  the  osmotic  pressure.  It  is  then  found, 
if  the  substances  dissolved  be  not  decomposed  or  dissociated  by  the  act  of 
solution,  that  the  pressure  exerted  by  the  dissolved  substance  is  the  same  as 
would  have  been  exercised  by  its  molecules  if  it  had  been  reduced  to  a  gas 
at  the  temperature  and  volume  of  the  solution;  and  that  this  pressure  is 
proportional  to  the  absolute  temperature  (p.  364),  and  is  independent  of  the 
nature  of  the  septum.  In  saline  solutions,  on  the  other  hand,  the  phenomena 
are  of  precisely  the  same  kind,  with  this  exception,  that  the  pressure  is 
mostly  greater  than  with  solutions  of  indifferent  substances  :  and  this  tends 
to  show  that  there  is  Dissociation  of  the  molecules,  which  dissociation  is 
more  complete  the  greater  the  dilution  or  the  higher  the  temperature. 
Where  the  septum  is  more  or  less  permeable  to  the  substance  dissolved,  as 
well  as  to  water,  we  have  the  phenomena  of  ordinary  dialytic  osmosis,  as 
through  parchment  paper  or  animal  membrane. 

2.   THE  STATICS  OF  LIQUID  MASSES. 

Liquids  are  incapable  of  resisting  a  change  of 
shape  when  acted  on  by  force  which  is  not  equally  applied 
over  the  whole  surface,  and  they  flow  when  thus  acted  on, 
unless  supported  on  all  sides. 

All  soft  masses  which  cannot  in  the  aggregate  permanently 
resist  a  change  of  shape  are  practically  liquids^  and  are  subject 
to  hydrostatical  laws. 

Dilatancy .  —  Granular  masses,  such  as  loose  sand,  alter  in  volume  when 
their  shape  is  changed.  If  their  volume  cannot  alter,  neither  can  their  shape : 
they  are  then  rigid.  If  they  have  been  well  shaken  up,  they  occupy  the  least 


XL]  DILATANCY.  289 

possible  volume  ;  and  any  change  of  shape  involves  increase  of  volume.  If 
water  lie  between  the  granules,  the  water  may  fail  to  fill  the  spaces  between 
the  granules  if  the  volume  of  the  whole  be  thus  increased  :  and  the  mass 
becomes  rigid  whenever  any  change  of  shape  would  thus  result  in  a  ten- 
dency to  a  vacuum  between  the  granules.  When  footprints  are  impressed 
upon  wet  sand,  the  change  of  shape  under  the  foot  is  enabled  to  go  on  by 
drawing  water  from  the  neighbouring  sand,  which  becomes  dry.  (Osborne 
Reynolds.) 

It  is  often  convenient,  in  discussing  the  equilibrium  of 
liquids,  to  imagine  little  elements  of  the  liquid,  floating  in  and 
forming  part  of  the  liquid,  to  become  solidified  or  otherwise  to 
become  separately  recognisable,  while  not  altering  their  other 
relations  to  the  surrounding  mass.  Then,  if  the  liquid  as  a 
whole  be  at  rest,  each  of  these  little  elements  of  mass  must  also 
be  at  rest. 

This  being  so,  the  forces  acting  on  each  little  element  of 
mass  must  be  in  equilibrium,  and  their  resultant  must  be  nil. 
This  can  only  occur  (since  each  fluid  element  is  subject  to  pres- 
sure on  all  sides,  as  may  be  understood  by  considering  the  rush 
of  fluid  from  all  sides  that  would  occur  if  the  little  element  of 
mass  were  suddenly  annihilated)  if  the  pressure  on  all  sides  be 
equal ;  and  since  the  element  may  be  reduced  to  a  material 
point,  the  proposition  follows  that  at  any  point  in  a  liquid  the 
pressure  in  all  directions  is  equal. 

The  pressure  at  any  point  of  the  surface  of  a  liquid  at  rest 
must  be  at  right  angles  to  the  surface.  If  it  were  not  so,  it  must 
be  oblique ;  being  oblique,  it  would  be  resoluble  into  a  compo- 
nent at  right  angles  and  one  parallel  to  the  surface.  The  latter 
could  not  fail  to  act,  the  surface  being  that  of  a  liquid ;  hence 
the  liquid  would  not  be  at  rest ;  whence  there  is  no  such  com- 
ponent, and  the  pressure  is  at  right  angles  to  the  surface.  Con- 
versely, when  a  liquid  is  at  rest,  the  pressure  which  it  exercises 
on  the  vessel  containing  it  is  at  right  angles  to  the  walls  of  the 
vessel,  for  the  walls  of  the  vessel  coincide  in  aspect  with  the 
surface  of  the  liquid. 

If  in  a  liquid  at  rest,  expressly  supposed  to  be  not  under  the 
influence  of  gravity,  two  elements  were  imagined  to  be  in  con- 
tact, and  yet  to  be  subject  to  different  pressures,  there  would  at 
the  point  or  surface  of  contact  be  a  relative  difference  of  pres- 
sures which  would  necessarily  cause  movement  of  the  liquid ; 
but  the  liquid  is  supposed  to  be  at  rest ;  hence  there  can  be  no 
difference  between  the  pressures  of  any  two  contiguous  elements, 
and  the  pressure  throughout  a  weightless  liquid  at  rest  is  every- 


290  OF  LIQUIDS.  [CHAP. 

where  the  same  (Pascal's  Principle),  and  is  the  same  in  all 
directions.  It  is  the  same  within  the  liquid  as  it  is  at  right 
angles  to  the  surface ;  and  therefore,  instead  of  considering  p 
the  pressure  per  sq.  cm.  between  the  liquid  and  the  vessel  con- 
taining it,  and  at  right  angles  to  the  surface  of  the  liquid  or  of 
the  vessel,  we  may  replace  this  by  the  pressure  p  per  sq.  cm., 
numerically  equal  to  p,  but  exerted  within  the  liquid  in  all 
directions.  This  is  called  the  Hydrostatic  Pressure. 

If  a  pressure  be  applied  from  without  to  some  of  the  parti- 
cles of  a  liquid,  and  if  that  liquid  be  free  to  change  its  shape, 
it  will  do  so ;  if  it  be  not  free  to  flow,  the  particles  pressed  on 
will  press  against  contiguous  particles,  and  these  against  their 
neighbours ;  thus  the  pressure  becomes  equalised  throughout 
the  whole  of  the  liquid.  This  is  the  principle  of  the  so-called 
Transmissibility  of  Fluid  Pressures.  The  pressure 
applied  to  any  area  of  the  surface  of  a  liquid  not  free  to  flow 
becomes  equally  felt  over  every  equal  area  of  the  surface. 

If  a  wide  cylinder,  with  a  piston  whose  area  is  a  sq.  cm.,  be 
placed  in  communication  by  a  tube  with  another  cylinder,  nar- 
rower, and  provided  with  a  piston  whose  area  is  b  sq.  cm.,  and  if 
both  cylinders  and  the  communicating  tube  be  completely  filled 
with  water,  a  total  effective  pressure  P  applied  to  the  smaller 
piston  will  produce  an  equal  pressure  P  on  every  b  sq.  cm. 
of  the  surface  of  the  fluid,  and  therefore  on  every  b  sq.  cm. 
of  the  larger  piston,  and  a  proportionately  greater  pressure, 
P'=P-a/6,  on  the  whole  surface  (a  sq.  cm.)  of  the  greater 
piston.  This  is  the  principle  of  the  Hydraulic  Press,  by  which 
a  smaller  force,  P,  acting  on  a  smaller  piston,  may  produce  a 
greater  force,  P'  =  (P  •«/&),  distributed  over  the  inner  surface 
of  a  larger  piston  ;  and  as  the  area  a  may  bear  any  proportion 
to  the  area  6,  the  force  obtained  may  bear  any  proportion  to  the 
force  applied.  The  principle  of  the  Conservation  of  Energy 
holds  good,  however;  the  volume  of  water  remains  constant, 
and  if  the  smaller  piston  move  through  a  space  s,  the  larger 
piston  moves  through  a  shorter  space  s'  =  (s  •£/«).  The  work 
done  upon  the  smaller  piston,  total  force  x  displacement  =  Ps  ; 
that  done  by  the  larger  piston,  PV  =  (P  •  a/H)  x  (s  •  5/a),  gives 
the  same  product,  Ps. 

An  analogous  action  takes  place  in  an  aneurism.  A  small  aperture  of 
communication  with  the  artery  allows  the  arterial  blood-pressure  to  be  com- 
municated to  the  whole  interior  of  the  aneurismal  sac ;  the  total  pressure 
exerted  is  very  great,  the  rate  of  distension  comparatively  slow. 


XL]  HYDROSTATICS.  291 

If  the  action  of  a  hydraulic  press  be  reversed,  a  great  total 
pressure  applied  to  the  larger  piston  will  have  the  effect  of  pro- 
ducing a  smaller  total  pressure  on  the  inner  surface  of  the  small 
piston.  A  small  resistance  applied  to  the  smaller  piston  will 
have  the  effect  of  checking  the  onward  motion  of  the  larger 
piston  under  the  influence  of  the  powerful  force.  If  a  bladder 
full  of  water  be  connected  with  a  narrow  upright  glass  tube, 
heavy  weights  placed  on  the  bladder  will  be  able  to  uphold  only 
a  very  small  quantity  of  liquid  in  the  tube,  this  arrangement 
being  in  fact  a  hydraulic  press  worked  backwards.  If  the  tube 
be  shortened  down  so  as  to  form  simply  the  neck  of  the  bladder, 
the  total  expulsive  pressure  exerted  by  the  bladder  upon  the  con- 
tents of  the  neck  may  seem  to  be  very  small  when  compared  with 
the  total  pressure  exerted  over  the  walls  of  the  bladder  upon  the 
whole  contents.  Here  we  have  apparent  destruction  of  force. 

Heavy  Liquids.  —  Let  us  suppose  a  cylindrical  vessel,  filled 
with  liquid,  to  stand  upon  a  plane  base ;  the  area  of  the  base  is 
A  sq.  cm. ;  the  height  of  the  liquid  is  h  cm. ;  the  density  of  the 
liquid  is  p ;  and  the  local  acceleration  of  gravity  is  g.  The  quan- 
tity of  matter  standing  on  the  base  is  Ahp,  and  the  weight  of 
that  mass  is  Ahpg.  The  total  pressure  on  the  base  is  therefore 
P  =  Ahpg,  and  the  pressure  per  unit  of  area  of  the  base  is 

P  =  %• 

If  the  unit  of  area  on  which  the  pressure  is  to  be  found  be 
plane,  but  not  horizontal,  it  may  be  considered  to  lie  at  an  aver- 
age depth  equal  to  the  depth  of  its  centre  of  figure.  Then  the 
pressure  p  on  a  plane  of  unit-area,  chosen  anywhere  in  the  fluid 
and  looking  in  any  direction,  is  equal  to  the  product  of  pg  into 
the  vertical  distance  h  between  the  surface  of  the  liquid  and  the 
centre  of  figure  of  that  plane ;  and  if  the  plane  have  any  area  A 
other  than  unity,  the  pressure  is  the  product  of  the  area  A,  x  h 
the  vertical  depth  of  the  centre  of  figure,  x  p  the  density,  x  g. 

For  all  points  in  the  same  horizontal  layer  the  depth  h  is 
the  same,  and  therefore  in  a  heavy  fluid  the  pressure  is  the  same 
throughout  the  same  indefinitely-thin  horizontal  layer.  The 
lateral  pressure  on  the  rim  of  the  stratum  is  equal  to  the  ver- 
tical pressure  at  that  level  —  i.e.,  p  =  hpg  per  unit  of  area. 

In  Fig.  102  let  A,  B,  C  represent  three  vessels,  each  having 
a  base  whose  area  is  A  square  centimetres,  and  each  filled  with 
water  to  a  height  of  h  cm.  The  whole  pressure  on  the  base  is 
the  same  (P  =  Ahpg)  in  all  the  cases,  though  the, weights  of 
the  masses  of  water  differ  greatly. 


292 


OF  LIQUIDS. 


[CHAP. 


In  the  first  case  the  lateral  pressure  against  the  walls  of  the  cylinder 
produces  a  reaction  which  has  no  vertical  component  and  does  not  affect  the 
pressure  on  the  base.  In  the  second  we  may  isolate  a  cylinder  of  the  fluid 
in  the  fluid ;  the  lateral  parts  of  the  fluid  have  a  certain  weight :  the  walls 
of  the  vessel  are  exposed  to  a  certain  pressure  P  which  is  equal  to  the  prod- 
uct of  their  area  x  the  depth  of  their  centre  of  figure  (=  \li)  into  pg.  This 
pressure  may  be  resolved  into  a  horizontal  and  a  vertical  component,  to  each 
of  which  the  corresponding  reactions  of  the  walls  of  the  vessel  are  equal  and 
opposite :  the  one  reaction  resists  outward  yielding,  the  other  supports  the 

Fig.102. 
B 


'  weight  of  the  fluid.  It  will  be  found  that  the  upward  reaction  of  the  sloping 
walls  of  vessel  B  is  exactly  equal  to  the  Weight  of  the  fluid  overlying  them ; 
the  walls  support  the  whole  weight  of  the  lateral  masses.  In  vessel  C  the 
reaction  of  the  walls  of  the  vessel  may  be  found  in  the  same  way  and 
resolved  into  horizontal  and  vertical  components.  The  latter  acts  down- 
wards upon  the  fluid,  and  will  be  found  to  be  precisely  equal  to  the  weight 
of  that  quantity  of  fluid  that  would  lie  vertically  above  the  base  if  the 
column  of  fluid  were  perfectly  cylindrical  and  of  the  height  A,  but  which, 
owing  to  the  form  of  the  vessel,  does  not  so  lie. 

The  total  pressure  on  the  base  of  a  vessel  containing  liquid 
depends  on  the  height  (Ji)  of  the  liquid  and  the  area  (A)  of 
the  base,  the  density  p  of  the  liquid,  and  g  the  local  acceleration 
of  gravity,  but  does  not  depend  on  the  actual  Mass  or  Weight 
of  the  liquid  employed.  It  is  P  =  Ahpg.  If  a  flask  filled  with 
water  be  fitted  with  a  cork  in  which  a  long  narrow  tube  is  fixed 
upright,  a  very  small  quantity  of  water  poured  into  the  tube  will 
be  competent  to  burst  the  flask. 

This  proposition  —  that  the  same  amount  of  water  may  pro- 
duce widely-differing  amounts  of  pressure  on  the  vessel  in  which 
it  is  contained,  these  amounts  depending  on  the  form  of  that 
vessel  —  is  said  to  be  a  Hydrostatic  Paradox  ;  the  only  paradox- 
ical element  about  it  is,  however,  its  discrepancy  with  a  certain 
uninformed  intuitional  belief  in  the  Conservation  of  Force. 

A  slack  bag  containing  liquid,  and  set  to  xest  upon  a  plane 
surface,  exerts  a  pressure  upon  that  surface  which  is  equal  to 
the  product  of  the  area  of  contact  x  the  height  of  the  centre  of 
gravity  of  the  liquid.  So  for  semi-fluid  masses. 

When  the  surface  of  a  liquid  is  exposed  to  the  atmospheric 


XI.] 


HYDROSTATICS. 


293 


pressure  of  760  mm.  or  76  cm.,  it  bears  on  each  sq.  cm.  of  sur- 
face the  weight  of  76  cub.  cm.  of  mercury,  or  1033*3  grammes  ; 
this  is  equal  to  (1033-3  x  981)  dynes  :  or  if  the  barometer 
stand  at  x  cm.,  the  pressure  p  on  each  sq.  cm.  of  surface  is 
(13-596a;  x  981)  dynes.  This  number  of  units  of  force  per  sq. 
cm.  may  be  expressed  by  the  symbol  n  .  Then  the  Total  Atmos- 
pheric Pressure  on  area  A  sq.  cm.  is  11=  An.  The  liquid  pressure 
on  the  area  A  at  the  mean  depth  h  cm.  would  have  been  Ahpg 
if  there  had  been  no  pressure  at  the  surface.  When  the  atmos- 
pheric pressure  acts  at  the  surface  of  a  liquid,  the  total  pressure 
on  any  plane,  whose  area  is  A  and  whose  mean  depth  below  the 
surface  is  A,  amounts  to  (An  -f- 


When  the  human  body  (as  in  ordinary  circumstances)  has  the  head  in 
the  highest  position,  the  blood  in  the  head  is  exposed  to  the  ordinary  atmos- 
pheric pressure.  If  the  head  be  downwards,  the  pressure  on  the  blood- 
vessels of  the  head  is  increased  by  the  weight  of  the  column  of  blood  in  the 
inverted  body,  and  hence  there  is  congestion.  If  the  body  float  submerged 
in  a  liquid  of  its  own  sp.  density,  head  up,  the  pressure  on  the  blood  vessels 
of  the  head  is  the  ordinary  atmospheric  pressure  increased  by  the  weight  of 
the  column  of  liquid  immediately  overlying  the  head  ;  but  if  the  head  be 
suspended,  though  the  increased  depth  causes  a  correspondingly-increased 
external  pressure  on  the  head,  yet  the  equally-increased  internal  pressure  of 
blood  balances  this  effect,  and  there  is  no  congestion.  This  may  be  illus- 
trated by  a  loop  of  thin  indiarubber-tubing  filled  with  water  :  suspended  in 
air,  the  depending  part  is  distended  :  suspended  in  water,  it  is  relieved  from 
distension. 

Communicating  Vases.  —  "Water  seeks  its  own  level."  If 
there  be  two  communicating  vessels  containing  the  same  liquid, 
the  lowest  part  of  the  communicating  channel  may  be  consid- 
ered as  a  common  base  :  its  area  is  A.  Regarding  it  as  the  base 
of  vessel  C  (Fig.  103),  we  see 
that  the  pressure  P  on  it  must 
be  Ahpg,  and  the  height  of  the 
liquid  in  C  is  h;  regarding  it 
as  the  base  of  vessel  B,  the  pres- 
sure (which  must  be  the  same, 
for  the  liquid  is  at  rest)  is  equal 
to  Ahtpg  :  whence  h  =  A;,  the 
height  of  the  liquid  in  the  two 
vases  must  be  the  same,  and  the 
level  must  be  the  same  in  two  communicating  vases,  whatever  be 
the  shape  of  the  communication,  so  long  as  the  communication- 
pipe  is  continuously  filled  with  liquid.  This  implies  that  suffi- 
cient time  for  assuming  equilibrium  is  allowed. 


F1&.103. 


294  OF  LIQUIDS.  [CHAP. 

If  the  liquids  in  the  two  communicating  columns  be  not  of 
the  same  density,  the  effect  is  an  inequality  in  the  heights  of  the 
columns,  which  vary  inversely  as  the  density. 

The  two  pressures  are  AJitp,g  and  Ahpg  ;  these  are  equal ;  .•.  hp  =  h,pfl 
or  h  :  ht : :  p/ :  p. 

In  Fig.  104  the  column  of  water  ab  and  that  of  mercury  cd  balance  one 
another  because  they  produce  an  equal  pressure  on  the  base  e.  If  a  U-tube 
contain  water,  of  which  that  in  one  limb  is  heated  while 
that  in  the  other  remains  cool,  the  liquid  in  the  hotter  limb 
will  stand  at  a  higher  level  than  that  in  the  cooler.  The 
relative  specific  densities  of  fluids  may  be  estimated  by 
methods  based  on  this  principle. 

The  accuracy  of  the  " water-level"  may  be  in- 
terfered with  by  capillarity.  If  both  limbs  of  a 
U-tube  be  narrow,  but  unequally  so,  water  will 
stand  at  a  greater  height  in  the  narrower  limb. 

If  a  U-tube  be  taken,  of  which  the  narrower  limb  is  the 
shorter,  the  quantity  of  water  placed  in  the  tube  may  be 
regulated  so  as  to  afford  the  following  three  conditions :  — 
(1)  The  shorter  limb  filled  with  water,  the  upper  surface  of  which  is  con- 
cave, while  the  water  stands  at  a  lower  level  in  the  wider  tube ;  (2)  The 
shorter  limb  completely  filled  with  water  the  upper  surface  of  which  is  plane, 
and  the  concave  surface  of  the  water  in  the  wider  tube  at  nearly  the  same 
level,  but  a  little  higher ;  (3)  The  shorter  limb  completely  filled  with  water 
the  upper  surface  of  which  is  convex,  while  the  water  stands  at  a  higher 
level  in  the  wider  tube,  its  surface  being  concave. 

Every  liquid  tends  to  set  the  whole  of  its  free  surface  at 
right  angles  to  the  force  of  gravity. 

When  a  cylindrical  vessel  containing  a  liquid  is  rotated 
round  its  longitudinal  axis,  the  surface  of  the  liquid  assumes 
a  parabolic  form  which  is  maintained  constant  so  long  as  the 
rotation  is  uniform. 

Thus  the  form  of  the  free  surface  of  liquids  is  affected  by 
gravity,  by  molecular  forces,  and  by  rotation. 

Archimedes'  Principle.  —  If  an  element  of  mass  of  a  liquid 
be  supposed  to  be  solidified,  this  will  not  affect  its  equilibrium 
in  the  midst  of  the  fluid  of  which  it  had  previously  formed  part ; 
it  will  neither  rise  nor  sink.  Even  though  its  nature  be  altered, 
provided  that  it  do  not  become  either  lighter  or  heavier,  it  will 
neither  sink  nor  rise:  it  has  apparently  lost -its  weight.  If  it 
become  heavier  than  the  liquid  it  will  sink ;  if  it  become  lighter 
it  will  rise.  Gravity  has  no  effect  in  making  a  body  rise  or  sink 
in  a  liquid  except  in  so  far  as  there  is  a  difference  between  the 
density  of  the  liquid  and  that  of  the  body  suspended  in  it.  This 


XI.] 


HYDROSTATICS. 


295 


leads  to  Archimedes'  principle:  —  "A  body  suspended  in  a 
fluid  apparently  loses  as  much  weight  as  is  equal  to 
the  weight  of  the  mass  of  fluid  which  it  displaces." 
The  application  of  this  principle  to  the  study  of  specific  density 
we  have  already  seen. 

A  body  lighter  than  water  may  be  loaded  with  just  so  much 
mass  as  will  sink  the  light  body  without  that  additional  mass 
itself  entering  the  liquid ;  the  whole  will  then  float,  the  lighter 
body  displacing  a  bulk  of  water  equal  to  its  own  bulk;  the 
weight  opposing  the  buoyancy  of  the  water  is  the  weight  of 
the  body  plus  that  of  the  load  placed  on  it ;  and  the  ratio 

weight  of  body 


weight  of  (body  4-  load) 


=  specific  density  of  the  floating  body. 


That  a  body,  even  though  sufficiently  light  to  float,  tends  to  sink  in  water 
until  the  weight  of  the  water  displaced  becomes  equal  to  the  weight  of  the 
whole  body,  may  be  shown  by  a  very  simple  experiment.  Take  two  similar 
phials,  two  small  elastic  bands,  and  four  nails  which  must  not  be  too  heavy. 
With  these  may  be  constructed  a  couple  of  rough  models  representing  a 
person  with  his  arms  kept  down  by  his  sides,  and  a  person  whose  arms  are 
elevated  above  his  head.  On  putting  these  models  into  water  the  difference 
in  floating  capacity  will  be  very  obvious. 

Measurement  of  Pressure.  —  The  pressure  to  which  the 
surface  of  a  liquid  is  exposed  can.be  measured  by  the  height 
of  the  liquid  column  which  that  pressure  can  support.  If  in 
Fig.  105  the  water  contained  in  the  cylinder  AB  be  exposed  to 
a  certain  pressure  communicated  by  a  piston  at  A,  and  if  a  side 


Writlnr  Point—. 


To  Reffistennr  Appara 


tube  (a  piezometer  tube)  placed  at  C  be  in  communication 
with  the  liquid,  water  will  rise  in  the  tube  until  there  is  equi- 
librium. This  equilibrium  is  between  the  Pressure  of  the  fluid 
in  AB  (together  with  the  atmospheric  pressure  acting  through 
A),  tending  to  push  upwards  the  column  of  water  QD,  and  the 
downward  pressure  upon  C,  due  to  the  Weight  of  that  column, 


296  OF  LIQUIDS.  [CHAP. 

which  (together  with  the  atmospheric  pressure  acting  on  D) 
tends  to  make  it  sink  back  into  the  cylinder.  The  whole  out- 
ward Pressure  P  exerted  by  the  liquid  on  the  orifice  C  must  be 
equal  to  the  downward  pressure  due  to  the  Weight  of  the  col- 
umn CD. 

The  latter  is  (if  the  area  of  the  orifice  at  C  be  A,  and  h  the  height  of 
the  column)  equal  to  g  x  the  mass  of  the  column  =  Ah- p- g.  As  this  is  dis- 
tributed over  an  area  A,  on  every  unit  of  area  of  the  surface  of  the  fluid  its 
amount  must  be  li  •  pg.  Whence  at  C  the  outward  pressure  exerted  by  the 
liquid  is,  per  unit  of  area,  p  =  hpg. 

Let  us  suppose  that  the  column  CD  is  one  of  water,  13-596  cm.  high ; 
the  pressure  per  unit  of  surface  is  hpg  =  (13-596  x  1  x  981)  dynes  per  sq.  cm. 

If  at  B  a  U-tube  (a  manometer  tube)  be  fixed,  contain- 
ing in  its  bend  a  quantity  of  mercury,  the  mercury  will  stand  at 
the  same  level  in  both  branches  so  long  as  the  internal  pressure 
and  the  external  are  equal ;  but  if  the  internal  pressure  be 
increased,  the  mercury  will  be  depressed  in  the  branch  nearer 
the  cylinder,  and  will  rise  in  the  other. 

In  the  case  supposed  it  would  (setting  aside  any  difference  of  pressure 
due  to  difference  of  level  between  C  and  B)  sink  through  £  cm.  in  the 
nearer  and  rise  through  f  cm.  in  the  farther  limb :  a  difference  of  1  cm. 
of  mercury  being  thus  established.  This  column  of  mercury  is  that  whose 
weight  balances  the  internal  pressure :  its  weight  is  (1  cub.  cm.  x  13-596  x 
981)  dynes,  acting  upon  every  square  centimetre.  Hence  — 

The  pressure  on  the  surface  of  the  liquid  in  the  cylinder, 
AB  of  Fig.  105,  may  be  equally  well  represented  in  brief 
phraseology  as  a  pressure  of  say  13-596  cm.  of  water,  or  one  of 
1  cm.  of  mercury. 

Exploration  of  the  pressure  in  the  interior  of  a  sta- 
tionary liquid  mass.  —  In  Fig.  105  let  there  be  an  aperture  in 
the  walls  of  the  cylinder,  at  O  ;  through  this  aperture  pass  a  tube 
which  exactly  fits  it.  The  inner  end  of  this  tube  is  furnished 
with  a  flexible  and  elastic  cap.  The  outer  end  is  connected 
directly  or  by  means  of  indiarubber  —  or,  better,  of  leaden  — 
tubing,  first  with  a  stopcock  (the  bore  of  which  is  the  same  as 
that  of  the  tube),  and  then  with  a  manometer  tube.  Before 
the  tube  is  passed  through  the  orifice  O,  the  level  of  the  mer- 
cury in  the  manometer  must  be  adjusted.  This  is  done  while 
the  stopcock  is  open,  by  pouring  mercury  into  the  manometer 
tube  and  bringing  it  to  an  exact  level  l>y  the  addition  or  sub- 
traction of  mercury  in  the  outer  limb;  the  stopcock  is  then 
closed,  and  the  tube  adjusted  with  its  elastic  closed  end  in  the 
body  of  the  cylinder.  The  stopcock  is  then  opened ;  the  pres- 


XL]  MEASUREMENT   OF  PRESSURE.  297 

sure  of  the  fluid  in  the  cylinder  on  the  indiarubber  cap  (if  it 
differ  from  the  atmospheric  pressure)  alters  the  shape  of  the 
cap,  and  the  mercury  in  the  manometer  assumes  a  difference  of 
level  which  indicates  the  pressure  in  the  interior  of  the  cylin- 
der. If  the  cap  be  so  small  that  it  is  collapsed  by  a  given  pres- 
sure JP,  it  cannot  be  used  to  record  pressures  of  greater  amount 
than  p.  This  defect  can  be  remedied  either  by  using  a  larger 
cap  or  else  by  using  capillary  manometers  of  uniform  bore,  in 
which  the  displacement  of  a  very  small  quantity  of  mercury 
(and  therefore  a  small  compression  of  the  indiarubber  cap)  will 
serve  to  indicate  high  differences  of  pressure.  If  the  cap  be  at 
all  inflated  before  it  is  inserted  within  the  cylinder,  the  elastic 
recoil  of  the  cap  adds  an  unknown  quantity  to  the  internal  fluid 
pressure,  and  the  readings  of  the  instrument  are  untrustworthy, 
unless  special  contrivances  are  made  use  of  for  ascertaining  the 
exact  effect  due  to  this  cause. 

Fig.  105  F  shows  the  essential  parts  of  another  instrument 
by  which  the  pressure  in  the  cylinder  AB  may  be  measured ; 
it  is  substantially  identical  with  Bourdon's  Steam  Gauge. 
A  hollow  tube  of  elastic  metal  having  an  elliptical  cross-section, 
bent  into  the  shape  of  a  3,  and  filled  with  liquid  (alcohol, 
glycerine,  water,  or  oil),  suffers  changes  of  shape  under  the 
influence  of  changes  of  pressure  in  the  contained  fluid.  When 
the  internal  pressure  increases,  the  3  straightens  out ;  when  it 
decreases,  it  becomes  more  curved. 

The  pressure  increasing,  the  cross-section  tends  to  become  more  circular 
(the  circle  being  a  figure  of  greatest  area  for  least  circumference)  :  the 
surface  and  the  mean  curvature  are  constant ;  the  curvature  across  the  tube 
increasing,  that  along  the  tube  diminishes,  and  the  tube  straightens  out. 
The  same  principle  is  applied  in  some  Aneroid  Barometers,  in  which  a  coil 
of  elliptical  tubing  tends  to  straighten  out  when  the  external  pressure 
diminishes ;  and  vice  versa,  tends  to  flatten  and  curl  up  when  it  increases. 

Such  a  tube  is  continuous  with  a  box  or  cavity  containing 
liquid,  which  may  in  its  turn  be  continuous  with  the  liquid  of 
the  cylinder  when  the  surface-pressure  has  to  be  found,  or  may 
be  connected  merely  with  an  indiarubber  cap  like  that  inserted 
as  an  explorateur  in  orifice  O  of  the  same  figure. 

• 

For  physiological  work  this  principle  is  applied  in  Fick's  Federmano- 
meter,  in  which  the  Q-tube  is  filled  with  alcohol,  and  the  tubes  which 
intervene  between  it  and  those  blood-vessels  in  which  the  blood-pressure 
has  to  be  determined  are  filled  with  a  solution  of  bicarbonate  of  soda  of  a 
sp.  gr.  of  1-083;  this  being  (Cyon)  the  strength  of  solution' which  most 
markedly  checks  any  tendency  to  coagulation. 


298  OF   LIQUIDS.  [CHAP. 

A  given  amount  of  bend  of  the  Q-tube  may  be  interpreted  as 
signifying  exposure  to  a  certain  amount  of  pressure,  if  the  instru- 
ment be  previously  graduated  by  rinding  the  relation  between 
certain  known  pressures  and  the  distortions  produced  by  them. 

The  instrument  S  in  Fig.  105  is  the  sphygmoscope  of 
Marey.  The  tube  a  is  closed  by  an  elastic  cap  which  projects 
into  the  lumen  of  the  wider  tube  b  ;  a  and  its  cap  are  filled  with 
liquid,  which  is  continuous  with  that  of  the  cylinder ;  the  pres- 
sure within  the  cylinder  forces  the  fluid  into  the  cap  until  the 
elasticity  of  the  cap  and  the  pressure  of  the  liquid  are  in  equi- 
librium :  the  air  in  the  tube  b  is  compressed,  and  the  pressure  is 
communicated  to  a  manometric  capsule  or  other  registering 
apparatus,  the  displacement  of  the  lever  of  which  may  be  made 
by  preliminary  graduation  to  indicate,  in  terms  of  mercury- 
column,  the  value  of  the  pressure  to  be  measured. 

The  instrument  M  is  the  manometre  mdtallique  in- 
scripteur  of  Marey.  An  elastic  metallic  capsule  filled  with 
liquid,  which  is  continuous  with  that  of  the  cylinder  AB,  plays 
in  this  instrument  a  part  which,  in  principle,  is  exactly  the 
same  as  that  of  the  elastic  cap  in  the  sphygmoscope  S. 

Measurement  of  Variable  Pressure.  —  If  the  pressure  in 
the  cylinder  AB  of  Fig.  105  be  variable  —  as,  for  example,  if 
the  piston  A  oscillate  —  the  various  manometers  represented 
in  the  figure  will  give  oscillating  readings.  The  manometers 
at  B  or  O  and  the  piezometer  at  C  are  subject  in  action  to  the 
defect  that,  when  a  single  momentary  increase  of  pressure  pro- 
duces a  rise  of  the  liquid  or  of  the  mercury  in  the  column,  the 
column  does  not  return  promptly  to  its  mean  position  when 
the  additional  pressure  is  taken  off,  but  oscillates  like  a  pen- 
dulum for  a  period  of  time  more  or  less  protracted,  until  at 
length  friction  and  viscosity  bring  it  to  rest.  If  the  piston  A 
oscillate,  its  movements  are  not  faithfully  reproduced  by  the 
oscillations  of  the  mercury  manometer,  for  the  latter  depend 
on  (1)  the  weight  of  the  column  of  liquid  lifted  at  each  dis- 
placement from  the  mean  position ;  (2)  the  variations  of  inter- 
nal pressure  tending  to  make  the  column  assume  new  mean 
positions ;  and  (3)  on  friction ;  and  they  are  the  result  of  the 
composition  of  two  sets  of  oscillations,  the  one  due  to  the  vari- 
ations of  pressure  in  AB  (and  agreeing  with  these  variations 
in  period,  but  not  in  form  or  amplitude  or  phase),  while  the 
other  set,  the  pendulum-oscillations  of  the  manometer-column 
(which  may  even  overpower  the  former  if  the  mass  of  mercury 


XL]  MEASUREMENT   OF   PRESSURE.  299 

be  great  or  if  the  tubes  be  wide  and  offer  little  resistance),  are 
due  to  the  inertia  of  the  mercury,  but  vanish  if  the  frictional 
resistance  be  very  great.  The  oscillations  of  the  mercury  may 
be  checked  by  making  one  part  of  the  manometer-tube  capillary 
(Marey's  manometre  compensateur),  or  by  interposing 
a  stopcock  (Setschenow)  the  orifice  of  which  can  be  narrowed 
till  all  oscillations  are  cut  off,  the  instrument  then  recording 
merely  the  slow  variations  of  mean  pressure. 

Kick's  instrument  F  is  damped  (prevented  from  oscillat- 
ing in  virtue  of  its  own  elasticity)  by  connecting  with  the 
writing  levers  a  disc  immersed  in  glycerine,  as  shown  in  the 
figure :  the  viscosity  of  the  glycerine  causes  all  secondary  oscil- 
lations rapidly  to  die  away.  The  result  is  that  the  Federmano- 
meter  is  very  trustworthy  as  a  recorder  of  the  general  form  of 
the  variations  of  pressure  in  AB.  The  sphygmoscope  S  and 
the  metallic  inscriptor  M,  not  having  much  inertia  to  combat, 
render  accurately  the  general  form  of  the  variations  of  pressure, 
especially  if  in  the  liquid  surrounding  the  elastic  capsules  in  the 
latter  instrument  there  be  lightly  packed  a  number  of  bits  of 
sponge  to  check  elastic  vibrations  of  the  capsules ;  but  all  the 
different  forms  of  pressure-indicators,  with  the  exception  of 
those  shown  at  O,  C,  and  B,  require  preliminary  graduation 
before  their  indications  can  be  held  to  denote  the  absolute  value 
of  the  pressures ;  and  further,  this  preliminary  graduation  must 
be  frequently  repeated. 

3.   THE  KINETICS  OF  LIQUID  MASSES. 

Streams.  —  When  a  liquid  flows  in  a  stream,  its  particles  do 
not  become  separated  from  one  another  to  any  perceptible  extent, 
and  the  liquid  usually  preserves  its  mean  density.  The  liquid 
moves  as  a  whole  and  has  inertia,  as  may  be  seen  in  a  rapid  and 
full  stream  leaping  over  a  chink  into  which  a  slow  or  meagre 
stream,  would  be  pulled  by  gravity. 

This  principle  is  sometimes  made  use  of  in  order  to  prevent  an  excess  of 
rain-water  entering  drain-pipes ;  a  sloping  gutter  has  chinks  in  it,  opening 
into  the  drainage  system  :  when  the  gutters  become  flooded,  the  water  rushes 
over  these  chinks,  and  the  comparatively  pure  water  is  directed  elsewhere 
than  into  the  sewage,  the  excessive  dilution  of  which  may  be  considered  as 
a  commercial  evil. 

When  once  a  steady  stream-flow  has  been  set  up,  it  can  in 
general  be  maintained  by  the  maintenance  of  the,' supply  of 
liquid  and  of  the  propelling  force. 


300  OF  LIQUIDS.  [CHAP. 

Steadiness  of  flow  is  favoured  in  actual  cases  by  viscosity,  by  a  free 
bounding  surface,  by  converging  solid  boundaries,  by  a  stream  passing  round 
a  curve  with  its  greatest  velocity  externally.  (Osborne  Reynolds.) 

In  any  steady  stream  there  may  be  drawn  a  series  of  imagi- 
nary lines,  which  represent  the  direction  of  movement  of  the  ele- 
ments of  liquid  through  which  they  pass.  These  lines  are  called 
Stream-lines,  or  Lines  of  Flow.  So  long  as  a  stream 
retains  the  same  breadth  and  form,  these  lines  may  be  con- 
sidered parallel  to  one  another ;  if  the  stream  widen  out  they 
diverge  ;  if  it  contract  they  converge.  Such  lines  are  shown  in 
Fig.  106,  which  represents  a  steady  stream  of  frictionless  liquid 
flowing  either  from  A  towards  C,  or  from  C  towards  A. 

Law  of  Continuity.  —  If  we  consider  successive  sections, 
equal  or  unequal,  taken  across  a  liquid  stream,  it  is  plain  that 
the  amount  of  liquid  which  crosses  each  section,  during  any 
given  interval  of  time,  is  equal  in  each  case :  otherwise  there 
would  be  congestion  at  some  part  of  the  stream.  In  Fig.  106 
the  amount  of  liquid  which  crosses  A  or  C  in  a  second  must  be 


Fig.106. 


equal  to  that  which  crosses  B  :  in  other  words,  the  Amount  of 
Flow  across  all  sections  of  a  liquid  stream  is  the  same.  This 
may  be  otherwise  expressed  by  saying  that  at  any  part  of  a 
stream  the  velocity  varies  inversely  as  the  area  of  section 
at  that  part :  if  the  stream  be  broadened  out  so  as  to  have  a  ten- 
fold cross-section,  its  velocity  is  decreased  to  one-tenth.  The 
statement  of  this  law  is  due  to  Lionardo  da  Vinci. 

Forces  producing  Flow.  —  If  a  perfect  liquid,  exercising  no 
intermolecular  friction  and  no  friction  on  the  walls  of  the  canal 
or  tube  conveying  it,  were  once  set  in  motion  (say  in  a  closed 
circuit  or  circular  tube),  it  would  go  on  moving  without  the 
continued  application  of  force.  The  energy  actually  expended 
on  the  liquid  in  producing  its  movement  would  remain  in  the 
fluid-mass,  the  velocity  of  which  would  consequently  remain 
unaltered.  Such  a  liquid  might  be  exposed  to  a  severe  hydro- 
static stress  —  as,  for  example,  if  such  a  perfect  closed  stream 
were  contained  in  a  continuous  flexible  tube  exposed  to  the 
weight  of  a  mass  of  liquid  in  which  it  was  deeply  immersed  at 


xi.]  FLOW.  301 

the  one  level  —  and  yet  the  flow  would  not  be  affected.  A  hang- 
ing- loop  of  tubing  containing  a  circulating  liquid,  of  which  the 
lower  part  is  exposed  to  a  greater  pressure  than  the  upper,  will 
present  a  turgidity  of  the  lower  part  of  the  loop  if  the  tubing  be 
distensible,  while  if  it  be  rigid  there  will  be  no  expansion  of  the 
stream ;  in  the  latter  case  the  flow  will  not  be  affected ;  in  the 
former  the  expansion  of  the  stream  affects  the  local  velocities, 
and  therefore  the  distribution  of  the  energy  of  the  system,  but 
the  mean  velocity  may  remain  constant.  In  the  case  of  a  sus- 
pended loop  of  distensible  tubing  the  indirect  effect  of  gravity 
is  thus  to  diminish  the  velocity  of  the  lower  part  and  to  increase 
that  of  the  upper  part,  both  of  the  descending  and  of  the  ascend- 
ing parts  of  the  stream ;  but  on  the  amount  of  flow  it  may  pro- 
duce no  effect. 

Flow,  on  the  one  hand,  and  Hydrostatic  Pressure  uniformly 
applied,  on  the  other  hand,  are  thus  seen  to  be  perfectly  distinct 
conceptions,  and  in  a  perfect  fluid  they  might  be  independent  of 
one  another ;  but  in  every  physical  fluid  viscosity  and  friction 
come  into  play,  and  flow  can  only  be  kept  up  by  maintaining  a 
difference  of  pressure  within  the  fluid,  considered  as  a 
whole  from  end  to  end.  As  the  flow  is  kept  up,  so  is  it  started : 
liquid  in  equilibrium  may  be  made  to  flow  by  locally  increasing 
the  pressure  or  by  locally  diminishing  it. 

If  the  pressure  at  a  point  A  be  jo',  and  that  at  a  point  B  be  />",  the  dif- 
ference of  pressure  between  these  two  points  is  p'  —  p".  The  difference  of 
pressure  per  unit  of  distance  is  (/>'  —  p")/AB.  The  force  producing  the 
flow  depends  on  this  ratio;  and  the  greater  this  ratio,  the  greater  (but  not 
in  direct  proportion,  see  p.  311)  is  the  velocity  produced.  Liquid  tends  to 
flow  in  the  direction  in  which  the  pressure  falls  off  most  rapidly ;  and  the 
Force,  acting  on  a  cubic  cm.  of  its  volume,  is  numerically  equal  to  the  rate 
of  decrease  of  pressure,  per  linear  cm.  in  that  direction  —  that  is,  to  the 
Pressure-Gradient  or  Pressure-Slope  in  that  direction. 

Small  velocities  are  associated  with  small  gradients  of 
pressure ;  or,  in  other  words,  with  relatively  great  distances 
between  points  whose  difference  of  pressure  is  equal  to  any 
predetermined  quantity,  say  a  unit  of  force.  When  the  veloc- 
ity is  great,  the  points  between  which  the  difference  of  pressure 
is  unity  are  relatively  near  to  one  another.  The  theory  of 
Flow  from  this  point  of  view  resembles  that  of  Potential :  sur- 
faces of  equal  pressure  correspond  to  equipotential  surfaces ; 
Lines  of  Flow  or  Stream-Lines  correspond  to  Lines  of  Force. 

When  the  liquid  is  driven  through  a  long  uniform  tube 
there  is,  at  the  orifice  of  inflow,  a  certain  initial  pressure ;  at 


302  OF  LIQUIDS.  [CHAP. 

the  other,  the  orifice  of  outflow,  there  is  no  pressure  at  all.  If 
the  liquid  be  driven  by  an  equal  force  through  a  shorter  tube,  the 
pressure  vanishes  in  the  same  way,  but  does  so  more  rapidly, 
and  —  since  a  greater  difference  of  pressure  per  unit  of  length 
is  associated  with  greater  velocity  —  the  velocity  is  greater  than 
in  the  longer  tube.  The  shorter  the  tube  the  greater  the  veloc- 
ity, other  things  being  equal.  The  shortest  tube  possible  would 
be  a  plain  aperture  in  the  side  of  the  vessel  from  which  the 
liquid  issues.  In  this  case  the  liquid  at  once  assumes  the 
greatest  velocity  which  it  can  acquire  under  the  action  of  a 
given  pressure. 

"Head"  of  Liquid.  —  In  the  case  of  a  vessel  containing 
liquid  which  passes  out  through  an  aperture,  the  pressure  driv- 
ing the  particles  through  the  orifice  is  the  hydrostatic  pressure 
on  that  orifice ;  it  is  therefore  equal  (if  the  area  of  the  orifice 
be  A  arid  the  height  of  the  surface  of  the  liquid  above  the 
centre  of  figure  of  the  orifice  be  H)  to  A-H-p(/  =  P  over 
the  whole  area,  or  p  ==  H/#  per  unit  area.  The  height  H  is 
known  as  the  Head  of  the  liquid  producing  the  pressure ;  and 
the  "  Head  of  Water  "  is  a  term  familiar  to  hydraulic  engineers. 
Head  of  liquid  may  be  real,  as  where  the  flow  is  fed  from  a  cis- 
tern ;  or  it  may  be  virtual,  as  where  an  equivalent  pressure  is 
produced  by  mechanical  means.  In  the  latter  case,  H  =  P/Apg 
—  P/P9-*  where  P  is  the  total  pressure  on  the  orifice,  and  p  the 
pressure  per  unit  area. 

Any  pressure  exerted  on  a  liquid  may  be  stated  in  terms  of 
Head  H  of  the  same  liquid;  for  p  =  H^.  Then,  g  being 
known  and  p  also  known,  it  is  sufficient  to  specify  the  pressure 
by  stating  the  value  of  H ;  or  vice  versd. 

The  pressure  produced  by  compression,  as  in  pressing  home 
a  syringe,  the  negative  pressure  produced  by  rarefaction,  as  in 
pulling  up  the  handle  of  a  syringe,  may  all  be  measured  in  the 
same  way. 

Torricelli's  Law. —  If  v  be  the  constant  velocity  of  out- 
flow of  a  stream  passing  out  of  a  vessel  under  the  pressure  of  a 
constant  head  H,  v  =  V2^rH.  If  the  aperture  be  in  the  sides  of 
the  vessel,  the  liquid  issues  with  velocity  v  at  right  angles  to 
the  walls  of  the  vessel ;  this  velocity  becomes'  combined  with  a 
new  downward  fall  due  to  gravity,  and  the  liquid  travels  in  a 
parabolic  path,  forming  a  continuous  parabolic  Jet.  The  form 
of  the  parabola  indicates  the  proportion  between  v  and^;  and 
thus  v  is  found  to  differ  very  little  (one  per  cent)  from  Torn- 


XL]  TORRICELLI'S  LAW.  303 

celli's  value,  v  =  V2#H.  It  is  somewhat  greater  the  more  con- 
vex the  wall  of  the  vessel.  The  amount  of  outflow  per  unit  of 
time  is  not,  however,  the  product  of  the  area  of  the  aperture 
into  the  velocity ;  it  is  only  about  -ff$  of  that  amount. 

Since  H  =  p/pg,  v  =  V2#H  =  V2/>/p. 

In  general,  whenever  any  fluid  is  acted  upon  by  a  number 
of  pressures  corresponding  to  the  respective  heads  of  the 
same  fluid  H,  H;,  H/;,  etc.,  the  velocity  of  outflow  of  the 
fluid  through  an  orifice  is  v  =  V2</(H  +  H;  +  H/y  +  etc.). 

Example.  —  The  pressure  at  the  bottom  of  a  column  of  water  1033-3 
cm.  deep  is  equal,  per  sq.  cm.,  to  the  weight  of  1033-3  grammes.  The 
atmospheric  pressure  on  the  surface  of  a  liquid  is  equal  to  the  same.  Hence 
the  atmospheric  pressure  is  equal  to  that  of  a  head  of  water  of  1033-3  cm. 
Water  at  a  depth  of  1033-3  cm.  under  a  water-surface  exposed  to  the  atmos- 
phere is  exposed  to  as  much  pressure  as  if  it  lay  at  a  depth  of  2066-6  cm.  of 
water  under  a  free  surface  not  exposed  to  the  atmosphere.  In  a  vessel  filled 
with  water,  1033-3  cm.  deep,  provided  with  an  aperture  in  its  lower  surface, 
this  aperture  communicating  with  a  vacuum,  and  the  upper  surface  of  the 
liquid  communicating  with  the  atmosphere,  the  velocity  of  outflow  will, 
according  to  Torricelli's  law,  be  \/2  x  981  x  2066-6  =  2013-6  cm.  per  sec., 
while,  if  the  lower  aperture  were  also  in  communication  with  the  atmosphere, 
the  effective  head  of  water  would  be  1033-3  cm.,  and  v  =  V2  x  981  x  103-3-3 
=  1423-8  cm.  per  sec. 

Torricelli's  law  shows  that  the  velocity  with  which  a 
liquid  issues  through  an  aperture  varies  as  the  square  root  of 
the  Head  of  that  liquid  or  of  the  pressure;  or  that  the  Head  of 
liquid,  or  the  pressure  jt?,  necessary  to  produce  a  certain  veloc- 
ity in  a  free  stream  of  liquid,  subject  to  no  resistances,  is  pro- 
portional to  the  square  of  the  velocity. 

For  a  given  head  H,  the  velocity  of  outflow,  V2^H,  does  not 
depend  upon  p,  the  density  ;  all  liquids  —  ether  and  mercury  — 
issue  with  equal  velocities  under  the  action  of  equal  heads  of 
their  own  substance  :  but  for  a  given  pressure  jt>,  the  velocity  of 
outflow,  V2j9/jO,  is  inversely  proportional  to  the  square  root 
of  the  density  of  the  liquid. 

Energy  of  Jet.  —  If  Torricelli's  law  held  perfectly  good, 
that  v  =  v2^H,  the  velocity  would  be  the  same  as  if  every 
particle  had  fallen  from  the  surface  of  the  liquid  to  the  orifice, 
and  had  passed  out  of  the  orifice  with  a  velocity  due  to  its  fall 
through  the  height  H. 

The  outflowing  jet  would  thus  convey  with  it  kinetic  energy  equal  to 
u2/2,  or  to  #H  '=  p/p,  ergs  per  gramme ;  or  to  p  ergs  per  cub.  cm.,  where  p  is 
the  pressure,  in  dynes  per  sq.  cm.,  on  the  orifice.  ^ 

This  would  be  absolutely  the  case  were  it  not  for  friction 


304  °F  LIQUIDS.  [CHAP. 

and  viscosity.  If  the  level  of  liquid  be  maintained  constant  by 
a  continued  supply,  the  velocity  is  constant.  At  the  instant 
when  the  whole  of  the  original  liquid  has  passed  out  through 
the  orifice,  the  experiment  may  be  stopped.  The  liquid  which 
has  passed  out  has  conveyed  with  it  energy  =  |-mv2  =  rn^H  if 
Torricelli's  law  be  true.  At  the  commencement  of  the  experi- 
ment it  had  potential  energy  (mass  m  at  an  average  height  of 
JH)  of  Irn^H  only.  It  has  therefore  gained  energy  =  Jm^H. 
This  energy  has  been  lost  by  the  liquid  which  has  replaced  it, 
and  sunk  from  the  surface  to  an  average  depth  of  ^H  below  the 
surface,  thus  losing  potential  energy  ^mc/H.  without  any  com- 
pensating gain  of  energy  in  any  other  form. 

The  same  principle  is  illustrated  in  the  following  experiment.  One  cork 
of  a  Woultt'  's  bottle  completely  filled  with  water  is  fitted  with  a  piece  of  glass 
tube  drawn  out  so  as  to  form  a  jet ;  the  other  cork  admits  a  tube  leading 
from  a  vessel  containing  mercury ;  the  mercury  is  caused  to  fall  into  the 
bottle.  Some  of  the  water  which  already  fills  the  bottle  is  driven  out  with 
great  velocity  in  a  thin  stream.  The  mercury  sinking  through  the  water 
loses  energy  proportional  to  its  density  (mgK  =  bp#H) ;  the  water  forced 
out  acquires  this  energy,  and  hence  has  a  great  velocity  imparted  to  it. 

The  Vena  Contracta.  —  The  issuing  jet  may  be  observed 
(especially  when  it  is  directed  upwards)  not  to  be  perfectly 
cylindrical,  but  to  diminish  in  diameter  from  the  aperture  to  a 
spot  called  the  vena  contract  a,  whose  position  is  sometimes 
somewhat  difficult  to  define.  This  conical  form  is  due  to  the 
fact  that  the  onward  flow  of  liquid  is  not  confined  to  that  part 
of  the  fluid  which  exactly  faces  the  aperture,  because  the  lateral 
parts  of  the  liquid  converge  on  the  orifice ;  thus  the  most  exter- 
nal stream-lines  of  the  jet,  which  are  at  first  tangential  to  the 
wall  of  the  vessel,  assume  a  direction  at  right  angles  to  this 
wall,  changing  their  direction  gradually,  and  therefore  present- 
ing a  curved  form  as  shown  in  Fig.  107.  From  the  vena  con- 
Fig.io?.  tracta  onwards  the  jet  is  approximately 

cylindrical,  and  presently  breaks  up  into 
drops,  which  (especially  if  any  vibration 
affect  the  vessel  from  which  the  jet 
issues)  are  found  to  be  oscillating  in 
form,  each  becoming  alternately  a  pro- 
late and  an  oblate  spheroid.  The  un- 
aided eye  cannot  perceive  these  separate 
drops,  but  recognises  the  vein  as  contin- 
uous though  troubled.  When,  however,  the  jet  is  instantane- 
ously illuminated  by  the  electric  spark,  and  its  momentary 


XL]  VENA  CONTRACTA.  305 

shadow  upon  a  screen  observed,  the  existence  not  only  of  these 
separate  drops,  but  also  of  others  of  a  smaller  size  occupying 
intermediate  positions,  may  be  demonstrated  with  ease ;  for  the 
instantaneous  impression  on  the  retina  persists  for  the  sixth  part 
of  a  second,  and  the  shadow  of  the  jet  appears  stationary  on  the 
screen.  The  jet  may  also  be  looked  at  through  a  Stroboscopic 
Disc,  a  rotating  disc  provided  with  equidistant  narrow  aper- 
tures. Through  each  aperture,  as  it  passes  the  eye,  a  glimpse  is 
caught  of  the  jet  in  a  certain  position.  If  the  rate  of  rotation  of 
the  disc  be  properly  adjusted,  each  successive  glimpse  is  caught 
just  when  each  falling  drop  has  had  its  place  taken  by  its  suc- 
cessor; and  thus,  on  the  whole,  under  such  a  succession  of 
glimpses  the  jet  appears  to  be  stationary. 

This  phenomenon  is  one  of  free  fall  in  the  air,  for  the  break-up  into  drops 
depends  greatly  on  surface-tension ;  a  liquid  cylinder  of  excessive  length  and 
with  a  free  surface  first  assumes  an  undulating  contour,  and  then  breaks  up 
into  separate  vibrating  drops,  as  Plateau  has  shown.  The  vibrations  of 
liquids  in  tubes  are  therefore  not  to  be  explained  as  phenomena  of  this  kind. 

Ajutages.  —  The  amount  of  outflow  from  an  aperture  in 
the  wall  of  a  vessel  is  greatly  influenced  by  the  form  of  the 
ajutage  or  mouthpiece  through  which  the  liquid  passes.  This 
may  be  made  so  as  to  present  the  same  form  as  the  jet  itself, 
and  if  it  be  prolonged  just  as  far  as  the  vena  contracta,  the 
amount  of  outflow  becomes  equal  to  the  product,  area  x  v  x 
time ;  not  because  the  outflow  is  itself  altered,  but  because  the 
area  of  the  orifice  of  outflow  is  reduced  so  as  to  become  equal 
to  {amount  of  outflow/y^,  the  terms  of  this  ratio  being  unal- 
tered. If  the  ajutage  project  inwards,  the  outflow  and  velocity 
are  materially  diminished.  If  the  ajutage  project  outwards, 
being  cylindrical,  the  cylinder,  if  its  walls  be  wetted  by  the 
liquid,  is  completely  filled  by  it,  the  jet  is  cylindrical,  and  the  out- 
flow is  greater  than  when  there  is  no  such  ajutage.  The  liquid 
is  drawn  towards  the  sides  of  the  cylinder,  and  conversely,  the 
sides  of  the  cylinder  are  drawn  towards  the  liquid.  Hence  there 
is  no  pressure  exerted  on  the  walls  of  the  tubular  ajutage ;  on 
the  contrary,  there  is  suction,  and  if  any  part  of  the  walls  of  the 
tube  be  mobile,  it  will  be  drawn  into  the  stream. 

The  peculiarly  beautiful  forms  presented  by  jets  under  various  circum- 
stances are  described  and  figured  by  Savart  in  the  Annales  de  Chimie  et  de 
Physique,  vols.  54  and  55. 

If  two  vessels  having  an  aperture  in  each  of  the  same  size  and  shape,  and 
at  the  same  level,  be  so  arranged  that  these  apertures  are  exactly'opposite  one 
another  and  close  together :  if  liquid  be  poured  into  the  one  vessel,  it  will 


306  OF  LIQUIDS.  [CHAP. 

ran  into  the  other.  The  vessels  may  then  be  removed  to  a  certain  distance 
from  one  another,  and  the  liquid  will  continue  to  pass  from  the  one  vessel 
into  the  other,  through  a  tube  formed  of  its  own  superficial  film,  until  the 
same  level  is  nearly  attained ;  then  the  liquid  begins  to  flow  out  of  both 
vessels,  and  the  two  jets,  meeting,  spread  out  into  a  sheet  which  is  driven 
back  and  fore  between  the  two  orifices  as  the  liquid  in  the  one  or  the  other 
vessel  stands  for  the  moment  at  the  higher  level. 

Recoil.  —  The  law  of  action  and  reaction  perfectly  applies 
to  liquid  jets  and  to  the  vessels  from  which  they  issue.  The 
Hydraulic  Tourniquet  is  an  example :  a  cistern  containing 
water  and  capable  of  rotating  on  an  axis:  pipes  ending  obliquely 
issue  from  its  sides:  water  runs  out  of  these  pipes:  and  by 
reaction  they  are  driven  backwards.  Since  they  are  not  fitted 
to  an  immovable  cistern,  but  to  one  free  to  rotate,  the  whole 
rotates,  and  thus  the  contrivance  may  be  used  to  convey  water- 
power,  the  water  constantly  running  into  the  rotating  cistern, 
and  running  out  of  the  obliquely-set  exit  pipes. 

Resistances.  —  When  a  fluid  stream  passes  through  a  tube 
or  a  channel  it  experiences  different  retarding  resistances,  which 
convert  energy  of  motion  into  heat,  and  of  which  the  following 
are  the  chief:  —  Surface  Adhesion,  Surface  Friction,  Inequali- 
ties of  the  Surface  of  the  bounding  solid,  Eddies,  and  Fluid 
Viscosity. 

Surface  Adhesion.  —  If  a  liquid  wet  the  walls  of  the  tube 
or  channel  through  which  it  passes,  the  layer  of  liquid  which  is 
in  contact  with  the  walls  does  not  change  except  by  molecular 
diffusion  and  exchange.  It  remains  in  situ  while  the  liquid 
flows  past;  in  other  words,  there  is  infinite  friction  between 
this  layer  and  the  walls  wetted  by  it.  This  surface  adhesion, 
when  once  the  flow  has  been  set  up,  does  not  directly  cause  any 
waste  of  energy.  While  the  walls  are  being  wetted  there  is  a 
slight  liberation  of  heat,  due  to  the  satisfaction  of  the  mutual 
molecular  attractions  between  the  liquid  and  the  walls. 

Surface  Friction.  —  If  the  liquid  do  not  wet  the  tube 
through  which  it  passes,  the  surface  of  the  moving  liquid  and 
the  walls  of  the  vessel  rub  against  one  another,  and  energy  is 
lost  in  overcoming  this  friction.  Loss  of  kinetic  energy  is  also 
caused  by  roughnesses  on  the  walls  of  the  tubes  or  channels, 
which  give  rise  to  little  eddies  or  whirlpools. : 

Eddies  are  produced  when  a  moving  fluid  is  subjected  to 
unsymmetrical  retardations.  The  cases  in  which  eddies,  whirl- 
pools, vortex-rings,  rolling  and  tumbling  water,  and  the  like, 
are  produced  are  extremely  numerous.  Water  flowing  in  a  tube 


XL]  FLOW.  307 

which  suddenly  widens  or  suddenly  narrows  generally  presents 
such  eddies  at  the  point  of  sudden  enlargement  or  contraction. 

The  production  of  eddies  is  favoured  by  mobility  of  the  liquid,  by 
variations  of  velocity  at  different  parts  of  the  cross-section  of  the  stream, 
by  rigid  bounding  walls,  by  diverging  boundaries,  by  curvature  with  the 
greatest  velocity  internally.  (Osborne  Reynolds.) 

Viscosity.  —  When  a  disc  or  cylinder  suspended  in  a  liquid 
is  caused,  by  twisting  the  supporting  wire  or  wires,  to  enter 
into  oscillation,  it  is  found  that  the  oscillations  soon  die  away ; 
though  they  continue  isochronous,  their  amplitude  diminishes ; 
and  the  amplitudes  of  any  two  successive  oscillations  stand  to 
one  another  in  a  constant  proportion.  If  the  disc  or  cylinder 
be  wetted  by  the  liquid,  the  layer  immediately  in  contact  with 
the  solid  remains  in  contact  with  it ;  this  film,  moving  with  the 
solid,  sets  in  motion  the  film  next  in  contact  with  it,  and  that 
in  its  turn  sets  the  next  in  motion.  Each  film  goes  through  a 
displacement  somewhat  less  extensive  and  more  retarded  than 
the  one  gone  through  by  the  film  which  sets  it  in  motion.  Con- 
tinuous rotation  of  the  disc  or  cylinder  would  in  time  cause  the 
whole  liquid  to  rotate ;  but  the  influence  of  an  oscillating  disc 
travels  a  very  short  distance,  for  half  an  inch  away  from  the 
disc  the  liquid  remains  undisturbed.  Within  this  small  distance 
the  liquid  performs  oscillations  which  in  period  resemble  those 
of  the  oscillating  disc,  but  which  in  amplitude  are  less,  and  in 
phase  more  retarded,  the  greater  the  distance  from  the  disc. 
This  lagging  behind  on  the  part  of  the  liquid  has  the  effect  of 
dragging  on  the  disc  and  of  gradually  bringing  it  to  rest. 

If  the  disc  be  wetted  the  retardation  is  independent  of  the 
nature  of  the  material  of  the  disc,  for  there  is  no  velocity  lost 
by  friction  between  the  solid  and  the  liquid.  If  the  disc  be  not 
wetted,  there  is  distinct  friction  (external  friction)  in  addition 
to  the  viscosity  (internal  friction). 

The  Coefficient  of  Viscosity  serves  as  the  means  of 
measuring  the  viscosity  of  a  substance.  We  have  already  seen 
(p.  227)  that  it  is  equal  numerically  to  the  force  which  is  nec- 
essary to  maintain  a  flow  of  one  laj^er  of  one  unit-area  past 
another  of  the  same  area  with  a  relative  velocity  of  one  unit, 
the  distance  between  the  layers  being  unity,  and  the  space 
between  them  continuously  filled  with  the  viscous  substance. 

If  F  be  the  total  force  required  to  keep  up  the  flow  of  two  layers  past 
each  other,  their  area  being  each  A,  their  respective  distances'from  a  plane 
of  reference  being  dt  and  d,n  and  their  distance  from  each  other  therefore 


308  OF  LIQUIDS.  [CHAP. 

^  _  dit  •  if  their  respective  velocities  be  vt  and  v/t,  and  their  relative  velocity 
vt  —  vlt ;  and  if  the  coefficient  of  viscosity  be  rj,  F  =  rj  •  A  (v,  —  vit}/dt  —  dtl 
=  i]  •  A  •  tan  6./t,  where  tan  6  is  the  total  shear  effected  in  time  t. 

The  Activity  required  to  keep  up  this  flow  is  the  product  of  the  force  F 
acting,  into  the  mean  relative  velocity,  4  (v,  —  vn},  of  the  moving  liquid ; 
that  is,  it  is  equal  to  ^77  •  A(d,  —  dtl)  •  (tan  0./*)2>  or>  Per  cu^«  cm.,  to  ly 
(tan0./02. 

The  dimensions  of  rj  are  [M/LT]. 

In  the  case  of  water  at  0°-6  C.  this  coefficient  is  0-0173,  at 
45°  C.,  0-005833,  at  90°  C.,  0-00339  (Meyer),  while  that  of  air,* 
which  obeys  the  same  laws,  is  -00017  (1  +  0-00733*),  wh'ere  £is 
the  C.  temperature ;  all  expressed  in  C.G.S.  units. 

Though  the  density  of  air  is  ^th  that  of  water,  its  viscosity  is  as  much 
as  Ti-0th  that  of  water.  For  brass,  17  is  about  300,000,000000.  Moist  air 
is  more  viscous  than  dry  air :  hot  air  is  more  viscous  than  cold  air. 

Hot  water  is  less  viscous  than  cold.  Most  saline  solutions 
are  more  viscous  than  water,  saltpetre  solution  being  an  excep- 
tion. Most  saline  solutions  are  more  viscous  the  more  concen- 
trated they  are,  saltpetre  solution  being  again  an  exception 
(Meyer). 

The  experimental  determination  of  the  coefficient  rj  by  means  of  obser- 
vations made  with  the  aid  of  an  oscillating  disc  involves  much  mathematical 
computation,  and  it  is  often  quite  sufficient  to  record  the  so-called  Logarith- 
mic Decrement  or  log.  dec.  special  to  each  liquid.  Let  us  suppose  that  the 
oscillating  disc  or  cylinder  first  turns  through  the  angle  8;  that  at  the  next 
oscillation  its  deviation  from  its  mean  position  is  T9o90-  8 ;  that  at  the  third  it 
is  TV<5-  x  TV7  x  8;  and  so  forth.  Then  each  successive  angle  is  equal  to  the 
one  immediately  preceding  multiplied  by  T9797 ;  its  log.  is  equal  to  the  log.  of 
the  preceding  angle  of  oscillation  plus  that  of  T9o9^,  or  minus  the  log.  of  YTT  5 
that  is,  minus  -0043648.  Such  a  constant  difference  in  the  logarithms  of  the 
successive  angles  of  oscillation  is  the  log.  dec.  for  the  particular  substance 
whose  viscosity  it  measures.  Under  Poiseuille's  Law  (p.  315)  we  shall  find 
a  simple  method  of  measuring  the  value  of  17. 

Effect  of  Viscosity  on  a  stream  of  liquid.  —  The  external 
layer  is  at  rest.  The  axial  parts  of  the  stream  are  less  influ- 
enced by  viscosity.  The  velocity  of  the  axial  part  of  the  stream 
is  greater  than  that  of  the  peripheral;  the  fall  of  pressure  is 
therefore  greatest  in  the  centre  of  the  current.  The  pressure 
being  least  in  the  centre,  the  external  parts  of  the  stream  tend 
to  move  into  the  centre,  and  to  have  their  velocity  accelerated. 
In  capillary  tubes  the  axial  stream  travels  with  a  greater  speed 
than  the  average  as  determined  by  Poiseuille's  Law,  to  be 
presently  stated. 

*  O.  E.  Meyer,  Pogg.  Ann.,  vol.  cxlviii.,  1873. 


XI.] 


FLOW. 


309 


Fis.108. 


Lateral  diminution  of  pressure.  —  If  through  a  tube  of 
the  form  shown  in  Fig.  108  there  pass  a  current  of  liquid  in  the 
branch  AB  under  a  pressure  which 
is  barely  sufficient  to  keep  up  a 
stream  filling  the  tube,  the  mutual  *" 
attraction  of  the  walls  of  AB  and 
the  liquid  will  put  the  liquid  in  AB 
in  a  state  of  tension  and  diminish 
the  pressure  in  AB.  In  the  side 
tube  CD  a  certain  column  of  liquid 
can  be  supported  in  consequence  of 
the  diminished  pressure  in  AB.  If 
this  rise  to  the  point  C,  the  upper  layers  of  the  column  DC  will 
be  constantly  carried  off  by  the  stream  BA,  and  thus  a  stream  is 
set  up  in  the  direction  DC.  If  the  pressure  in  the  main  pipe 
AB  be  too  great,  liquid  will  be  driven  down  CD. 

The  former  action  is  by  some  considered  as  explaining  the  flow  of  lymph 
up  the  thoracic  duct. 

The  same  kind  of  suction-effect  may  be  perceived  in  the  older  forms  of 
washhand-basins  connected  with  a  house  drain-pipe  by  a  simple  bent  tube  or 
trap ;  a  downrush  of  liquid  along  the  main  pipe  produces  a  deficiency  of 
pressure,  which  allows  the  atmospheric  pressure  communicated  through  the 
basin  to  drive  the  liquid  which  seals  the  trap  into  the  drain-pipe,  and  thus 
to  leave  a  channel  patent  to  the  entry  of  sewer  gas. 

Constant  flow  through  uniform  rigid  pipes.  —  The  pres- 
sure which  is  necessary  to  keep  up  a  continuous  flow  of  water 
in  a  uniform  pipe,  EF  in  Fig.  109,  may  be  produced  by  a  total 

Fig.109. 


I  *„ 


head  H  of  water  in  a  vessel  (a  pressure-vessel,  ABCD  in  the 
figure),  this  height  H  being  maintained  constant.  TJ>e  water  is 
observed  to  issue  from  F  with  a  constant  velocity  v' ;  this  veloc- 


310  OF  LIQUIDS.  [CHAP. 

ity  would  (if  there  had  been  no  resistances)  have  corresponded 
to  a  head  Hv  =  v'2/2g;  this  may  be  considered  (so  far  as  the 
velocity  and  the  kinetic  energy  of  the  outflowing  stream  at  F 
are  concerned)  to  be  the  effective  head  of  water  at  F,  the  orifice 
of  exit:  it  may  be  called  the  "  velocity-head. "  This  velocity- 
head,  GJ,  is  equal  in  all  parts  of  the  tube. 

The  hydrostatical  pressure  in  the  immediate  neighbourhood 
of  F  is  necessarily  null ;  that  at  E,  just  within  the  pipe  EF,  is 
less  than  the  pressure  ( =  Hpg  per  unit  of  area)  corresponding 
to  H,  the  original  head  of  water ;  it  corresponds  to  a  head  H^, 
(the  Pressure-head),  which  differs  from  H  in  the  first  place  in 
consequence  of  a  certain  slight  waste  of  head  caused  by  the 
formation  of  eddies  between  D  and  E,  and  in  the  second  place 
differs  from  H  by  the  amount  of  the  velocity-head  itself.  If  we 
neglect  the  effect  of  these  eddies  we  may  say  that  the  velocity- 
head  and  the  pressure-head  are  together  equal  to 
the  total  head:  Hv  +  Hp  =  H. 

The  hydrostatic  pressure  in  the  tube  (if  the  tube  be  uni- 
form) dies  away  uniformly,  as  is  shown  by  the  level  assumed 
by  the  water  in  the  successive  piezometer-tubes  of  Fig.  109. 

If  the  tube  were  lengthened  there  would  be  a  similar — but 
necessarily  a  slower  —  dying  away  of  the  pressure ;  the  velocity 
would  be  less  throughout  the  tube ;  the  velocity-head  being  less, 
the  pressure-head  would  be  greater:  there  would  therefore  be 
a  greater  pressure  at  E. 

The  hydrostatic  pressure  at  any  part  of  a  stream  measures 
the  resistance  which  has  yet  to  be  overcome.  If  there  were  no 
resistance  (as  in  the  imaginary  case  of  a  perfect  liquid)  there 
would  be  no  lateral  pressure,  no  pressure-head  ;  and  the  whole 
of  the  original  total  head  H  would  be  taken  up  in  producing  a 
velocity  v  =  V2#H. 

The  greater  the  velocity  of  a  stream,  the  greater  the  resist- 
ance encountered  by  it  within  a  tube  of  given  dimensions.  The 
resistance  at  any  point  thus  depends  not  only  on  the  dimensions 
of  the  tube  between  that  point  and  the  orifice  of  outflow,  but 
also  on  the  velocity  of  the  stream. 

The  relation  is  U  -  I  (av'*/r  +  bv'/r2)  (Haagen) ;  U  being  the  meas- 
ure of  the  resistance,  I  and  r  the  length  and  radius-  of  the  tube  yet  to  be 
traversed  by  the  stream,  a  and  b  constants  to  be  found  by  experiment. 

U  is  not  a  number  of  units  of  force,  but  it  is  the  height  (in  crn.)  of  a 
lateral  column  of  water  whose  weight  can  be  supported  by  the  stream-resist- 
ance. Its  weight  is  \Jg  dynes  per  sq.  cm.,  and  the  local  Resistance  at  any 
point  of  the  stream  is  therefore  R  =  U</  dynes  of  force  per  sq.  cm.  of  trans- 
verse section  of  the  uniform  stream  passing  that  point. 


XL]  FLOW.  311 

Given  that  the  tube  has  a  certain  length  Z,  and  internal 
radius  r,  and  a  certain  constant  driving  head  of  water  H,  the 
velocity  v'  must  so  adjust  itself  that  the  three  equations  H  =  Hp 
+  Rv-,  H,  =v'*/2y;  and  R/g  =  U  =  Hp  =  l(av^/r  +  bv'/r*),  shall 
all  hold  good.  If  the  tube  be  exceedingly  long,  the  resistance 
becomes  proportionally  very  great  and  the  velocity  very  small : 
yet  in  a  tube  of  any  assignable  length  there  would  be  a  constant 
velocity,  and  the  pressure  would  uniformly  (though  slowly) 
diminish  from  one  end  of  the  tube  to  the  other.  The  pres- 
sure-line, GF  of  Fig.  109,  would  in  such  a  case  —  a  case  of 
low  velocity — have  a  gentle  slope.  When  the  tube  is 
very  short,  the  resistance  is  initially  small  and  rapidly  falls  ; 
thus  great  velocity  is  associated  with  steep  slope  of  the 
pressure-line. 

If  the  driving  pressure  increase  or  diminish  (the  dimen- 
sions of  the  tube  remaining  unchanged),  the  velocity  produced 
by  it  and  the  resistance  brought  into  play  both  increase  or  both 
diminish.  If  the  dimensions  of  the  tube  be  altered  while  the 
driving  pressure  remains  unchanged,  the  resistance  and  the 
velocity  will  vary  in  contrary  senses :  increased  resistance, 
diminished  velocity;  diminished  resistance,  increased  velocity. 
If  the  resistance  be  increased  by  increasing  the  length  or  lessen- 
ing the  diameter  of  the  tube,  the  velocity  and  the  amount  of 
flow  cannot  remain  constant  unless  the  driving  pressure  be  also 
increased.  (Hypertrophy  of  the  heart  when  the  placental  is 
added  to  the  ordinary  circulation.) 

If  the  hydrostatic  pressure  be  found  to  have  increased 
(higher  columns  being  supported  in  the  piezometers),  the  plain 
inference  is  either  that  the  driving  pressure  has  been  increased, 
or  else  that  the  peripheral  resistance  has  been  increased  by  nar- 
rowing or  lengthening  or  perhaps  by  roughening  the  tube.  If 
the  pressure  be  found  to  have  been  diminished,  either  the  driv- 
ing power  or  the  resistance,  or  both  these,  must  have  been  also 
diminished. 

If  more  than  one  of  these  elements  vary,  the  result  may  be 
either  accumulation  or  compensation  of  effects.  Higher  head  or 
narrowed  tubes  both  increase  the  pressure  ;  with  lowered  driving 
pressure  on  the  one  hand  and  narrowed  or  lengthened  tubes  on 
the  other,  the  pressure  may  remain  the  same,  though,  in  this 
case,  the  velocity  is  diminished.  Hence  it  is  necessary  to  observe 
both  the  pressure  and  the  velocity  in  order  to  investigate  the 
local  condition  of  any  stream. 


312  OF  LIQUIDS.  [CHAP. 

Flow  due  to  variable  pressure  in  uniform  rigid  tubes.  — 
If  the  driving  pressure  be  reduced,  the  pressure-head  becomes  a 
greater,  and  the  velocity-head  a  less,  fraction  of  the  reduced  total 
head ;  the  velocity-head  is  thus  lessened  not  only  in  proportion 
to  the  diminution  of  driving  pressure,  but  in  a  still  greater  ratio. 
Conversely,  if  the  driving  pressure  be  increased,  the  velocity- 
head  is  increased  in  a  greater  ratio.  Since  the  velocity  is  pro- 
portional to  the  square  root  of  the  velocity-head,  it  is  not  the 
velocit}^  but  the  square  of  the  velocity  which  is  a  little 
more  than  doubled  by  doubling  the  driving  force.  Hence  a 
curve  indicating  the  variations  of  velocity  agrees  in  general 
form,  but  not  in  its  amplitudes,  with  a  curve  indicating  the 
variations  of  driving  pressure. 

Interrupted  flow  through  uniform  rigid  pipes.  —  a.  The 
driving  force  may  be  applied  intermittently,  and  may  cease 
during  the  intervals.  A  perfect  incompressible  fluid,  treated  in 
this  way,  would  move  like  a  solid  rod  struck  endwise  by  a 
hammer:  all  its  particles  would  move  simultaneously,  and  liquid 
would  pass  through  the  orifice  of  exit  without  any  interval  of 
time.  A  physical  liquid  is  hurled  upon  itself,  compresses  itself, 
and  resiles.  Thus  the  suddenness  of  outflow  at  the  orifice  of 
exit  is  somewhat  modified ;  but  even  with  physical  liquids  the 
more  rigid  the  tube  the  more  abrupt  is  the  onflow.  (Athero- 
matous  arteries.) 

b.  The  pressure  being  continuous,  the  flow  may  be  suddenly 
stopped  by  an  obstruction,  say  by  a  stopcock  suddenly  closed. 
Beyond  the  stopcock  the  liquid  runs  on  somewhat  and  rarefies 
itself,  or  even  produces  a  vacuum  near  the  stopcock ;  it  returns 
and  oscillates  until  it  comes  to  rest.  Between  the  driving  pres- 
sure and  the  stopcock  there  is  a  sudden  increase  of  pressure.  If 
a  house  water-tap  be  suddenly  turned  off  when  water  is  running 
from  it,  a  jar  or  jolt  may  be  given  to  the  water  in  the  pipes, 
which  may  be  audibly  perceived  throughout  a  large  building. 
This  jolt  is  due  to  the  sudden  stoppage  of  the  water,  which  has 
already  acquired  momentum.  The  water  compresses  itself, 
rebounds  and  oscillates,  producing  waves  of  condensation  and 
rarefaction  which  travel  back  into  the  mains.  The  pressure  in 
the  pipes  is  greatly  increased  by  this  mode  of  treatment ;  an 
original  pressure  of  30  Ibs.  per  sq.  in.  may  be  raised  to  one  of 
120  or  130  Ibs. 

This  principle  is  economised  in  the  Hydraulic  Ram.     A  stream  of 
water  is  alternately  cut  off  and  allowed  to  flow  :    every  cut-off  enables  the 


XI.] 


FLOW. 


313 


stream,  whose  pressure  is  thereby  greatly  increased,  to  force  a  valve  which 
it  could  not  otherwise  force,  and  water  is  thus  driven  into  a  small  chamber 
containing  a  limited  volume  of  air.  This  air  is  compressed,  and  its  elasticity 
enables  it  to  force  the  water  out  through  a  narrow  jet,  at  a  pressure  nearly 
equal  to  the  greatest  pressure  experienced  by  the  liquid  during  the  cut-off. 

Flow  through  bent  tubes.  —  Bends  increase  the  resistance 
and  diminish  the  proportionate  velocity.  The  forward  momen- 
tum of  the  liquid  is  destroyed,  the  reaction  of  the  walls  is  called 
into  play,  and  by  the  elasticity  of  the  liquid  arid  of  the  walls  a 
new  path  is  given  to  the  liquid.  Energy  is  consumed  in  this 
process,  particularly  in  producing  eddies  in  the  stream,  and  the 
piezometer-tubes  show  that  the  pressure  in  the  water,  which  is 
about  to  meet  the  obstacle,  is  much  greater  than  in  that  which 
has  just  left  it  (Fig.  110).  If  the  driving  pressure  be  applied 


Fig.llO. 


intermittently,  the  liquid  between  the  driving  apparatus  and  the 
rigid  bend  may  be  sharply  compressed  before  it  can  pass  round 
the  bend ;  it  is  driven  against  the  bend  like  a  solid,  and  if  the 
bend  be  at  all  extensible  it  is  driven  forward.  (Locomotive 
pulse.) 

Flow  in  tubes  not  of  uniform  diameter.  —  We  are  apt  to 
think  that  when  a  fluid  passes  from  a  wide  into  a  narrow  tube 
the  pressure  is  increased ;  and  converse!}',  when  a  fluid  runs  out 
of  a  narrow  into  a  wide  tube,  that  it  is  relieved  of  pressure.  The 
reverse  is  the  case.  To  understand  this  we  must  consider  the 
flow  as  already  set  up  and  constant.  The  law  of  continuity 
shows  that  when  a  rapid  stream  passes  into  a  wide  channel,  it 
travels  more  slowly.  The  velocity-head  suffers  a  diminution, 
and  the  pressure-head  increases  :  the  kinetic  energy  possessed  by 
the  rapidly  entering  narrow  stream  is  partly  spent  in  dashing 


314  OF  LIQUIDS.  [CHAP. 

that  stream  against  the  comparatively  stationary  layers  in  the 
wider  channel,  and  is  thus  partly  converted  into  potential  energy. 
A  certain  degree  of  compression  is  thus  produced,  and  a  corre- 
sponding pressure,  which  is  additional  to  the  hydrostatic  pres- 
sure already  existing.  Conversely,  when  a  stream  narrows  it 
runs  more  rapidly :  its  kinetic  energy  becomes  greater  (mass  for 
mass),  and  there  is  a  tendency  to  stretch  or  rarefy  the  narrowed 
and  accelerated  stream.  This  tendency  to  rarefaction,  or  even 
to  tearing  asunder  the  stream,  corresponds  to  a  defect  of  pres- 
sure in  the  narrower  tube ;  and  it  has  been  utilised  in  Venturi 
water-meters,  which  consist  of  manometers  connected  with  a 
wider  and  a  narrower  part  of  the  same  water-pipe.  The  read- 
ings are  different  at  different  velocities. 

In  the  case  of  a  liquid  passing  from  a  narrower  channel  into 
a  wider,  we  have  a  flow  from  a  place  of  lower  pressure  into  one 
of  higher.  This  apparently  anomalous  flow  is  explained  by  the 
fact  that  the  pressure,  even  in  the  wider  channel,  can  never  (on 
account  of  the  gradual  disappearance  of  pressure-head  in  the 
production  of  heat)  exceed,  but  must  always  be  less  than,  that 
corresponding  to  the  difference  between  the  original  head  H 
and  the  local  velocity-head. 

Flow  in  branched  rigid  tubes.  —  If  the  total  cross-sectional 
area  of  the  branches  do  not  exceed  that  of  the  main  tube  from 
which  they  spring,  the  parietal  surface-area  of  the  stream 
is  increased ;  this  increases  the  resistance,  and  the  velocity  falls. 
If  the  total  cross-sectional  area  do  exceed  that  of  the  main 
tube,  the  channel  is  widened  and  the  resistances  are  relatively 
diminished:  they  may  even  be  diminished  by  this  cause  more 
than  they  are  increased  by  the  increase  of  the  total  surface. 
The  resistances  are,  on  the  whole,  absolutely  diminished  in  this 
case,  and  the  velocity  of  the  whole  system  may  be  greater  than 
that  in  an  unbranched  tube  of  corresponding  length. 

If  we  compare  two  branched  systems :  the  one  large,  con- 
taining many  branches,  each  of  which  would,  if  the  stream  were 
driven  through  it  alone,  offer  much  resistance,  but  all  together 
affording  the  stream  a  wide  bed  for  its  flow ;  the  other  system 
small,  containing  few  branches,  of  which  each  is  capable  of  offer- 
ing only  a  small  resistance,  but  which,  by  their  small  number, 
cause  the  stream  to  flow  in  a  narrow  bed ;  it  is  possible  that  the 
driving  pressure  necessary  to  produce  a  given  velocity  may,  in 
these  two  cases,  be  the  same.  The  advantages  of  the  aggregate 
wide  channel  in  the  first  system  are  neutralised  by  the  great 


XL]  FLOW.  315 

resistances  ;  the  advantages  of  the  small  resistances  in  the  second 
system  are  counteracted  by  the  narrowness  of  the  channel. 

Thus  both  small  and  large  animals  have  approximately  the  same  blood 
pressure  in  the  aorta. 

Where  branches  are  given  off,  the  pressure  either  increases 
or  begins  to  fall  off  less  rapidly,  because  the  velocity  diminishes ; 
where  the  branches  reunite,  the  pressure  rapidly  falls  off.  If 
the  whole  system  be  quite  symmetrical,  the  pressure  in  the 
middle  of  the  system  is  greater  than  half  the  initial  pressure. 

The  pressure  in  the  capillaries  is  more  than  half  the  initial  pressure  in 
the  aorta,  though  their  joint  sectional  area  is  very  great  and  their  resistance 
accordingly  very  small ;  and  the  pressure  very  rapidly  falls  as  the  veins 
unite.  At  the  same  time  the  velocity  increases  as  the  sectional  area  dimin- 
ishes, and  the  amount  of  flow  into  the  auricles  is  equal  to  that  from  the 
ventricles. 

When  in  a  system  of  branched  tubes,  some  of  the  branches 
are  relatively  shorter  or  wider,  the  amount  of  flow  through  these 
is  to  some  extent  relatively  more  rapid. 

When  in  such  a  system  the  flow  is  once  fairly  established, 
if  the  branches  as  a  whole  become  narrowed,  the  resistance  is 
increased  and  the  velocity  falls.  If  some  only  be  narrowed, 
while  the  driving  pressure  remains  the  same,  the  velocity  in  the 
remaining  branches  may  be  increased,  for  the  channel  is  nar- 
rowed, being  wholly  or  partly  restricted  to  the  latter.  The 
pressure  in  those  which  are  narrowed  is  increased;  but  the 
pressure  in  the  uniiarrowed  branches  may  also  be  increased,  for 
the  peripheral  resistance  of  the  system  as  a  whole  is  rendered 
greater,  and  the  velocity-head  is  diminished.  The  effect  on  the 
unnarrowed  vessels  may  thus  be  the  same  as  if  the  driving  power 
had  been  increased. 

Flow  through  capillary  tubes.  —  Poiseuille  found  that  the 
law  regulating  the  velocity  of  the  flow  of  liquids  through  tubes 
is  materially  altered  when  the  diameter  is  very  small.  Through 
capillary  tubes  he  found  that  the  volume  of  liquid  flowing  in 
unit  of  time  is  fr/£  =  k  •  r4H/7,  where  b  is  the  whole  volume 
observed  to  flow  in  the  course  of  time  £,  r  and  I  the  radius  and 
length  o£  the  tube,  H  the  head  of  liquid,  and  k  a  constant  to 
be  determined  by  experiment.  This  determination  is  effected 
by  actually  observing  the  number  of  seconds  taken  to  drive  a 
given  volume  of  liquid  through  a  capillary  tube  of  given  length 
and  diameter.  The  constant  k  does  not  depend  onAhe  nature 
of  the  walls  of  the  tube,  if  the  walls  be  wetted  by  the  liquid; 


316  OF  LIQUIDS.  [CHAP. 

it  depends  only  on  the  nature  of  the  liquid  and  on  the  tempera- 
ture. Water  near  the  boiling  point  passes  five  times  as  rapidly 
through  capillary  tabes  as  water  near  its  freezing  point.  We 
can  do  no  more  here  than  assert  with  a  reference*  that  theory 
indicates  that  (on  the  assumption  that  the  layer  of  liquid  in 
contact  with  the  solid  wall  is  at  rest,  an  assumption  verified  by 
the  fact  that  it  is  immaterial  what  is  the  nature  of  the  substance 
of  the  tube,  provided  that  it  be  wetted  by  the  liquid)  the  coeffi- 
cient k  =  777x7/877:  whence  it  is  easy  to  find  the  value  of  77,  the 
coefficient  of  viscosity,  for  any  liquid  at  any  given  temperature. 

Since  k  =  irpg/8r),  the  Volume  of  liquid  passing  through  a  capillary  tube 
in  time  t  is  ij  —  7rpg-r*R-t/Sr)l  or  7rr4-t-p/Srjl,  where  p  is  the  pressure,  in 
dynes  per  sq.  cm.,  upon  the  surface  of  the  liquid  as  it  is  delivered  into  the 
capillary  tube  ;  the  pressure  at  the  other  end  of  the  tube  being  nil. 

The  Mean  Velocity  v'  =  Volume  of  fluid  flowing  across  any  section  in 
unit  of  time  H-  Area  of  that  section.  Hence,  in  a  capillary  tube,  v'  =  ij/7rr2/  = 


The  flow  in  capillary  tubes  is  proportional  not  to  the  square, 
but  to  the  fourth  power  of  the  radius  ;  the  velocity  is  propor- 
tional not  to  the  square  root  of  the  pressure,  but  to  the  pressure 
itself. 

The  resistance,  R  =  p,  in  capillary  tubes  varies  directly  as 
the  velocity  ;  in  wide  tubes  approximately  as  the  square  of  the 
velocity.  This  seems  discrepant  ;  it  is  due  to  the  formation  of 
eddies  in  the  wider  tubes  ;  in  a  capillary  tube  the  flow  is  steady. 

But  what  is  a  "  capillary  "  tube  ?  For  water  it  is  a  tube 
under  1/50  inch  in  diameter.  Would  it  be  the  same  for  treacle? 
No  ;  a  long  inch-  wide  tube  behaves  with  treacle  as  a  1/50  inch 
tube  does  with  water;  the  flow  of  treacle  through  it  obeys 
Poiseuille's  Law.  Professor  Osborne  Reynolds  has  made  the 
very  important  discovery  that  steadiness  of  flow  arid  obedience 
to  Poiseuille's  Law  cease  only  when  the  expression  (Velocity  of 
stream  x  Width  of  stream  -s-  Viscosity  of  fluid)  has  reached 
a  certain  critical  value.  Too  great  a  velocity,  too  wide  a  stream 
—  in  either  case  the  stream  breaks  up  into  eddies  and  the  move- 
ment is  like  that  of  water  in  a  wide  tube  ;  but  even  in  a  wide 
tube  —  not  too  \vide  —  the  effect  of  great  viscosity  may  keep 
the  above  expression  down  below  its  critical.  value  and  the  flow 
may  be  steady  like  that  of  water  in  a  capillary  tube.  If  the 
stream  be  wide  enough  the  above  expression  will  be  above  its 

*Helmholtz  and  Piotrowski,  Sitzungsberichte  d.  Wien.  Acad.,  Math  .-naturw  .  CL 
XL.  18HO;  and  O.  E.  Meyer,  Wiedemann's  Ann.  d.  Physik  u.  d.  Chemie,  1877,  vol. 
ii.,  and  literature  there  cited. 


XL]  FLOW.  317 

critical  value ;  a  Mississippi  of  treacle  would  flow  turbulently 
round  any  obstruction  ;  lava-streams  flow  like  water.  One  and 
the  same  tube  may  be  made,  by  increasing  or  diminishing  the 
velocity  of  flow  through  it,  to  play  the  part  of  a  wide  tube  with 
eddies  or  of  a  narrow  tube  with  steady  flow. 

Measurement  of  the  pressure  at  any  point  of  a  stream. 
—  This  cannot  be  effected  by  cutting  the  tube  and  fixing  a 
manometer  in  it.  The  result  in  that  case  would  be  a  determi- 
nation of  the  original  driving  pressure  or  head  of  liquid,  for  the 
condition  becomes  statical  and  the  liquid  seeks  its  level.  The 
manometer  must  be  fixed  laterally  and  at  right  angles,  and 
the  flow  must  be  allowed  to  proceed  without  any  hindrance. 
This  being  seen  to,  any  one  of  the  forms  of  manometer  already 
described  may  be  used. 

Measurement  of  the  velocity  of  a  stream.  —  It  is  needless  to 
point  out  that  the  velocity  cannot  be  calculated  from  a  single  observation 
of  the  pressure  at  any  point. 

The  velocity  may  be  observed  by  direct  or  by  indirect  methods  with 
more  or  less  accuracy. 

A.  Direct  Methods.  —  a.  Optical.  —  The  velocity  of  bodies  floating 
in  the  stream  may  be  measured  by  observing  the  distance  traversed  by  one 
of  them  in  a  given  time. 

Objection  to  this  method.  —  The  velocity  of  floating  bodies  does  not  neces- 
sarily represent  the  velocity  of  the  stream.* 

A  body  of  the  same  sp.  density  as  the  liquid  moves  in  the  axial  stream : 
the  larger  it  is,  the  more  it  is  delayed  by  the  peripheral  layers  and  the  more 
slowly  it  moves.  It  moves  without  rolling,  unless  it  gets  into  the  periph- 
eral layers  and  is  twisted  out  of  them  into  the  centre  of  the  stream. 

A  body  heavier  or  lighter  than  the  liquid  is  pressed  against  the  upper 
or  lower  wall.  It  rolls  in  the  peripheral  layers,  for  it  is  urged  forward 
by  unsymmetrical  forces;  and  it  is  retarded.  Within  certain  limits,  the 
larger  it  is,  the  less  it  is  retarded ;  but  it  always  travels  more  slowly  than 
a  body  of  the  same  density  as  the  liquid. 

Of  two  heavy  bodies  the  heavier  moves  more  slowly ;  of  two  light  bodies 
the  lighter  moves  more  slowly ;  an  effect  due  to  rolling  friction. 

A  disc  which  rolls  travels  at  the  same  rate  as  a  sphere  of  the  same 
density  and  radius ;  if  it  travel  in  the  axial  stream  the  velocity  is  the  same 
as  that  of  a  sphere  or  cylinder  of  the  same  radius.  If  a  heavy  or  light  disc 
lie  flat  against  the  wall,  the  friction  is  increased  and  the  speed  diminished. 

Bodies  nearly  filling  'a  tube  approximate  more  nearly  in  speed  than 
when  they  are  small  in  relation  to  the  tube ;  if  heavy  they  tend  to  check 
the  stream  and  to  permit  the  pressure  to  accumulate  behind  them :  if  light 
they  tend  to  roll  rapidly,  and  thus  to  diminish  the  pressure  behind  them. 

If  the  density  of  the  liquid  be  diminished,  the  bodies  (e.g.,  red  corpuscles), 

*  For  the  facts  mentioned  in  these  paragraphs  I  am  indebted  to  the  kindness  of 
Prof.  Hamilton  of  Aberdeen,  who  rejects  Schklarewsky's  assertion,  that  the  same 
granular  substance  may  float  in  the  axial  or  in  the  peripheral  stream,  according  to 
the  nature  of  the  other  granules  with  which  it  is  associated  in  the  stream. 


318  OF  LIQUIDS.  [CHAP. 

which  had  been  just  light  enough  to  float  in  the  unaltered  liquid,  sink  and 
roll  in  the  lower  part  of  the  stream.  If  it  be  increased,  all  float.  In  either 
case,  or  if  the  sp.  gr.  of  the  floating  bodies  be  altered,  these  bodies  block  the 
bends  of  the  tube  and  check  or  slow  the  stream  (albuminuria,  cholera,  per- 
nicious anaemia,  fatty  embolism,  injection  of  milk  into  the  veins).  When 
the  stream  is  slowed,  if  the  particles  be  viscid  they  adhere  to  the  sides  of 
the  tube  and  the  stream  flows  past  them :  after  a  while  they  may  be  torn 
oft'  and  proceed. 

b.  Chemical.  —  This  is  a  method  principally  employed  in  physiological 
work  and  suggested  by  He  ring.     A  soluble  chemical  substance  easily  recog- 
nised is  introduced  into  the  stream  :   at  a  certain  distance  the  stream  is 
tapped,  and  samples  of  the  liquid  are  there  collected  at  regular  intervals 
of  time.     The  interval  of  time  which  elapses  before  the  chemical  substance 
can  be  detected  in  the  liquid  at  a  given  distance  affords  a  datum  from  which 
the  mean  velocity  can  be  roughly  calculated. 

c.  Volumetric.  —  1.    The  vessel  may  be  cut  and  the  amount  of  outflow 
measured.     This  is  objectionable  (1)  because  the  resistance  is  diminished 
and  the  velocity  increased,  and  (2)  if  the  stream  be  a  closed  circuit,  opening 
into  it  causes  loss  of  liquid  and  a  modification  of  the  driving  pressure. 

2.  A  tube  rilled  with  liquid  may  be  placed  in  the  course  of  the  stream. 
The  liquid  in  the  tube  is  driven  into  the  stream,  liquid  from  the  stream 
taking  its  place.     The  time  taken  to  empty  the  tube  is  observed.     The 
objections  are  (1)  resistance  interposed,  and  (2)  difficulty  of  recognising 
the  exact  instant  at  which  the  liquid  is  wholly  replaced. 

3.  In  Ludwig  and  DogiePs  Stromuhr,  used  by  physiologists,  there  are 
two  large  cavities;  the  one  nearer  the  heart  is  filled  with  oil;  the  cavity 
nearer  the  periphery  is  filled  with  defibrinated  blood  (the  introduction  of 
which  into  the  animal's  circulation  does  little  harm).     When  the  stream 
flows,  the  oil  passes  from  the  proximal  chamber  into  the  peripheral  one  :  the 
defibrinated  blood  of  the  peripheral  chamber  passes  into  the  animal:  the 
proximal  cavity  of  the  stromuhr  becomes  filled  with  the  fresh  blood  of  the  ani- 
mal.    Then  by  a  play  of  stopcocks  (effected  just  when  the  oil  is  on  the 
point  of  escaping  into  the  vessels  of  the  animal)  the  stream  in  the  instru- 
ment is  reversed,  and  the  animal's  blood  flows  into  the  chamber  now  occu- 
pied by  the  oil,  the  oil  passing  back  into  the  chamber  which  it  had  originally 
occupied,  and  the  blood  which  had  freshly  entered  that  chamber  being 
returned  into  the  circulation.     The  oil-chamber  is  always  functionally  in 
the  rear.     This  may  be  repeated  several  times,  and  thus  the  amount  of 
time  taken  to  pass  a  certain  large  volume  of  blood  through  the  instrument 
may  be  ascertained.     The  animal  suffers  on  the  whole  no  loss  of  blood,  and 
there  is  no  material  increase  (2  to  5  mm.)  of  resistance  in  the  circuit  of 
fluid;    while   if   the   stream   be  periodically  reversed   with  attention  and 
promptitude,  relatively  great  accuracy  is  attainable  in  the  determination 
of  the  mean  velocity. 

B.  Indirect  Methods.  —  The  mechanical  effects  of  a  stream  of  liquid 
are  derived  from  its  forward  momentum. 

a.  "  Hydrostatic  Pendulum."  —  If  some  object  be  attached  by  its  upper 
end  to  the  walls  of  the  tube,  and  swing  freely  in  the  stream,  the  quicker  the 
flow  the  more  will  the  lower  free  end  be  displaced.  A  box  containing  such 
a  contrivance  (the  so-called  hydrostatic  pendulum  of  engineers)  may 
be  inserted  (as  in  Vierordt's  Haemotachymetre)  in  the  course  of  a 
stream;  or  the  pendulum  itself  may  consist  simply  of  a  needle  thrust 


XT.]  VELOCITY   OF   FLOW.  319 

through  the  elastic  walls  of  the  tube  or  (as  in  Chauveau's  Haemodro- 
mo  metre)  through  elastic  parts  of  the  wall  of  the  tube.  In  the  latter 
case,  as  the  immersed  end  of  the  needle  is  driven  by  the  stream  in  one 
direction,  the  external  end  moves  in  the  opposite  direction,  and  the  elastic 
walls  of  the  tube  exercise  a  constant  pressure  upon  it,  tending  to  adjust  it 
in  its  normal  position  at  right  angles  to  the  wall  of  the  tube  or  to  the 
axis  of  the  stream.  The  whole  arrangement  is  very  sensitive  to  variations 
of  velocity. 

Chauveau's  instrument  has  been  so  modified  that  at  one  and  the  same 
part  of  the  circulation  the  pressure  may  be  found  by  a  sphygmoscope.  and 
the  velocity  ascertained  by  a  hsemodromometer  which,  being  coupled  with 
self  -registering  apparatus,  has  acquired  the  name  of  hsemodromograph. 
The  condition  of  a  stream  cannot  be  thoroughly  investigated  unless  the 
pressure  and  the  velocity  are  both  ascertained. 

b.  Pitofs  Tubes.  —  If  a  piezometer-tube  be  prolonged  into  the  axis  of  a 
stream,  and  if  it  be  bent  at  the  submerged  end  so  that  its  lower  orifice  faces 
the  stream  directly,  the  liquid  will  be  forced  up  in  it  to  a  certain  height, 
varying  with  the  velocity,  and  greater  than  that  which  corresponds  to  the 
lateral  pressure  at  the  same  part  of  the  tube.     If  its  lower  orifice  be  turned 
away  from  the  stream,  the  column  of  liquid  is  lower  than  it  would  have 
been  in  a  plain  piezometer.     If  two  such  lateral  tubes  be  fixed  near  to  one 
another  in  the  walls  of  a  main  tube,  the  mouth  of  one  facing,  that  of  the 
other  turned  away  from  the  stream,  there  will  be  set  up  a  difference  in  the 
heights  of  the  columns  in  these  tubes  ;  this  difference  depends  entirely  on 
the  velocity,  and  varies  with  it.     This  principle  has  been  applied  by  Marey 
(7  raw.  du  Lab.,  1875,  p.  347)  in  the  formation  of  a  registering  instrument 
of  great  excellence,  but  in  physiological  work  unfortunately  not  suitable, 
because  coagulation  is  promoted  by  the  projection  of  the  lateral  tubes  into 
the  lumen  of  the  main  tubes,  the  vessels  through  which  the  blood  passes. 

c.  Venturi-meters.  —  See  p.  314. 

All  instruments  in  which  indirect  methods  are  applied  must  be  graduated 
by  exposing  them  to  the  action  of  streams  of  various  known  velocities,  and 
marking  the  corresponding  positions  at  which  the  index  stands. 

Work  done  in  keeping  up  a  stream.  —  The  initial  head 
H  would  (if  no  energy  were  wasted  in  overcoming  resistances, 
etc.)  produce  a  mean  velocity  v  =  V2#H,  and  the  kinetic  energy 
imparted  to  a  mass  m  of  fluid  would  be  ^mv*  =  m^H.  This  is 
the  whole  energy  lost  by  the  water  falling  out  of  the  cistern, 
and  this  is  independent  of  the  amount  of  the  resistances.  The 
work  done  in  keeping  up  a  stream  is  therefore  independent  of 
the  length  of  the  pipes  ;  the  pressure  at  any  point  is  not  so.  If 
we  know  m  and  H  it  is  easy  to  calculate  the  total  work  done  by 
the  driving  apparatus  :  if  we  do  not  know  H,  but  do  know  Hp 
(the  height  of  the  maximum  piezometer  column),  the  equation 


Work  done  (=  mgH~)  =  mg]^p  -f  ^mv'2 

+ 

enables  us  to  find  it  when  we  know  v',  the  velocity  of  outflow. 


320  OF  LIQUIDS.  [CHAP. 

This  equation  is  arrived  at  by  combining  the  two  equations,  H  =  H^,  +  H^ 
and  Hw  =  v'*/2g. 

If  the  left  ventricle  of  the  human  heart  propel  at  each  systole  180 
grammes  of  blood  at  a  mean  pressure  equal  to  that  of  12-8  cm.  of  mercury 
(that  is,  the  sp.  gr.  of  blood  being  1-055,  at  a  head  H  of  165  cm.  of  the 
same  liquid,  blood),  during  each  systole  the  left  ventricle  does  work  equal 
to  mgU.  =  180  x  981  x  165  =  29,128000  ergs  =  weight  of  29,692  grammes 
raised  1  cm.  or  -29(59  kilos,  lifted  through  1  metre.*  Otherwise,  if  p  be  the 
mean  pressure  in  dynes  per  sq.  crn.,  the  energy  expended  =  mgH  —  mp  p  = 
pb ;  and  p= weight  of  12-8  cub.  cm.  of  mercury  =  (12-8  x  13-596  x  #);  whence 
Work  =  (12*8  x  13-596  x  981)  x  (180  H-  1-055)  =  29,128000  ergs  during  each 
systole. 

If  the  liquid  leave  the  tubes  with  actual  velocity  vf,  its  kin- 
etic energy  is  then  only  ^mv'2.  The  difference  (m^H  —  ^wv'2), 
having  been  spent  in  overcoming  resistances,  has  been  trans- 
formed into  heat. 

Streams  in  elastic  tubes.  —  If  the  inflow  be  continuous 
the  internal  pressure  expands  the  tube,  and  continues  to  do  so 
until  the  tube  exerts  a  contractile  restitution-pressure  equal  to 
the  expansile  pressure  of  the  liquid.  Then  the  stream  flows  on 
as  in  a  rigid  tube. 

If  the  inflow  be  intermittent  the  case  is  different.  We  may 
first  suppose  the  liquid  to  be  injected  instantaneously,  and  the 
tube  to  expand  as  a  whole.  In  such  a  case,  a  sudden  inflow 
creates  a  pressure  which  expands  the  walls  of  the  tube,  in  addi- 
tion to  forcing  onwards  a  certain  quantity  of  the  liquid.  Work 
is  thus  done  upon  the  walls  of  the  vessel.  These,  being  elastic, 
tend  to  restore  the  work  done  upon  them.  When  the  pressure 
due  to  the  inflow  is  relieved,  the  primitive  form  of  the  tube  is 
gradually  resumed :  the  potential  energy  of  the  stretched  walls 
is  transferred  to  the  liquid  in  the  form  of  kinetic  energy.  The 
stream  is  thus  kept  up  until  the  original  form  of  the  tube  is 
restored. 

If  there  be  a  quick  succession  of  influxes,  each  successive 
inflowing  quantity  may  enter  the  elastic  tube  before  its  prede- 
cessor has  left  it.  If  the  frequency  of  inflow  be  sufficiently 
great,  the  outflow  may  be  uninterrupted  though  variable  in 
velocity  and  amount.  The  rate  of  succession  necessary  for  con- 
tinuous outflow  depends  on  the  width  of  the  tube  —  being  greater 
as  this  is  greater  —  and  also,  in  the  reverse  sense,  on  the  exten- 
sibility of  the  walls  of  the  tube  and  on  the  mechanical  resistance 
offered  to  the  onflow.  The  greater  the  resistances,  or  the  greater 

*  Nearly  150  foot-pounds  per  minute,  or  ^  horse-power  on  the  average ;  about 
sV  h.-p.  during  contraction :  its  own  weight  raised  about  22,000  feet  per  hour. 


XL]  FLOW   IN  ELASTIC   TUBES.  321 

the  extensibility  of  the  tube,  the  greater  will  be  the  proportionate 
size  of  the  dilatation  or  pouch  of  the  elastic  tube,  and  the  more 
continuous  will  be  the  outflow:  the  more  deliberate,  therefore, 
may  be  the  succession  of  influxes  necessary  to  keep  up  a  con- 
tinuous outflow. 

Primary  waves  in  elastic  tubes.  —  The  tube  does  not  in 
fact  dilate  and  contract  as  a  whole,  nor  does  the  liquid  at  each 
inflow  enter  instantaneously.  The  pouching  is  local,  and  the 
inflow  more  or  less  gradual.  The'  more  gradual  the  inflow,  the 
less  the  width  and  the  greater  the  length  of  the  pouch  produced  : 
the  more  abrupt  the  inflow,  the  wider  and  shorter  will  the  pouch 
be.  The  pouch  contracts  and  drives  the  liquid  onwards  :  this 
action  dilates  the  walls  of  the  tube  beyond  the  pouch  :  the  dila- 
tation travels  onwards,  and  liquid  travels  with  it..  The  con- 
traction of  a  pouch  can  never  produce  another  quite  equal  to 
itself  in  width  :  and  so,  as  the  dilatation  travels  along,  it  becomes 
narrower  and  longer.  In  this  case  the  direction  of  movement 
of  the  liquid  as  a  whole,  and  that  of  the  dilatation,  are  the  same. 

If  instead  of  a  sudden  inflow  of  liquid  due  to  pressure 
there  were  a  sudden  outflow  due  to  suction,  there  would  be  a 
local  collapse  of  the  walls  of  the  tube.  The  walls,  returning 
to  their  original  form,  will  cause  a  stream  to  be  set  up,  which 
travels  towards  the  orifice  of  suction.  The  contracted  form  of 
the  tube  will  travel  in  a  direction  opposed  to  that  of  this  stream. 

The  dilatation  or  the  contraction  of  the  tube,  as  it  travels, 
forms  a  wave,  the  so-called  Pulse-Wave  —  positive,  and  trav- 
elling in  the  same  direction  as  the  liquid,  in  the  case  of  an 
inflow  and  dilatation  ;  negative,  and  travelling  in  the  opposite 
direction,  in  the  case  of  a  suction  of  the  liquid  and  a  contraction 
of  the  elastic  tube. 

The  farther  down  the  tube,  the  later  the  arrival  of  this  pulse- 
wave.  The  velocity  of  propagation  of  a  wave  of  this  kind 
depends  on  g,  the  elasticity-coefficient,  and  on  a,  the  thickness, 
of  the  wall  :  the  greater  these  are,  the  greater  is  that  velocity. 
It  also  depends  on  d,  the  diameter  of  the  tube,  and  on  p,  the 
density  of  the  liquid:  the  greater  these  are,  the  less  is  v,  the 
velocity  of  propagation,  the  distance  traversed  by  the  wave  in 
one  second. 


The  law  is  (Moens  :  Die  Pulscurve,  Leiden,  1878)  v  =  0-9  Vr 


The  elasticity  varies  in  the  case  of  arteries  ;  the  more  expanded  an  artery 
is,  the  more  elastic  it  is  ;  and  therefore  a  full  pulse  travels  more  rapidly  than 
one  of  small  expansion.  The  length  of  such  a  wave  =  time  of  inflow  x 


322  OF  LIQUIDS.  [CHAP. 

rate  of  propagation.  In  the  case  of  the  pulse  the  former  is  |  sec.,  the  latter 
is  from  10  to  18  metres  per  sec.  The  length  of  the  dilatation  in  the  arteries 
would  be  3-33  to  6  metres  if  the  arterial  system  were  long  enough  to  con- 
tain a  whole  wave.  The  arterial  system  is  never  during  life  wholly  relieved 
from  pressure,  and  is  in  a  state  of  permanent  though  variable  distention. 
The  elasticity  and  self-contraction  ("  arterial  tension  ")  of  the  arteries  are 
opposed  to  the  expansile  internal  blood-pressure,  and  at  each  instant  these 
are  either  equal  to  it  or  are  in  the  act  of  coming  into  equilibrium  with  it. 

The  form  of  a  simple  pulse- wave.  —  The  more  abrupt  the 
disturbance,  the  steeper  will  be  the  front  of  the  resulting  pulse- 
wave,  and  the  more  abrupt  will  be  the  expansion  of  any  part  of 
the  tube  at  which  the  pulse-wave  arrives.  The  greater  the  elas- 
ticity, the  less  will  be  the  height  of  the  wave  ;  the  greater  the 
length  of  the  wave,  the  gentler  will  be  the  pulse-rise.  The 
greater  the  resistance,  the  more  abrupt  will  be  the  dilatation, 
and  the  more  slowly  will  it  disappear. 

Secondary  waves  in  elastic  tubes.  —  When  by  a  sudden 
access  and  sudden  removal  of  pressure  a  primary  wave  is  pro- 
duced in  an  elastic  tube,  the  distal  end  of  which  offers  no  resist- 
ance to  outflow,  the  liquid  is  not  found,  on  removal  of  the 
pressure,  to  stop  just  when  it  has  regained  its  position  of  equilib- 
rium. In  virtue  of  its  inertia,  it  overshoots  that  mark  and  passes 
beyond  that  position,  leaving  the  tube  somewhat  collapsed  behind 
it.  The  tube  being  elastic  regains  its  form,  and  thereby  exer- 
cises suction :  a  back-rush  occurs  which  in  its  turn,  and  for  the 
same  reason,  is  again  excessive ;  and  a  system  of  secondary 
waves  is  thus  set  up,  of  which  usually  only  the  first  is  very 
important. 

The  form  of  the  physiological  pulse-wave.  —  The  pulse-wave  pre- 
sents first  a  sudden  rise,  a  steep-fronted  primary  wave,  due  to  a  rapid  con- 
traction of  the  ventricle ;  then  a  series  of  equidistant  secondary  waves,  of 
which  there  are  commonly  two,  seldom  three,  sometimes  only  one,  and  some- 
times that  one  (Moens)  so  strongly  marked  as  to  resemble  the  primary  wave 
in  heigut  ("  dicrotic  pulse"),  a  result  specially  observed  when  the  tension 
of  the  vessels  is  small  and  their  "coefficient  of  elasticity  "  great.  Between 
the  primary  wave  and  the  first  secondary  wave  there  is  a  sudden  sinking  and 
recovery  of  pressure  which  give  the  appearance  of  an  interpolated  undula- 
tion. This  is  (Moens)  due  to  the  cessation  of  the  ventricular  pressure  and 
to  the  folding  back  of  the  valves  under  the  influence  of  the  pressure  in  the 
elastic  blood-vessel  (aorta  or  pulmonary  artery). 

In  arteries,  the  higher  a  wave  the  faster  it  travels :  the  primary  wave 
travels  faster  than  the  secondary ;  and  the  distance  between  the  primary  and 
the  secondary  waves  increases  as  they  travel.  In  caoutchouc  tubes,  the 
coefficient  of  elasticity  does  not  vary  with  the  distention  as  it  does  in  arte- 
ries, and  there  is  no  such  relative  retardation  observed. 

Reflected  waves  in  elastic  tubes.  —  If  the  tube  be  wide 


XL]  WAVES  IN  ELASTIC   TUBES.  323 

down  to  the  orifice  of  outflow,  which  is  itself  narrow,  pouching 
takes  place  at  the  distal  end  of  the  tube.  There  will  be  greater 
or  less  recoil,  which  is  the  greater  the  greater  the  elasticity  of 
the  tube ;  and  this  produces  reflected  waves  in  the  stream ;  but 
the  outflow  through  the  narrow  distal  orifice  may  under  such 
circumstances  be  singularly  uniform. 

The  reflected  wave  returning  may  meet  the  hinder  part  of  the  same 
on-coming  wave,  and  may  complicate  its  form  with  secondary  undulations 
(see  Marey,  La  Circulation  du  Sang,  1881). 

This  does  not  wholly  explain  secondary  oscillations  of  the  pulse,  because 
the  capillary  system  (which  presents  a  wide  channel  for  the  onflowing  stream) 
does  not  present  this  kind  of  resistance. 

Amount  of  outflow  from  distensible  elastic  tubes. — 

Marey  showed  that  when  an  intermittent  inflow  was  distributed 
by  a  j^-tube  and  divided  between  a  rigid  and  a  distensible-elastic 
tube  (the  proximal  ends  of  which  were  provided  with  valves  to 
check  regurgitation,  and  the  distal  orifices  of  which  were  nar- 
rowed to  increase  the  resistance),  the  flow  from  the  distensible 
tube  was  (if  the  intermittent  inflow  were  sufficiently  frequent) 
not  only  continuous,  but  also  absolutely  greater  in  amount  than 
that  from  the  rigid  tube.  This  has  been  explained  as  depending 
on  the  diminished  mean  resistance  offered  by  the  widened  tube ; 
consequently  a  given  initial  total-head,  H,  may  have  a  larger 
proportion  of  its  own  amount  devoted  (as  velocity-head)  to 
imparting  movement  to  the  liquid,  though,  as  a  result  of  the 
widened  area,  the  actual  mean  rate  of  flow  across  each  unit  of 
sectional  area  may  be  less  than  that  in  the  rigid  tube. 

If  the  distensible  elastic  vessel  became  rigid  it  would  be  necessary,  in 
order  to  keep  up  the  same  onflow  in  it,  to  increase  the  driving  power. 
[Hypertrophy  of  left  ventricle  in  atheroma.] 


CHAPTER   XII. 

OF   GASES. 

Density.  —  The  standard  of  density  of  gases  is,  for  chemical 
purposes,  Hydrogen  =  1 :  sometimes  air  is  taken  as  the  standard, 
in  which  case  the  density  of  hydrogen  is  Tf || J-g-.  It  is  for  most 
physical  purposes  better  to  adhere  to  the  C.G.S.  system,  accord- 
ing to  which  air  has  a  density  p  =  -0012932,  and  hydrogen  a 
density  =  .0000895682. 

As  a  rule  the  density  of  a  gas  is  determined  by  first  weigh- 
ing a  vessel  —  a  glass  or  copper  vessel  or  a  collapsed  indiarubber 
bag  —  devoid  of  contents,  and  by  again  weighing  it  when  it 
contains  a  known  volume  of  the  gas  in  question. 

Elasticity.  —  Gases,  as  we  have  seen,  have  elasticity  of  vol- 
ume alone ;  and  in  this  they  are  perfect.  In  gases,  subjected  to 
a  compressing  force,  but  not  allowed  to  vary  in  temperature,  the 
Pressure  outwards  (acting  across  each  unit  of  area)  is  equal  to 
the  Resistance  to  the  compressing  Force  acting  inwards  (across 
the  same  area)  ;  the  Coefficient  of  Elasticity  of  Volume,  k,  is,  for 
every  temperature,  so  long  as  that  temperature  is  kept  constant 
(see  p.  368),  numerically  equal  to  the  Pressure  per  sq.  cm. ;  k=p. 

The  elasticity-coefficient,  ft,  is  any  small  increment  of  pressure,  p, 
divided  by  the  compression,  ij/ij,  produced;  it  is  therefore  equal  to  b-p/ij. 
When  there  is  no  change  of  temperature,  job  =  p"ti'  =  a  constant  for  any 
given  mass  of  gas  ;  hence  (p  +^)(iJ  —  ij)  =  pb  :  or,  omitting  the  product  jtrtj, 
which  vanishes  when  p  and  ij  are  infinitesimal,  tip  —  pb  =  0,  or  jo/ij  =  /?/b; 
whence  the  Elasticity-Coefficient  at  constant  temperature,  ft,  =  ij  -jo/ii  =  p  ',  it 
is  numerically  equal  to  the  Pressure  per  sq.  cm.  When  the  gas  is  com- 
pressed, but  at  the  same  time  no  heat  is  allowed  to  escape  from  it,  the 
compression  is  said  to  be  "  adiabatic  "  ;  and  the  law  of  the  relation  between 
the  volume  and  the  pressure  of  a  given  mass  of  gas,  under  these  circum- 
stances, is  stated  by  the  adiabatic  equation  (p.  395)  pW/0  =  const.,  where  k 
and  c  are  the  two  "thermal  capacities  "  (p.  367).  On  treating  that  equation 
in  the  same  way  as  the  former,  we  find  p/i}  =  k/c-p/b',  whence  the  Elas- 
ticity of  Volume  in  Adiabatic  Compression  is  not  ft  =  p,  but  ft'  =  p-  k/c :  and 
the  ratio  between  the  two  Elasticity-Coefficients,  ft  and  ft',  of  a  gas  is  the  same 
as  that  between  k  and  c,  the  two  Thermal  Capacities.  (See  note,  p.  370.) 

324 


CHAP.    XII.] 


PRESSURE. 


325 


Fig.lll. 


The  Pressure  within  a  gas  is  hydrostatic. 

Compressibility.  —  The  extent  to  which  a  gas  can  be  com- 
pressed is  indefinite,  provided  that  its  temperature  be  above 
Andrews's  critical  point  (p.  233) ;  if  the  temperature  be  below 
this  point,  compression  may  liquefy  the  gas. 

Boyle's  Law,  already  stated,  is  that  the  volume  of  a  gas 
and  the  pressure  acting  on  it  vary  inversely. 

Float  on  water  a  little  glass  bulb,  which  contains  an  adjusted  quantity 
of  air,  and  the  interior  of  which  communicates  by  an  aperture  with  the 
liquid  on  which  it  floats;  if  the  pressure  on  the  surface  of  the  water  be 
increased,  water  passes  into  the  bulb  and  compresses  the  contained  air ;  the 
bulb  as  a  whole  becomes  heavier  and  sinks  (Descartes'  Diver). 

In  a  fish  the  air  bladder  acts  during  muscular  relaxation  as  a  float ;  dur- 
ing contraction  of  the  muscular  walls  of  the  bladder  the  contained  air  is  com- 
pressed, the  mean  density  of  the  whole  body  is  increased,  and  the  fish  sinks.* 

Manometers  can  be  constructed  so  as  both  to  illustrate 
and  to  apply  Boyle's  Law.  If  a  tube,  bent  and  containing  mer- 
cury as  shown  in  Fig.  Ill,  and  enclosing  a  certain  volume  of 
air  within  the  space 
AB,  be  exposed  to  an 
additional  pressure  act- 
ing through  C,  that 
additional  pressure  will 
be  partly  spent  in  sus- 
taining the  weight  of 
the  column  EF  of  mer- 
cury raised  in  the  tube, 
partly  in  maintaining 
a  compression  of  the 
air  AB  within  the  space 
FB.  By  preliminary 
graduation  such  an  instrument  may  be  made  to  act  as  a  manom- 
eter, and  may  be  added  to  those  of  Fig.  105. 

Boyle's  law  is  somewhat  departed  from  by  oxygen,  carbonic  oxide, 
nitrogen,  air,  hydrogen,  whose  bulk  at  increasing  pressures  is  greater  than 
that  law  would  indicate ;  while  sulphurous  acid,  carbonic  acid,  and  other 
easily  condensible  gases  shrink  in  volume  more  rapidly  when  exposed  to 
moderately-increasing  pressures  than  the  amount  of  pressure  alone  would 
lead  us  to  expect.  The  latter  gases  present  very  curious  aberrations  when 

*  The  local  contraction  of  one  end  of  the  air  bladder  causes  the  other  end  to  act 
alone  as  a  float,  the  head  or  tail  being  thus  tilted  up  or  down.  The  air  bladder  is  in 
many  cases  too  near  the  ventral  aspect  to  sustain  the  fish  in  its  normal  position  in 
the  water ;  the  action  of  the  tail  sustains  the  fish  in  its  position  of  unstable  equi- 
librium throughout  the  whole  period  of  life. 


326  OF   GASES.  [CHAP. 

extremely  high  pressures  —  bringing  the  gas  to  the  verge  of  liquefaction  — 
are  applied. 

The  Tendency  of  Gases  to  Indefinite  Expansion  is 
utilized  in  the  Air-pump.  The  primitive  type  of  an  air-pump 
is  a  cylinder,  provided  with  a  piston  in  which  there  is  a  valve 
A  (Fig.  112),  opening  outwards.  The  cylinder  itself  is  con- 
nected with  C,  the  vessel  to  be  exhausted,  by  a  tube  closed  by 
the  valve  B,  opening  into  the  cylinder.  Suppose  the  piston 
drawn  out :  the  air,  within  C  and  the  connecting  tube  BC, 
expands,  thrusts  aside  the  valve  at  B,  and  fills  the  whole  space 
Fig  112<  open  to  it  —  that  is,  the 

^s\  cylinder,  the  tube,  and 
\  the  vessel  C.  The  pis- 
/  ton  returns :  the  valve 
y'  B  closes  and  the  valve 
- — -''  A  opens,  because  the 
air  between  A  and  B  is  compressed ;  the  air  in  AB  is  driven  out 
through  A.  By  repeating  this  process  often  enough  the  air  in 
C  is  greatly  reduced  in  quantity  and  becomes  of  correspondingly 
small  density.  This  simple  form  of  air-pump  is  liable  to  two 
objections :  —  (1)  It  is  tedious  to  pull  the  piston  against  an 
external  pressure  which  each  time  becomes  more  and  nearly 
equal  to  the  entire  atmospheric  pressure  of  15  Ibs.  per  square 
inch,  as  the  space  between  A  and  C  becomes  more  nearly  a 
vacuum ;  and  (2)  After  a  certain  number  of  strokes  the  expan- 
sion of  the  air  in  C  fails  to  lift  the  valve  B.  The  latter  objec- 
tion may  be  obviated  by  causing  the  piston  itself  to  lift  the 
valve ;  the  former  is  rendered  less  serious  by  connecting  with 
the  cavity  C  two  such  cylinders,  and  so  arranging  matters  that 
when  the  one  piston  is  being  driven  home  the  other  is  being 
drawn  out ;  the  whole  being  driven  by  a  large  and  heavy  wheel. 
Not  only  is  a  smaller  force  enabled  by  the  leverage  thus  gained 
to  resist  a  great  pressure,  but  the  inertia  of  the  flywheel  renders 
the  action  less  painful,  because  more  equable. 

Sprengel's  Air-pump  in  its  simplest  form  consists  of  a  long 
tube  AB,  Fig.  113,  provided  with  a  side-branch  EF,  which  com- 
municates with  a  vessel  C,  the  vessel  to  be  exhausted.  At  the 
upper  end  of  the  tube  AB  is  D,  a  supply  cistern  of  boiled  mer- 
cury, which  is  allowed  to  fall  down  AB.  As  it  passes  EF  the 
mercury  entangles  molecules  of  the  expansive  gas  in  EFC,  and 
these  are  continuously  removed  by  the  falling  stream  and  escape 
in  bubbles  at  B. 


xii.]  AIR-PUMPS.  327 

If  the  lower  end  of  the  tube  AB  be  bent  upwards,  a  vessel  filled  with 
mercury  may  be  inverted  over  the  upturned  end,  and  as  the  gas  issues  at  B 
it  can  take  the  place  of  the  mercury  in  that  vessel ;  the  Sprengel-pump  may 
thus  be  used  as  a  means  of  transferring 

small  quantities  of  gas  from  one  vessel  to      \  -?      Fig. 113. 

another. 

When  the  vacuum  in  C  is  toler- 
ably complete,  the  mercury  falls  as 
a  continuous  mass,  containing  no 
bubbles. 

If  a  long  rubber  tube  be  connected  with 
a  flask,  and  laid  across  a  table;  if  it  be 
squeezed  against  the  table  by  a  roller  pressed 
along  it,  away  from  the  flask  ;  the  air  in  the 
tube,  in  front  of  the  roller,  is  squeezed  out, 
and  part  of  the  air  in  the  flask  follows  the 
roller,  as  the  tube  regains  its  form  behind 
that  roller;  the  air  in  the  flask  thus  becomes  somewhat  rarefied.  If  this 
operation  be  repeated  often  enough,  without  ever  allowing  air  to  return 
from  the  external  atmosphere  into  the  rubber  tube,  a  considerable  degree  of 
rarefaction  may  be  attained  within  the  flask.  The  rubber  tube  should  be 
flexible  enough  for  the  roller  to  squeeze  it  flat,  but  at  the  same  time  rigid 
enough  to  withstand  the  atmospheric  pressure.  By  arranging  such  a  tube 
in  the  form  of  a  coil  inside  a  drum,  and  squeezing  it  by  a  continuously- 
rotating  roller,  very  considerable  rarefaction  can  be  easily  and  promptly  set 
up;  and  air-pumps  are  now  constructed,  acting  on  this  principle. 

The  Absorption  of  Gases  by  solids  is  sometimes  a  true 
solution,  as  in  the  case  of  the  alloy  of  metallic  hydrogen  *  with 
palladium.  This  is  produced  by  evolving  hydrogen  from  a 
palladium  electrode  in  the  electrolysis  of  water  (p.  658);  or  by 
heating  a  piece  of  palladium  in  vacuo,  and  allowing  it  to  cool  in 
an  atmosphere  of  hydrogen ,  or  even  by  heating  it  in  a  tube 
through  which  a  current  of  hydrogen  passes,  and  allowing  it  to 
cool  in  that  gas.  The  same  kind  of  colloid  solution  is  exemplified 
by  the  alloys  of  iron  and  hydrogen,  or  of  platinum  and  hydro- 
gen ,  or,  again,  by  the  carbonic  oxide,  which  (to  the  extent  of 
4:15  vols.)  is  retained  on  cooling  by  cast-iron,  or  the  carbonic 
dioxide,  of  which  a  half-per-cent  volume  may  be  retained  by 
indiarubber. 

Such  absorption  may,  on  the  other  hand,  be  due  to  surface 
attraction  and  condensation  within  the  pores  of  the  solid, 
—  as  in  the  case  of  animal  charcoal,  which  can  absorb  so  much 

oxygen  or  so  much  ammonia,  that  these  gases  must  even  be 

. _^ 

*  This  seems  (Graham,  Physical  and  Chemical  Researches)  to  be  a  white  para- 
magnetic metal  of  density  T95  ;  diamagnetic  (Blondlot) . 


328  OF   GASES.  [CHAP. 

liquefied  within  its  pores,  or  which  can  absorb  both  oxygen  and 
oxidisable  gases,  and  bring  them  into  such  close  molecular  rela- 
tions that  they  combine,  as  in  charcoal  respirators  ;  or  in  the 
case  of  platinum-black,  which,  if  surrounded  by  a  mixture  of 
oxygen  and  hydrogen,  absorbs  both  gases  and  brings  their  mole- 
cules into  contact  so  close  that  they  combine,  and  do  so  with 
evolution  of  heat  so  great  as  to  cause  the  platinum  to  become 
incandescent  and  thus  to  ignite  the  remainder  of  the  gas,  as  in 
Dobereiner's  Hydrogen  Lamp. 

Chemical  affinity  may  also  promote  the  absorption  of  a  gas 
by  a  solid.  If  a  dish  containing  spirit  of  wine  be  suspended 
over  quicklime  within  a  confined  space,  the  mixed  vapours  of 
alcohol  and  of  water,  which  pass  by  evaporation  into  the  space 
above  the  quicklime,  are  discriminated  by  it :  the  water  is 
absorbed,  the  alcohol  not;  more  water,  but  not  more  alcohol, 
is  evaporated  from  the  spirit  of  wine,  and  is  again  absorbed  by 
the  quicklime.  The  result  is  dehydration  of  the  spirit,  which 
may  proceed  to  an  extreme  degree. 

Where  a  gas  is  dissolved  freely  by  a  solid,  that  gas  may  freely 
traverse  that  solid.  Thus  hydrogen  leaks  freely  through  a 
white-hot  palladium  or  platinum  tube ;  so  does  carbonic  oxide 
through  glowing  iron ;  *  so  do  carbonic  acid,  marsh-gas,  coal- 
gas,  and  oxygen  in  small  quantities  through  indiarubber,  and 
coal-gas  under  high  pressures  through  cool  steel.  The  solids  in 
which  this  effect  is  observable  are  as  a  rule  colloid,  or  (like 
non-crystalline  metals)  resemble  colloids,  and  they  behave 
towards  gas  just  as  liquid  films  do. 

A  solution  of  a  gas  in  a  Liquid  is  itself  a  liquid,  of  which 
the  gas-molecules  form  a  part.  Liquids  differ  from  one  another, 
and  saline  solutions  from  pure  liquids,  in  their  power  of  dissolv- 
ing the  same  gas ;  and  different  gases  are  differently  soluble  in 
the  same  liquids.  The  more  readily  a  gas  can  be  liquefied,  the 
more  freely  will  it,  in  general,  dissolve  in  water  or  alcohol. 

The  Coefficient  of  Absorption  of  a  gas  in  a  liquid  is  the  vol- 
ume of  the  gas  dissolved  in  1  vol.  of  the  liquid,  the  volume  of  the  gas  being 
reduced  to  0°  C.  and  76  cm.  barom.  pressure.  The  actual  volume  of  gas  dis- 
solved in  1  vol.  of  liquid  at  any  specified  temperature  and  pressure  is  called 
the  Solubility  of  the  gas  in  the  liquid  at  the  given  temperature  and 
pressure. 

When  a  gas  is  dissolved  in  a  liquid,  the  liquid  increases  in 

*  Cast-iron  stoves  when  red-hot  allow  carbonic  oxide  to  escape  into  the  air  of 
a  room;  carbonic  oxide  in  small  quantities  destroys  the  red  blood  corpuscles  and 
produces  anaemia.  Carbonic  oxide  forms  a  volatile  compound  with  iron. 


xii.]  SOLUBILITY.  329 

volume,  generally  in  proportion  to  the  volume  of  gas  dissolved. 
Thus  1  vol.  of  water,  at  0°  C.  and  at  atmospheric  pressure, 
dissolves  1049  vols.  of  ammonia  gas  and  becomes  1-487  vols.  of 
ammonia-solution. 

According  to  Prof.  Henry,  carbonic  dioxide,  oxygen,  nitro- 
gen, and  some  other  gases,  are  dissolved  in  water  in  the  exact 
ratios  of  the  pressures  under  which  they  are  exposed  to  the  sur- 
face of  the  liquid ;  at  five  atmospheres'  pressure  five  times  as 
much  carbonic  acid  by  weight  can  be  dissolved  in  water  as  can 
be  dissolved  at  one  atmosphere.  The  volume  of  each  gas  dis- 
solved by  a  given  quantity  of  water  at  a  given  temperature  is 
thus  always  the  same.  Henry's  law  cannot  be  stated  as  a  uni- 
versal one  with  perfect  numerical  accuracy,  though  it  is  approxi- 
mately adhered  to  by  all  gases  in  relation  to  all  liquids :  the 
divergences  are  greatest  when  the  gas  is  very  soluble  in,  or  chem- 
ically unites  with  the  liquid,  or  acts  on  a  salt  dissolved  in  it. 

When  a  Mixture  of  gases  is  exposed  to  a  liquid,  each  gas 
is  dissolved  independently  of  the  rest:  each  is  dissolved  (if 
Henry's  law  be  obeyed)  in  proportion  to  the  partial  pressure 
exerted  by  it. 

Thus,  if  water  be  exposed  to  air  at  the  pressure  of  76  cm.  of  mercury, 
the  total  pressure  is  made  up  of  —^  x  76  =  15-884  cm.  pressure  due  to  oxy- 
gen, and  \-^-  x  76  =  60-116  cm.  pressure  due  to  nitrogen.  The  Solubility 
of  oxygen  in  water  at  10°  C.  is  -03250;  that  of  nitrogen  is  -01607;  both  at 
the  pressure  of  76  cm.  Hg.  It  happens  that  Henry's  law  applies  to  both 
these  gases.  Thus  the  volume  of  oxygen  dissolved  by  1  vol.  of  water  is 
±^±  x  -03250  =  -0067925  vol.;  that  of  nitrogen  is  ^^  x  -01607  =-0127114 
vol.  Hence  the  air  dissolved  in  water  —  that  which  subserves  the  respira- 
tion of  fishes  —  contains  oxygen  and  nitrogen  in  the  ratio  of  -0067925  to 
•0127114,  or,  in  percentages,  34-82  oxygen  to  65-18  nitrogen. 

The  solubility  of  gases  in  liquids  not  only  diminishes  with 
diminished  pressure,  but  also  with  increased  temperature.  The 
gases  dissolved  in  a  liquid  may,  if  they  form  with  it  a  simple 
solution, — as  in  an  aqueous  solution  of  ammonia,  — be  entirely 
removed  either  by  diminution  of  pressure  or  by  increase  of  tem- 
perature. In  some  cases  there  is  a  chemical  union  between  the 
gas  and  the  liquid,  or  some  constituent  of  it.  Thus  a  solution 
of  hydrochloric-acid  gas,  when  heated,  first  loses  some  gaseous 
HC1,  and  then  boils  over  as  a  whole  :  a  solution  of  bicarbonate  of 
soda,  when  the  pressure  is  greatly  reduced,  somewhat  suddenly 
loses  half  its  carbonic  acid  :  blood,  when  the  pressure  is  gradu- 
ally diminished,  first  loses  the  carbonic  acid  and  the  oxygen 
which  it  holds  in  simple  solution,  and  then,  at  a  very  low  pres- 


330  OF   GASES.  [CHAP. 

sure,  those  quantities  of  these  gases  which  it  holds  in  feeble 
chemical  combination  are  suddenly  given  off. 

A  gas  will  traverse  a  liquid  diaphragm  with  great  rapidity 
if  it  be  soluble  in  it :  it  is  dissolved,  diffuses,  and  emerges  on  the 
other  side.  A  soap  bubble  or  a  wet  bladder,  containing  hydro- 
gen and  surrounded  by  carbonic  acid,  absorbs  the  latter  gas  and 
enlarges  in  size,  although,  as  we  shall  see,  hydrogen  runs  with 
the  greater  speed  through  dry  or  indifferent  membranes,  and  a 
dry  bladder  under  similar  circumstances  would  collapse. 

It  seems  that  there  may  be  something  of  the  nature  of  a 
solution  of  a  gas  in  a  gas.  Oxygen  evolved  from  chlorate  of 
potash  may  contain  no  chlorine  or  any  oxides  of  chlorine  recog- 
nisable by  any  chemical  test ;  yet  if  it  be  passed  through  a  red- 
hot  tube  it  will  be  found  that  chlorine  can  now  be  detected  in  it 
(Schiitzenberger).  And  further,  a  gas  or  vapour  may  dissolve 
a  solid ;  boracic  acid  in  steam ;  naphthalene  in  coal-gas  rich  in 
hydrocarbon-vapours  ;  indeed  there  apparently  never  occurs  any 
evaporation  from  a  solution  entirely  without  this  result. 

Diffusion  of  Gases.  —  Two  gases  in  vessels  between  which 
a  free  communication  is  established  are  found  to  mix  freely,  and 
if  sufficient  time  be  allowed  the  mixture  will  become  uniform 
throughout.  The  rate  of  diffusion  is  somewhat  rapid.  Pure 
carbonic  acid  and  air  placed  in  communication  will  diffuse  at 
such  a  rate  that  the  air  at  a  distance  of  half  a  metre  will  be 
found  in  seven  minutes  to  contain  one  per  cent  of  carbonic  acid  : 
hydrogen  will  similarly  travel  a  third  of  a  metre  in  one  minute 
(Graham).  The  lighter  the  gas  the  more  rapidly  does  it  travel. 

This  process  is  molecular,  and  solid  particles  floating  in  either  gas  remain 
practically  at  rest ;  and  thus  mere  diffusion  is  not  sufficient  for  purposes  of 
ventilation. 

In  the  lungs,  diffusion  carries  air  out  of  the  air-cells  and  oxygen  into 
them ;  the  oxygen  tends  to  travel  more  rapidly  inwards,  and  hence  there  is 
a  small  force  tending  to  dilate  the  air-cells  (Graham). 

Effusion.  —  When  gas  is  caused  to  flow  through  apertures, 
such  as  pinholes  in  membranes,  the  law  of  Torricelli  is  obeyed, 
and  the  velocity  of  outflow  v  —  V2#H.  Air  rushing  through 
such  an  aperture  into  a  vacuum  will  do  so  as  if  the  atmos- 
phere consisted  of  a  uniform  layer  of  fluid,  of  uniform  density 
=  -0012932,  and  throughout  which  g  is  constant,  all  at  the 
freezing  temperature  and  at  the  pressure  of  76  cm.  of  mercury ; 
the  layer  having  a  depth  H  of  799022  cm.  (nearly  5  miles). 
Then  v  =  V%H  =  V2  x  981  x  799022  =  39595-2  cm.-per-sec. 


xii.]  EFFUSION.  331 

In  different  gases  at  the  same  pressure  the  height  H  will  vary  inversely 
as  their  densities ;  the  gases  will  therefore  pass  through  apertures  with 
velocities  inversely  proportional  to  the  square  roots  of  their  densities.  Thus 
oxygen  and  hydrogen,  whose  densities  are  16  : 1,  will  have  effusion-velocities 

— =  :  — — >  i-e->  1 : 4,  at  the  same  temperature  and  pressure. 
V16     VI 

In  any  gas  the  velocity  of  outflow  is  not  affected  by  changes  of  pres- 
sure. H  is  proportional  to  the  pressure  p;  it  also  varies  inversely  as  the 
density;  H  ocp/p.  If  the  pressure  be  increased,  Boyle's  law  shows  that  the 
density  is  increased  in  the  same  ratio ;  hence  p/p  is  constant.  Wherefore 
H  is  constant,  and  v(— V2^rH),  the  effusion-velocity  of  each  gas  (i.e.,  the 
volume  flowing  per  second  -=-  the  area  of  the  aperture),  is  constant  under  all 
circumstances  of  pressure ;  and  the  normal  rate  of  outflow  of  different  gases 
at  constant  temperatures  depends  only  on  the  nature  of  the  gases.  Under 
changes  of  temperature  at  constant  volume,  p  oc  r°  Abs.  (p.  370) ;  .-.  v  oc  Vr° 
Abs.  Under  changes  of  temperature  at  constant  pressure,  Hocl/p; 
.•.  v  <x  I/ Vp.  Perturbations  are,  however,  produced  by  variations  in  the  vis- 
cosity of  different  gases  at  different  temperatures ;  these  cause  slight  depar- 
tures from  this  law. 

This  phenomenon  of  outflow  or  effusion  is  one  of  masses, 
and  in  it  gases  act  as  fluids,  practically  continuous. 

If  a  gas  be  driven  under  pressure  through  a  substance 
which  is  porous,  but  whose  pores  are  too  small  to  allow  the 
mass  to  traverse  it  without  great  resistance,  the  result  is  the 
transpiration  of  the  gas,  a  slow  flow  under  resistance.  Trans- 
piration may  be  studied  by  driving  gases  through  long  capillary 
tubes,  or  even  through  tubes  which  are  not  capillary,  provided 
that  their  length  so  far  (4000  : 1)  exceed  their  diameter  that 
considerable  resistance  is  offered  to  the  onflow  of  the  gas.  It  is 
found  that  in  each  case  the  mass  of  gas  passing  per  second  is 
proportional  to  the  motive  pressure,  but  also  varies  inversely  as 
the  length,  directly  as  the  density  of  the  gas  (a  singular 
result),  and  further,  depends  on  a  constant,  the  Coefficient 
of  Transpiration;  m/t  =  k-pp/l.  A  film  of  gas  adheres  to 
the  sides  of  the  tube,  and  the  gas  flows  in  an  axial  stream  in 
each  channel. 

The  coefficient  of  transpiration  peculiar  to  each  gas  is  a  very  isolated 
factor,  and  does  not  seem  to  have  any  intelligible  relation  to  the  other  prop- 
erties of  gases.  The  transpiration-coefficients  of  nitrogen,  of  nitric  oxide, 
of  carbonic  oxide,  are  double  that  of  hydrogen  :  those  of  ether  and  of  hydro- 
gen are  the  same :  those  of  oxygen  and  nitrogen  are  related  to  one  another 
in  the  ratio  14  : 16,  so  that  equal  times  are  taken  by  equal  masses  of  these 
gases  to  pass  through  long  or  capillary  tubes. 

If  a  gas  be  heated  it  will  become  lighter,  and  its  transpiratipn-rate  will 
be  lessened  :  if  the  barometric  pressure  rise,  it  will  be  compressed  and  its 
transpiration-rate  will  be  increased. 


332  OF   GASES.  [CHAP. 

Membrane-Diffusion.  —  Gases  placed  on  opposite  sides  of 
an  indifferent  porous  membrane,  and  exposed  neither  to  the 
influence  of  a  difference  of  pressures  nor  to  that  of  a  difference 
of  solubilities  in  the  material  of  which  the  membrane  is  com- 
posed, will  pass  through  it  in  virtue  of  their  own  molecular 
motion. 

The  velocity  with  which  hydrogen  and  oxygen,  separated 
by  a  partition  of  plaster-of-Paris,  or  graphite,  or  biscuitware,  will 
traverse  that  partition  is  exceedingly  small  in  comparison  with 
the  rate  of  effusion  through  a  relatively  large  aperture  into  a 
vacuum ;  but  it  is  found  to  be  proportional  to  the  mean  velocity 
of  the  molecules  in  the  gas. 

We  have  already  seen  (p.  250)  that  the  mean  velocity  of 
the  particles  of  any  gas  is  inversely  proportional  to  the  square 
root  of  the  density  of  that  gas ;  and  hence  the  rate  of  diffusion 
of  any  gas  through  an  indifferent  membrane  is  inversely  pro- 
portional to  the  square  root  of  the  density  of  that  gas. 

A  dry  bladder  filled  with  hydrogen  and  surrounded  by  oxy- 
gen will  partially  collapse,  for  hydrogen  leaves  it  four  times  as 
fast  as  oxygen  enters  it. 

This  difference  of  diffusion-rates  may  be  made  to  effect  a 
partial  separation  of  gases.  If  a  long  porous  tube  be  fitted 
so  as  to  pass  through  a  vacuum  or  a  neutral  gas,  and  if  a  mix- 
ture of  gases  be  passed  through  the  porous  tube,  the  compo- 
nents of  that  mixture  will  escape  through  the  walls  of  the 
porous  tube  in  unequal  proportions.  If  the  vapour  of  chloride 
of  ammonium  be  passed  through  such  a  tube,  the  hydrochloric 
acid  (density  =18*25  when  H  =1)  and  ammonia  (density  =  8*5), 
into  which  the  chloride  is  dissociated  by  heat,  pass  through  in 

the  ratio  of to  — — :  the  ammonia  thus  passes  through 

VTM5       V8-5 

in  excess,  and  litmus  paper  placed  in  the  neighbourhood  of  the 
porous  tube  will  indicate  an  alkaline  reaction. 

A  gas  may  pass  through  the  pores  of  a  solid  by  liquefaction 
in  those  pores :  sulphurous  acid  may  pass  through  charcoal  and 
evaporate  on  the  farther  side. 

Diffusion  of  Gases  from  Liquids.  —  If  a  layer  of  liquid 
charged  with  gas  be  placed  upon  one  free  from  gas,  the  gas 
rapidly  permeates  the  whole  liquid.  If  the  two  layers  be  sepa- 
rated by  a  membrane  wetted  by  both,  the  diffusion  is  rapid.  If 
the  two  layers,  thus  separated  by  a  thin  membrane,  be  in  a  state 
of  relative  motion,  the  diffusion-rate  may  be  accelerated  if  the 


xii.]  DIFFUSION.  333 

velocities  be  not  too  great.  If  two  streams  so  separated  move 
in  opposite  directions,  they  may  completely  exchange  gases  ;  for 
suppose  two  such  streams  to  be  charged,  as  they  arrive  at  the 
opposite  ends  of  a  certain  tract  of  vessel,  with  gases  A  and  B  : 
then  throughout  the  whole  of  that  part  of  their  course  during 
which  they  are  contiguous,  the  A-charged  stream  passes  and 
diffuses  A  into  a  stream  which  is  at  every  point  poorer  in  A 
than  the  A-charged  stream  itself  is  at  the  same  point;  and  vice 
versd  :  so  that,  if  the  course  be  long  enough,  the  A-charged 
stream  may  lose  all  its  A,  and  the  B-charged  stream  all  its  B. 

The  Statics  of  Gases.  —  A  gas  always  fills  the  whole  space 
within  which  it  is  contained.  There  is  no  difference  in  respect 
of  statical  theorem  between  a  gas,  and  a  liquid  which  also  fills 
the  whole  space  within  which  it  is  contained.  Pascal's  prin- 
ciple, that  of  the  so-called  Transmissibility  of  Pressure,  that  of 
the  perpendicularity  of  the  pressure  exerted  by  a  fluid  upon  its 
bounding  surface  —  all  these  apply  equally  to  all  fluids  :  so  do 
the  principle  of  the  Hydraulic  Press  and  that  known  as  Archi- 
medes' Principle. 

The  last  must  be  kept  in  mind  when  accuracy  is  required  in  weighing. 
A  piece  of  brass  of  density  8  and  weighing  1  kilo,  in  vacuo  occupies  125  cub. 
cm.  (|  the  bulk  of  an  equal  mass  of  water).  It  apparently  loses,  when  weighed 
in  air,  the  weight  of  125  cub.  cm.  of  air  ;  that  is,  125  x  -0012932  grammes 
=  -16165  gramme.  The  substance  to  be  weighed  also  loses  weight,  but  if  it 
displace  more  air  than  the  counterpoising  mass  of  brass  does,  it  loses  more 
than  the  brass  does,  and  an  inaccurately  large  quantity  of  it  has  to  be  used 
in  order  to  counterpoise  the  metallic  kilogramme. 

Balloons  and  soap  bubbles  containing  coal-gas  or  hydrogen  rise  in  the 
air  ;  bulk  for  bulk  they  are  lighter  than  air.  The  lighter  they  are,  the  more 
rapidly  they  ascend  ;  and  they  can  be  loaded  until  they  weigh,  bulk  for  bulk, 
the  same  as  the  air  in  which  they  float.  If  a  balloon  with  its  contained  gas 
weigh  100  Ibs.,  and  the  bulk  of  air  displaced  by  it  weigh  120  Ibs.,  the  balloon 
will  rise  under  an  ascensional  force  equal  to  the  weight  of  20  Ibs.  applied  to 
a  mass  of  100  Ibs.  ;  its  upward  acceleration  will  be  equal  to  g  x  -f^  =  \g. 

The  pressure  on  the  walls  of  a  closed  vessel  containing  gas 
is  greater  the  lower  the  level  at  which  it  is  measured  :  the  law  is 
exactly  the  same  for  gases  as  for  liquids.  The  effect  is  seldom  per- 
ceptible, because  within  vessels  of  ordinary  size  the  mere  weight 
of  the  gas  adds  little  to  the  atmospheric  or  other  pressure  acting. 

With  vessels  of  ordinary  dimensions  a  manometer  applied 
laterally  at  any  part  will  indicate  the  internal  pressure  ;  strictly 
speaking,  in  gases,  as  in  liquids,  it  indicates  only  the  pressure  at 
the  horizontal  level  of  the  orifice  of  communication  between  the 
manometer  and  the  vessel. 


[WI7IRSITT] 

f*      «MT    _«V* 


334  OF   GASES.  [CHAP. 

Streams  of  Gas.  —  The  statements  made  in  the  discussion 
of  streams  of  liquid  in  Chap.  XI.  apply  also  to  streams  of  gas. 
The  Law  of  Continuity,  Torricelli's  Law,  the  distinction  between 
Velocity-head  and  Pressure-head,  the  gradual  disappearance  of 
the  latter,  coupled  with  the  simultaneous  heating  of  the  flowing 
fluid,  the  Lateral  Pressure  in  a  main  pipe,  and  the  propulsion  of 
the  fluid  up  piezometer  tubes  or  through  lateral  orifices,  —  all 
these  apply  to  gaseous  as  well  as  to  liquid  streams. 

In  calculations  based  on  Torricelli's  Law  it  is  necessary  to  find  H.  H  is 
the  height  of  that  column  of  the  outflowing  fluid  which  would,  if  acting 
alone,  produce  a  pressure  equal  to  that  actually  undergone  by  the  fluid  set 
in  motion.  If  the  gas  issue  from  a  vessel  in  which  the  pressure  is  such  as 
to  support  a  manometric  column  of,  say,  24  cm.  Hg  in  addition  to  the  atmos- 
pheric pressure;  and  if  the  atmospheric  pressure  at  the  time,  as  shown  by 
the  barometer,  be  76  cm.  Hg :  the  whole  pressure  on  the  gas  is,  for  each  sq. 
cm.  of  its  bounding  surface,  equal  to  the  weight  of  a  column  of  mercury  whose 
content  is  100  (=76  +  24)  cub.  cm.  This  is  equivalent  to  the  weight  of 
1359-6  cub.  cm.  of  water.  If  the  density  of  the  gas  be  ^  that  of  water,  the 
pressure  p  per  sq.  cm.  is  equal  to  the  weight  of  800  x  1359-6  =  1,087680  cub. 
cm.  of  that  gas,  the  same  gas  as  is  driven  out  in  a  jet.  This  column,  stand- 
ing on  a  sq.  cm.  base,  is  1,087680  cm.  high.  Hence,  for  the  gas  in  question, 
H  =  1,087680 ;  and  the  velocity  of  that  gas,  rushing  into  a  vacuum  under  a 
total  pressure  of  100  cm.  Hg  (so  long  as  the  pressure  is  maintained  at  that 
value),  is  v  =  V2^H  =  V2  x  981  x  1,087680  in  cms.  per  second;  while  into 
the  atmosphere  it  would  run  with  a  velocity  V2  x  981  x  (800  x  13-596  x  24) 
due  to  the  difference  (24  cm.  Hg)  between  the  internal  and  external 
pressures. 

Recoil.  — When  a  stream  of  gas  issues  from  a  jet  or  burner, 
the  reaction  is  equal  to  the  action,  and  there  is  a  tendency 
for  the  burner  itself  to  move  backwards.  This  tendency  we 
see  turned  to  account  in  certain  revolving  shop-window  gas- 
illuminations. 

Viscosity.  —  The  viscosity  of  gases,  which  is  due  to  diffu- 
sion, is  on  the  whole  similar  in  its  results  to  that  of  liquids. 

A  stream  of  air,  driven  through  air,  soon  comes  to  rest.  If 
it  have  a  great  velocity,  it  can  cut  its  way  through  air  to  a  greater 
distance  than  a  slower  stream  can. 

If  a  stream  of  air  be  introduced  into  a  room  through  a  funnel-shaped 
aperture,  the  broad  mouth  of  the  funnel  being  open  to  the  external  air,  it 
will  enter  the  room  through  the  narrow  orifice  with  great  velocity,  and  will 
pass  a  considerable  distance  (being  acutely  felt  as  a  draught),  until  at  length 
the  process  of  diffusion  between  it  and  the  surrounding  air  relieves  it  of  its 
relative  momentum.  If,  on  the  other  hand,  the  narrow  end  of  the  funnel 
be  presented  to  the  exterior  air,  the  stream  as  it  enters  will  (obeying  the 
Law  of  Continuity)  widen  out  in  accordance  with  the  shape  of  the  funnel, 
and  its  velocity  will  be  proportionately  diminished ;  the  result  being  that 


xii.]  VISCOSITY.  335 

a  considerable  amount  of  air  may,  through  ventilators  of  such  a  form,  be 
introduced  into  a  room  without  producing  a  perceptible  draught. 

All  objects  surrounded  by  air  bear  on  their  surface  an  adhe- 
rent film  of  air  which  is  almost  dustless.  When  a  body  moves 
in  air,  this  film  rubs  against  contiguous  layers  of  air,  and  the 
movement  of  the  body  is  retarded  by  internal  friction  in  the  air. 
Haughton  and  Emerson  Reynolds  found  that  a  granite  ball  sus- 
pended in  the  air,  and  swung  pendulum-fashion,  suffered,  on 
each  successive  swing,  a  diminution  of  amplitude  of  g-^g-  due 
to  this  cause. 

The  friction  within  a  gas  is  independent  of  its  density,  but  increases 
with  its  temperature  (Clerk  Maxwell,  Phil.  Trans.  1866). 

A  body  falling  in  vacua  is  not  retarded,  and  falls  with  a 
downward  acceleration  fully  equal  to  g  ;  falling  through  air,  it 
is  retarded,  because  the  viscosity  of  the  air  causes  friction.  A 
thick  piece  of  gold  and  a  piece  of  paper  fall  in  vacuo  at  the  same 
rate  :  through  air  the  gold  falls  more  rapidly,  because  it  presents 
less  surface  in  proportion  to  its  weight:  but  even  through  air 
a  piece  of  smooth  paper  and  a  piece  of  gold  leaf,  presenting  the 
same  total  surfaces  and  the  same  weights,  or  each  bearing  the 
same  proportion  between  its  surface  and  its  weight,  will  fall  side 
by  side. 

Falling  water  is  retarded  by  the  air ;  and  conversely,  air  is 
dragged  down  by  falling  water.  If  a  stream  of  water  be  made 
to  fall  through  a  closed  cavity,  the  water  will  drag  down  with  it 
a  considerable  volume  of  air ;  and  if  a  lateral  communication  be 
established  between  this  cavity  and  a  vessel  containing  air,  much 
of  the  air  in  that  vessel  may  be  extracted.  If  a  stream  of  air 
or  steam  be  maintained  through  a  cavity,  it  is  not  only  itself 
retarded,  but  the  surrounding  air  is  dragged  with  it,  and  the 
pressure  in  the  cavity  is  diminished. 

This  action,  due  to  viscosity,  is  independent  of  the  general 
diminution  of  pressure  experienced  by  fluids  in  motion.  A 
vibrating  tuning-fork  -brought  near  a  suspended  pith-ball  seems 
to  attract  it;  the  air  between  the  objects  vibrates,  the  pressure 
is  lessened,  and  the  exterior  atmospheric  pressure  urges  the  ball 
against  the  tuning-fork. 

When  the  density  of  the  vibrating  fluid  is  p,  at  a  point  in  the  fluid  where 
the  greatest  velocity  of  vibration  is  i>,  the  diminution  of  pressure  p  is  \pvz 
per  sq.  cm. ;  provided  that  the  cause  of  the  vibration  be  the  td-and-fro  move- 
ment of  solids  moving  within  a  finite  space  of  the  fluid  (Lord  Kelvin). 


336  OF  GASES.  [CHAP. 

Measurement  of  Flow.  —  The  amount  of  flow  of  gas 
through  pipes  may  be  measured  on  the  same  principles  as  the 
amount  of  flow  of  liquids. 

(a)  The  amount  of  gas  actually  passed  may  be  collected  and 
measured.  It  may  be  collected  in  a  balanced  bell-jar,  inverted 
over  water  like  a  small  gasometer  (Hutchison's  Spirometer),  or 
in  a  very  large  and  thin  flexible  caoutchouc  bag  (Boudin). 

(6)  It  may  be  made  to  drive  a  registering  train  of  wheel- 
work,  like  a  gas-meter,  as  in  Bonnet's  pneumatometer. 

(tf)  The  principle  of  the  hydrostatic  pendulum  or 
that  of  Pitot's  tubes  may  be  employed. 


Barlow's  Formula  for  the  flow  of  gases  in  pipes  is  Q  —  1350  d2  V/i^/x/, 
where  Q,  is  the  flow  in  cub.  ft.  per  hour,  d  the  diameter  in  inches,  h  the  pres- 
sure in  inches  of  water,  s  the  sp.  gr.  of  the  gas  (air  =  1),  and  I  the  pipe- 
length  in  yards.  In  C.G.S.  measures,  this  becomes  b  =  222-83  1 


=  7-115  t'Vpd5/pl,  where  p  =  dynes  per  sq.  cm.,  d  and  I  are  measured  in 
cm.,  p  is  the  density  (water  =  1)  and  b  is  the  number  of  cub.  cm.  passing  in 
time  t.* 

THE  PRESSURE  OF  THE  ATMOSPHERE. 

Most  of  our  experiments  and  observations  are  complicated 
or  affected  by  the  fact  that  we  live  at  the  bottom  of  an  atmos- 
pheric ocean  which  exerts  pressure  upon  every  surface  exposed 
to  it,  and  which  penetrates  even  into  the  recesses  of  everything 
porous,  and  there  also  exerts  pressure,  unless  special  appliances 
be  made  use  of  in  order  to  remove  it  wholly  or  in  part.  We 
live  at  the  bottom  of  such  an  atmosphere  without  inconvenience, 
just  as  deep-sea  fishes  live  at  the  bottom  of  the  ocean  :  so  long 
as  they  are  in  their  habitat,  the  internal  pressure  of  the  gases 
contained  and  dissolved  in  their  organisms  is  equal  to  and  is  in 
equilibrium  with  the  immense  external  pressure  exerted  by  the 

*  Reductions  of  this  kind  are  frequently  found  very  troublesome.  Here,  if  d=l  inch, 
A=l  inch,  ,s=l,  and  1=1  yard,  Q=1350  cub.  ft.  per  hour.  Similarly,  if  d=l  cm.  =  ^ 
inch,  h  =  1  cm.  =2^  inc^  and  I  =  1  cm.  =  ^^  yard,  Q  =  1350  X  (a.V*)2  X  V(2.54 
X  «7j-i/9iT44«)  =  (787-77  Ws  )  cub.  ft.  per  hour=  (787'77W«  X  (30-48)8-5-3600  sec.) 
=  (6196-4  -^  Vs)  cub.  cm.  per  second.  But  if  we  measure  the  density  of  the  gas  in 
terms  of  water  as  the  standard,  we  use,  instead  of  \A,  a  smaller  divisor  V/t> 
=Vo-Q012932g,  and  must  compensate  for  this  by  multiplying  the  numerator  6196'4  by 
VO'0012932.  The  number  of  cub.  cm.  per  second  is  then  (222-83+-  V/>)  or,  if  p  =  unity, 
222-83  simply.  This  is  the  numerical  factor  which  takes  the  place  of  the  original 
1350;  and  now  the  number  of  cub.  cm.  per  second  is  222'83  VdWpZ,  where  all  the 
terms  are  in  C.G.S.  units.  The  pressure  is  still  stated  in  terms  of  water-column;  to 
transform  it  to  dynes  per  sq.  cm.,  we  observe  that  when  the  manometer-column  has  a 
height  h,  p  =  hp'c/  where  p'  is  the  density  of  the  manometer-liquid;  but  in  a  water- 
column,  ^=JJ_and  h  =p/g  ;  whence  the  number  of  cub.  cm.  flowing  per  second  is 
222-83  v' 


xii.]  ATMOSPHERIC   PRESSURE.  337 

surrounding  water ;  but  when  they  are  brought  towards  the 
surface,  the  external  pressure  becoming  greatly  less,  the  gases 
contained  in  the  swim-bladder  and  throughout  the  tissues 
undergo  expansion,  and  the  fish  explode. 

The  pressure  within  our  organisms  cannot  be  less  than  the 
atmospheric  pressure,  that  exerted  by  the  atmosphere  on  the  sur- 
face, 1,013663  dynes  per  sq.  cm.,  or  a  pressure  equal  to  the 
weight  of  about  15  Ibs.  per  sq.  inch.  If  the  internal  pressure  in 
any  part  become  less  than  this,  the  fluids  or  the  semi-fluid 
tissues  or  masses  of  the  body  must  flow  towards  the  region  of 
diminished  pressure.  Hence  a  permanent  vacuum  within  the 
body,  total  or  partial,  is  impossible. 

The  abdominal  walls  are  closely  appressed  against  the  viscera :  the 
walls  of  these  are  pressed  against  one  another  as  far  as  their  contents  will 
allow. 

The  lungs  are  pressed  against  the  ribs  by  the  atmospheric  pressure  act- 
ing down  the  trachea  and  bronchi,  and  they  are  thereby  expanded  when, 
but  for  this  action,  the  expansion  of  the  ribs  would  tend  to  form  a  vacuum 
between  the  pulmonary  pleura  and  the  parietes  of  the  thorax.  This  expan- 
sion does  not  take  place  when  the  thorax  is  so  opened  by  a  wound  that,  on 
expansion  of  the  ribs,  air  can  pass  through  the  wound  into  the  pleural 
cavity,  and  can  thereby  equalise  the  internal  and  external  pressures  without 
the  aid  of  pulmonary  inflation.* 

The  atmospheric  pressure  acts  freely  upon  and  through  a 
mass  of  gas,  if  that  mass  be  free  to  expand  or  contract,  whatever 
be  its  temperature.  The  air  in  a  room  may  be  hot,  and  yet  the 
atmospheric  pressure,  acting  down  the  chimney  and  through 
all  the  chinks  and  orifices  of  the  room,  will  be  undiminished  in 
amount  and  in  effect. 

A  trap  in  a  wash-hand  basin  in  a  room  will  not  be  unable  to  prevent 
gases  from  being  forced  into  the  room  from  the  drains,  simply  because  the 
air  in  the  room  is  warm.  It  may  be  unable  to  do  so  if  the  pressure  within 
the  drains  become  excessive,  or  if  the  air  in  the  room  be  rarefied  by  a  strong 
draught  up  the  chimney,  especially  where  the  fittings  of  the  room  are  so 
air-tight  that  the  external  pressure  cannot  force  air  into  the  room  except 
through  the  trap. 

If  any  object  containing  gas  or  air  be  placed  in  a  region  of 
space  from  which  the  air  has  been  wholly  or  in  part  extracted 
—  such  as  the  bell  of  an  air-pump  —  the  internal  pressure  may 
overpower  the  external,  and  the  body  will  then  tend  to  become 
inflated  and  may  even  burst. 

*  In  such  a  case  some  of  the  air  in  that  cavity  can  be  expelled  by  an  expiratory 
effort  with  closed  glottis,  and  can  be  prevented  from  returning  by 'a  valve  opening 
outwards. 


338  OF  GASES.  [CHAP. 

A  little  indiarubber  balloon,  a  bladder  half  filled  with  air,  a  shrivelled 
apple,  a  dish  of  soapsuds,  present  under  the  air-pump  a  singular  appearance 
of  expansion.  If  a  loaded  piece  of  wood  be  put  in  a  dish  of  water,  and  the 
whole  placed  under  the  air-pump,  the  wood  will  appear  to  effervesce ;  the 
air  contained  in  its  pores  expands  and  forms  bubbles.  If  soda-water  already 
flat  be  subjected  to  similar  treatment  it  will  renew  its  effervescence. 

This  inflation  is  not  due  to  any  suction  on  the  part  of  the 
air-pump,  but  is  clue  to  the  expansion  of  the  contained  gas, 
which  always  tends  to  expand,  but  which  can  only  do  so  when 
the  resistance  offered  to  its  expansion  on  the  part  of  the  external 
pressure  is  diminished  or  removed.  The  gas  expands  until  the 
internal  pressure  of  the  expanded  gas  is  equal  to  the  pressure 
of  the  rarefied  air  or  gas ;  the  latter,  as  we  have  already  seen 
(Boyle's  Law),  suffers  diminution  in  the  same  ratio  as  the  density. 

If  the  pressure  within  an  object  or  a  cavity  exceed  or  be 
made  to  exceed  the  external  atmospheric  pressure,  there  is,  as  in 
all  such  cases,  a  tendency  to  establish  equilibrium  by  setting  up 
a  flow  from  the  place  of  greater  pressure  to  one  of  less.  Thus, 
if  a  bladder  containing  gas  and  provided  with  a  stopcock  be 
loaded  with  a  weight,  and  its  stopcock  opened,  the  atmospheric 
pressure  tends  to  drive  air  into  the  bladder,  but  it  is  over- 
powered by  the  greater  pressure  within  the  bladder,  and  there 
is  an  outward  flow  set  up,  due  to  the  difference  between  the 
internal  and  the  external  pressure. 

A  gasholder,  consisting  of  an  inverted  bell  floating  on  water,  may  be 
loaded  so  as  to  exercise  any  given  expulsive  pressure.  Thus  coal-gas  driven 
out  "  at  a  pressure  of  1  inch  of  water  "  is  subject  in  the  pipe,  when  the  stop- 
cock is  closed,  to  an  internal  pressure  =  atm.  pr.  +  "  1  inch  of  water,"  and 
to  an  exterior  pressure  at  the  burners  =  atm.  pr.  only. 

If  air  be  blown  into  a  flask  partly  filled  with  water,  partly  with  air,  and 
provided  with  a  narrow  open  glass  tube  passed  through  the  cork,  and  if  the 
flask  be  suddenly  inverted,  water  will  rush  out  through  the  nozzle  :  the  air 
has  been  compressed,  and  its  pressure  has  become  greater  than  the  atm. 
pr. ;  this  difference  of  pressures  causes  an  outward  flow,  a  jet  of  liquid. 

In  the  dome  of  the  fire-engine  air  is  compressed  in  the  same  way :  the 
inflow  is  intermittent,  the  outflow  continuous ;  for  the  air  never  ceases  to  be 
compressed,  and  it  exercises  a  continuous  pressure. 

If  a  gas-evolution  flask  containing,  say,  zinc  and  sulphuric  acid,  be 
fitted  with  an  ordinary  safety-funnel  dipping  into  the  liquid,  the  hydrogen 
evolved  will  pass  out  by  the  intended  channel :  the  liquid  of  the  flask  will 
be  observed  to  oscillate  a  little  in  the  safety  tube,  which  acts  as  a  manom- 
eter indicating  the  internal  pressure.  If  any  obstruction  offer,  the  gas 
accumulates  in  the  flask,  a  difference  is  set  up  between  the  internal  and  the 
external  pressure,  and  the  liquid  is  forced  up  the  safety  tube.  The  safety 
tube  should  dip  into  the  liquid  only  just  so  deeply  that  before  the  liquid 
forced  up  into  the  funnel  can  overflow,  the  level  of  the  liquid  in  the  flask 


xii.]  ATMOSPHERIC   PRESSURE.  339 

shall  have  been  so  far  depressed  that  nothing  but  gas  can  pass  out  through 
the  safety  tube. 

If  a  cistern  at  a  height  be  connected  by  a  tube  with  a  large  flask  con- 
taining air,  in  such  a  way  that  liquid  may  pass  from  the  cistern  into  the 
flask,  air  is  driven  out  of  the  flask :  it  may  be  driven  out  through  a  tube ; 
this  tube  may  be  connected  with  any  cavity  through  which  it  may  be  neces- 
sary to  drive  air.  This  is  one  form  of  Aspirator. 

The  same  principle  is  applied  in  the  plenum  method  of  ventilation: 
a  local  excess  of  pressure  is  set  up  by  forcing  air  into  a  building,  and  the 
air  is  left  to  find  its  own  way  out. 

When  the  thoracic  walls  contract,  air  is  driven  out  of  the  lungs,  and 
blood  out  of  the  thoracic  organs  in  general. 

When  the  abdominal  walls  contract,  a  general-contents-pressure  is  set 
up,  always  at  right  angles  to  the  general  surface  of  the  practically-fluid 
visceral  mass,  and  opposed  partially  or  completely  by  a  uniform  atmospheric 
pressure. 

When  the  external  atmospheric  pressure  exceeds  that  within 
an  object  or  cavity,  air  may  be  forced  into  it  or  it  may  be  com- 
pressed, or  if  these  effects  be  not  possible,  the  existence  of  the 
atmospheric  pressure  generally  becomes  in  some  way  strikingly 
manifest. 

The  Magdeburg  Hemispheres,  a  couple  of  hemispheres  fitting 
together  so  as  to  form  a  sphere,  and  ordinarily  separable  writh  ease,  but  when 
apposed,  and  the  air  extracted  from  between  them,  not  to  be  separated  with- 
out great  force;  the  boy's  leather  Sucker,  a  piece  of  moistened  leather 
closely  applied  to  any  object  and  pulled — :any  residual  air  still  remaining 
being  rarefied  —  the  pressure  of  air  between  the  sucker  and  the  object 
becoming  very  small,  and  the  sucker  being  thus  firmly  pressed  by  the  weight 
of  the  atmosphere  *  against  the  object  on  which  it  is  placed ;  the  difficulty 
experienced  in  pulling  the  handle  of  a  good  Syringe  when  the  nozzle  is 
stopped  up,  or  in  the  continued  working  of  a  reciprocating  Air-pump,  —  all 
these  clearly  point  out  the  part  played  by  atmospheric  pressure. 

In  the  experiment  previously  described,  in  which  gas  escaped  from  the 
pores  in  a  piece  of  wood  kept  under  water  and  exposed  to  the  action  of  the 
air-pump,  it  is  only  necessary  to  allow  the  atmospheric  pressure  again  to  act 
to  see  the  water  driven  by  that  pressure  into  the  pores  of  the  wood,  which 
thus  becomes  too  heavy  to  float. 

The  atmospheric  pressure  is  a  prime  agent  in  most  of  what 
we  usually  call  the  phenomena  of  Suction.  A  syringe  has  its 
nozzle  inserted  in  water  ;  the  handle  is  drawn  up  :  in  the  body  of 
the  syringe  there  would  arise  a  partial  vacuum  were  it  not  that 
the  external  atmospheric  pressure  overcomes  the  feeble 
internal  pressure,  and  pushes  the  liquid  through  the  nozzle 
into  the  body  of  the  instrument. 

s 

*  The  air  does  not  force  its  way  between  the  sucker  and  the  object  pulled  upon, 
for  the  intervening  film  of  moisture  is  held  in  place  by  adhesion. 


340 


OF   GASES. 


[CHAP. 


If  the  syringe  have  a  thin  closed  wooden  nozzle,  and  if  the  vacuum  in 
the  syringe  be  relatively  good,  the  atmospheric  pressure  can  force  water  or 
mercury  through  the  pores  of  the  wood. 

If  there  be  no  safety  tube  attached  to  a  gas-evolution  apparatus,  and 
if  the  evolution  of  gas  suddenly  cease  while  the  gas  still  continues  to  be 
absorbed  by  the  liquid  into  which  it  is  passed,  we  find  the  gas  diminishing 
in  amount,  and  the  external  atmospheric  pressure  forcing  the  absorbing 
liquid  back  into  the  gas-generating  flask.  If  there  be  a  safety  tube,  the 
very  short  column  of  liquid  at  its  lower  end  is  forced  down,  and  air  enters 
the  flask  until  the  internal  pressure  becomes  equal  to  the  external. 

Aspirators  are  generally  constructed  on  this  principle.  Water  flows 
from  a  large  flask  or  can,  Fig.  114  a:  air  must  take  its  place:  this  air  "is 
drawn,"  or  rather  is  pushed  by  the  atmospheric  pressure,  through  a  series  of 
flasks  which  it  must  traverse  on  its  way  from  the  outer  air  to  its  place  in 
the  aspirating  flask.  With  the  arrangement  b  of  Fig.  114,  the  flexible  tube 
between  A  and  B,  being  filled  with  water,  acts  as  a  siphon,  and  water  flows 


Fig.H4. 


out  of  A :  when  A  is  nearly  emptied,  disconnect  it  from  d,  and  place  it  at  a 
lower  level  than  B  ;  it  then  becomes  refilled. 

The  vacuum  method  of  ventilation  is  an  exhaust-method:  air  is 
removed  at  a  certain  point,  by  the  mechanical  action  of  a  fan  or  by  the 
ascent  of  heated  air  in  a  tall  chimney  or  shaft ;  air  then  finds  its  way  from 
different  parts  of  the  building  or  mine  towards  this  point. 

Filtration  may  be  assisted  by  connecting  the  filter  with  a  partial 
vacuum :  the  funnel  is  for  this  purpose  fixed  into  a  flask  by  a  cork  through 
which  there  also  passes  a  tube  leading  to  an  aspirator  of  any  kind,  a  Sprengel 
pump  worked  by  water,  and  called  a  Bunsen  pump,  being  frequently 
employed.  The  atmospheric  pressure  on  the  liquid  in  the  funnel  forces  it 
through  the  filter  into  the  partial  vacuum. 

Suction  nipples  and  bleeding  cups  illustrate  not  suction  but 
atmospheric  pressure :  the  pressure  within  them  is  less  than  the  external 
pressure ;  the  part  of  the  surface  of  the  body  exposed  to  their  action  suffers 
less  pressure  than  the  contiguous  parts  of  the  skin,  which  are  acted  upon  by 
the  full  atmospheric  pressure.  The  result  is  as  if  all  parts  of  the  surface 
except  the  area  operated  on  were  subjected  to  a  powerful  squeeze  :  the  fluids 
are  squeezed  by  the  atmosphere  towards  the  area  subjected  to  least  pressure. 

When  the  thoracic  walls  expand,  their  soft  parts  are  driven  inwards,  air 
is  driven  into  the  lungs,  and  blood  is  driven  into  the  thorax  from  the  parts 
of  the  body  acted  upon  by  the  full  atmospheric  pressure  ;  all  this  being  the 
consequence  of  the  so-called  negative  pressure  (i.e.,  pressure  less  than 
that  of  the  atmosphere)  in  the  thorax.  The  lungs  act  like  a  sphygmoscope 
(Fig.  105,  S)  :  they  are  dilated  by  internal  pressure  until  their  resistance  to 


XII.] 


ATMOSPHERIC   PRESSURE. 


341 


further  dilatation  is  equal  to  the  dilating  force.  The  less  extensible  they  are 
or  become,  the  sooner  will  this  limit  be  reached  :  if  their  extensibility  become 
so  small  that  the  limit  of  expansion  would,  if  the  ribs  expanded  to  their  full 
extent,  be  reached  before  the  pleural  cavity  is  filled,  then  the  blood  and  the 
thoracic  walls  themselves  are  pressed  inwards  and  the  chest-walls  lose  the 
habit  and  the  power  of  expansion .  If  while  the  chest  is  expanding,  there  be 
an  orifice  open  in  a  large  vein,  the  diminution  of  thoracic  pressure  allows  the 
atmospheric  pressure  not  only  to  drive  venous  blood  towards  the  heart,  but 
also  to  force  air  into  the  open  vein,  and  thus  into  the  circulation. 

If  a  test  tube  be  inserted  in  a  larger  test  tube  containing  water,  it  will 
float.  If  the  whole  be  inverted,  surface-tension  may  for  some  time  prevent 
the  escape  of  water ;  but  if  any  water  do  escape,  the  atmospheric  pressure 
pushes  the  smaller  tube  up  into  the  larger  one,  and  thus  causes  it  to  appear 
to  be  sucked  up. 

After  an  extreme  contraction  of  the  abdominal  muscles,  there  is  elastic 
restitution  of  position  of  the  abdominal  walls,  and  the  intra-abdominal  pres- 
sure sinks.  Apparent  suction  is  thus  exercised  on  the  pelvic  diaphragm. 

When  in  a  joint  the  bones  are  separated  by  extreme  flexion  or  extreme 
extension,  the  tendency  to  form  a  vacuum  between  them  permits  the  atmos- 
pheric pressure  to  press  skin  and  tissue  between  the  bones,  and  thus  to  form 
an  external  dimple. 

Columns  of  liquid  supported  by  the  atmospheric  pres- 
sure. —  If  a  vessel  filled  with  liquid  be  inverted  with  its  mouth 
beneath  the  surface  of  liquid  standing  in  a  larger  vessel,  we 
see  —  provided  that  the  inverted  vessel  do  not  exceed  a  cer- 
tain height,  about  33  feet  in  the  case  of  water,  about  30  inches 
in  that  of  mercury  —  that  the  liquid  does  not  fall  out  of  the 
inverted  vessel,  but  remains  in  position,  supported  by  the 
atmospheric  pressure.  If  in  Fig.  115  the  inverted  vessel  have  a 
mouth  whose  area  is  A,  and  if  the  height  of  the  column  of  fluid 
supported  be  CB,  while  that  of  the  whole  column  of  liquid  above 
the  orifice  is  AB ;  and  if  the  density  of  the  liquid  be  ^,  —  the 
whole  pressure  tending  to  drive  fluid  out  through  the  orifice  A 
is  Ampg  x  AB.  Opposed  to  this  we 
have  two  pressures:  —  (1)  the  at- 
mospheric  pressure  acting  through 
the  fluid,  equal  to  n  dynes  per 
unit  of  surf  ace,  and  therefore  equal 
to  A*n  over  the  mouth  of  the  ves- 
sel ;  and  (2)  the  water  pressure 
on  that  orifice  at  the  depth  AC  — 
that  is,  A* pg  x  AC.  The  whole 
pressure  tending  to  drive  water 
up  into  the  vessel  is  thus  A-n-f- 
(A-p#  x  AC).  Since  there  is  equilibrium  when  CB  has  the 
greatest  possible  height  —  equilibrium  brought  about  without 


342  OF   GASES.  [CHAP. 

bringing  into  play  the  elasticity  or  rigidity  of  the  upper  part  of 
the  vessel  —  we  can  find  the  greatest  free  height  CB  by  the 
equation  — 

A  •  pg  x  AB  =  A  •  II  +  A  •  pg  •  CA. 
A-^-BC^A-n. 

BC  =  u/pff  =  H. 

If  the  vessel  be  of  exactly  such  a  height,  or  be  immersed  just 
so  deeply,  that  its  own  free  height  BC  is  such  as  to  enable  it  to 
contain  a  column  of  the  "  barometric  "  height  H=  n/pg,  it  will 
be  exactly  filled. 

If  BC  =  ZT,  the  free  height  of  the  vessel,  exceed  K/pg,  it 
is  not  possible  that  the  column  of  liquid  supported  should 
extend  to  the  upper  limit  of  the  vessel ;  for  if  it  did,  the  weight 
of  that  column  would  exceed  the  atmospheric  pressure  which 
supports  it  against  gravity  —  an  evident  impossibility.  Hence 
the  column  actually  supported  cannot  have  a  height  greater 
than  u/pg,  and  the  space  between  the  top  of  the  column  of 
liquid  and  the  upper  limit  of  the  vessel  is  a  vacuum,  the  Torri- 
cellian vacuum. 

Thus,  if  the  free  internal  height  of  a  vessel  be  equal  to 
n/pg  or  greater  than  it,  the  height  of  the  liquid  column  sap- 
ported  against  gravity  by  the  atmospheric  pressure  can  never 
exceed  R/pg-,  but  will  be  equal  to  it,  whether  there  be  above  it 
a  vacuum  or  not,  and  whatever  be  the  size  of  that  vacuum. 

The  Barometer  is  in  its  simplest  form  a  tube  filled  with  liquid  and 
inverted  into  a  cistern.  If  the  tube  have  a  free  length  CB  greater  than 
n /pg  the  liquid  will  stand  in  it  at  a  free  height  H  equal  to  n/pg.  Thus,  if 
the  atmospheric  pressure  be  1,013663-376  dynes  per  sq.  cm.,  and  if  the  liquid 
employed  be  water  (p  =  1),  the  free  height  of  the  column  will  be  H  =  u/pg 
=  (1,013663-376  +  981)  =  1033-296  cm. ;  while,  if  the  liquid  employed  be  mer- 
cury (p  =  13-596),  the  free  height  of  the  barometric  column  will  be  n/pg 
=  (1,013663-376  -  (13-596  x  981)}  =  76  cm.  Hence  a  mercury  barometer  is 
much  more  convenient  an  instrument  than  a  water 
Fig. H6.  barometer,  for  the  height  of  the  column  in  the 

latter  is  over  33  feet. 

If  the  tube  be  tilted  obliquely,  its  lower  end 
being  kept  immersed,  the  liquid  will  move  upwards 
in  the  tube:  the  vertical  height  remains  unal- 
tered (Fig.  116). 

A  common  water-pump  'cannot  act  if  it  be  so 
deep  that  during  its  action  the  atmospheric  pres- 
sure would  have  to  support  a  greater  column  than 
one  of  about  33  feet :  a  vacuum  might  be  produced 
at  the  top  of  the  cylinder  of  the  pump,  and  yet  no  column  whose  height 
exceeded  H  =  u/pg  could  possibly  ascend  in  it.  The  Torricellian  vacuum 


xii.]  ATMOSPHERIC   PRESSURE.  343 

is  utilised  in  the  so-called  mercury  air-pump.  A  flask  is  filled  with  mer- 
cury :  this  flask  is  connected  with  a  flexible  tube  also  filled  with  mercury : 
this  mercury  is  continuous  with  that  in  a  cistern  into  which  the  flexible 
tube  dips.  The  flask  may  be  raised  to  a  certain  height  without  the  mercury 
leaving  it,  but  if  it  be  raised  so  high  that  the  upper  limit  of  its  cavity  comes 
to  an  elevation  greater  than  n/p^r  above  the  surface  of  the  mercury  in  the 
cistern,  a  Torricellian  vacuum  is  formed  by  some  of  the  mercury  leaving  the 
flask.  The  vacuum  may  be  laterally  connected  with  flasks  filled  with  liquids, 
the  gases  contained  in  which  are  to  be  extracted  for  analysis.  When  the 
flask  is  raised  and  a  vacuum  formed  in  it,  the  liquids  in  the  lateral  flasks 
effervesce  and  the  gases  previously  dissolved  in  them  ascend  into  the  mer- 
cury flask,  which  may  be  disconnected  and  removed  for  further  research. 

When  the  free  height  of  the  vessel  is  less  than  the  baromet- 
ric height  u/pg  =  H,  the  column  of  liquid  fills  the  vessel. 

If  a  card  be  laid  across  the  mouth  of  a  tumbler  completely  filled  with 
water,  the  whole  can  be  inverted ;  the  card  will  not  drop  off,  and  the  water 
will  not  drop  out  of  the  tumbler :  atmospheric  pressure  keeps  the  whole  in 
place.  It  is  important  to  observe  that  there  is  no  tendency  for  the  card  to 
become  bulged  in  any  sense. 

A  pipette  completely  filled  with  liquid  and  closed  by  the  thumb  will 
not  allow  the  contained  liquid  to  escape,  unless  the  lower  orifice  be  so  oblique 
or  irregular  as  to  permit  successive  portions  of  liquid  to  trickle  away.  If  it 
be  partly  filled  and  closed  by  the  thumb,  the  pressure  of  air  in  the  upper 
part  would  neutralise  the  effect  of  the  external  atmospheric  pressure,  and 
the  liquid  would  be  free  to  fall  were  it  not  in  the  first  place  for  the  surface- 
tension  at  the  lower  orifice,  which,  if  the  orifice  be  very  small,  may  be  able 
to  support  a  considerable  column  of  liquid,  and  in  the  second  for  the  rare- 
faction which  is  set  up  by  the  escape  of  some  drops  of  liquid. 

A  gas-holder  may  contain  a  certain  quantity  of  gas  above  and  of  water 
below,  and  even  though  an  orifice  be  made  in  the  walls  of  the  vessel  below 
the  level  of  the  water  —  provided  that  Fig. 117. 

it  be  not  too  large  —  none  of  the  gas 
will  escape,  for  the  atmospheric  pressure 
keeps  the  whole  in  place. 

It  is  often  of  importance  to  keep 
water  in  a  cistern  at  a  constant  level. 
The  arrangement  shown  in  Fig.  117 
enables  this  to  be  done.  The  instant 
that  the  level  of  the  liquid  passes  below 
that  of  the  orifices  of  the  nozzles  of  the 
flasks  A,  B,  C,  air  enters  these  flasks,  and 
water  passes  into  the  cistern.  The 
aggregate  delivering  power  of  the  flasks  must  not  be  less  than  that  of  the 
cistern  itself. 

When  a  column  is  supported  by  the  atmospheric  pressure,  its 
own  lateral  pressure  differs  at  different  altitudes.  This  is  illus- 
trated by  the  indications  of  the  lateral  manometers  of  Fig.  118. 

If  the  walls  of  the  tube  in  which  such  a  column  is  supported  be  rigid, 
these  walls  will,  on  account  of  differences  between  the  internal  pressures 


344 


OF   GASES. 


[CHAP. 


and  the  external  atmospheric  pressure,  be  subjected  to  stress ::  this  stress 
varies  from  point  to  point  according  to  the  altitude. 

Fi£r  118  If  some  parts  of  the  walls  be  flex- 

ible, mercury  will  leave  the  column, 
and  the  tube  will  yield  laterally  as  in 
Fig.  119 ;  this  it  will  do  until  the 
resistance  to  further  distortion  offered 
by  the  walls  is  equal  to  that  difference 
of  pressure  which  tends  to  produce  it. 
If  the  column  be  not  barometric 
but  closed,  and  if  in  the  same  way  the 
containing  vessel  have  local  flexibili- 
ties, the  upper  flexible  parts  of  it  will 
yield  inwards,  the  lower  will  bulge 
outwards  ;  in  each  case  equilibrium  is 
established  between  the  internal  pres- 
sure, the  atmospheric  pressure,  and 
the  elasticity  of  the  walls.  If  the 
whole  walls  be  flexible,  the  whole 
mass  becomes  pyriform  ;  here  the 
atmospheric  pressure  produces  no  spe- 
cial effect  in  the  determination  of  form,  for  it  is  equably 
yielded  to. 

If  the  upper  part  of  the  walls  be  rigid  while  the  lower  are 
flexible,  the  lower  part  will  bulge,  but  the  upper  will  be  com- 
pletely filled,  provided  that  the  whole  column  have  a  height 
not  greater  than  u/pg ;  if  the  height  be  greater,  there  will  be 
a  Torricellian  vacuum  produced.  If  the  upper  rigid  part  of 
such  a  vessel  become  flexible  in  whole  or  in  part,  it  will  col- 
lapse to  some  extent,  and  fluid  will  pass  into  the  lower  part 
of  the  column.  The  amount  of  collapse  of  the  upper  part 
depends  on  its  extensibility  :  equilibrium  will  be  established 
when  its  restitution-pressure  is  locally  equal  to  the  difference  between  the 
internal  and  external  pressures. 

Suspended  Loops,  —  A  suspended  loop  is  a  double  closed  column,  and 
it  presents  variations  in  pressure  and  in  distension  similar  to  those  of  a  sin- 
gle column.  The  pressure  at  any  altitude  is  determined  by  the  relative 
height  and  the  values  of  p  and  g:  the  amount  of  distension  at  any  altitude 
accommodates  itself  to  the  pressure.  A  loop  more  than  u/pg  cm.  deep 
must  either  present  a  vacuum  or  else  collapse  at  its  upper  part.  If  the 
ascending  part  of  the  loop  be  more  distensible  than  the  descending,  or  vice 
versa,  the  amount  of  distension  will  be  different  in  the  two  parts  of  the  loop, 
but  statically  the  pressures  at  equal  altitudes  in  the  two  parts  of  the  tube 
will  be  equal.  If  an  additional  quantity  of  fluid  be  forced  into  the  loop,  it 
will  settle  down  in  greater  quantity  in  the  more  extensible  parts  of  it.  If  a 
constant  flow  of  liquid  be  maintained  in  the  loop,  the  more  distensible  part 
will  contain  more  liquid,  but  (when  once  the  relative  quantity  of  fluid  in  the 
two  parts  of  the  loop  has  been  adjusted)  the  rate  of  passage  will  not  be 
affected  by  gravity.  If  an  intermittent  circulation  be  kept  up  in  such  a 
suspended  loop,  each  successive  increment  of  fluid  is  delayed  in  the  more 
extensible  part  according  to  the  relative  degrees  of  distensibility ;  but 
gravity  has  no  direct  effect  on  the  mean  velocity  of  the  stream. 


XII.] 


ATMOSPHERIC   PRESSURE. 


345 


Fi,,  120 


A  Siphon,  such  as  is  shown  in  Fig.  120,  is  an  inverted 
loop.  If  it  be  more  than  n/pg  in  free  height,  a  Torricellian 
vacuum  is  formed  in  its  upper  part.  The  maintenance  of  col- 
umns the  heights  of  which  are 
less  than  TL/pg  depends  on  atmos- 
pheric pressure ;  and  thus  a 
siphon  will  not  act  at  all  under 
the  air-pump.  In  Fig.  120  the 
tendency  of  the  column  AB  to 
fall  out  of  the  siphon  is  equal  to 
that  of  the  column  CD  to  fall 
towards  G ;  but  the  tendency  of 
the  column  EF  to  fall  towards  G 
is  uncompensated.  The  whole1 
mass  of  liquid  filling  the  siphon  at  any  moment  is  set  in  motion 
by  the  weight  of  the  liquid  column  whose  vertical  height  is  EF, 
and  its  cohesion  makes  it  move  as  a  whole. 

Woven  tissue  or  a  skein  of  thread  may  act  as  a  siphon,  as  in  the  drain- 
ing of  a  water  basin  by  a  towel,  one  end  of  which  is  left  in  the  water,  the 
other  hanging  over :  the  fibres  may  become  wetted  by  imbibition,  and  once 
wetted  they  allow  the  liquid  to  pass  over  in  tubes  whose  walls  in  part  con- 
sist of  the  fibres,  and  in  part  of  the  superficial  film  of  the  liquid  itself.  This 
siphon-action  is  impossible  under  the  air-pump. 

The  Common  Pump  (Fig.  121). —  By  an  upward  stroke  of 
the  piston  the  air  in  the  cylinder  AB  is  expanded  and  rarefied. 
Fig.isi.  The  atmospheric  pressure  drives  up  a  column  of 

liquid  along  DC.  The  piston  is  driven  downwards, 
or  else  descends  by  its  own  weight ;  the  valves  now 
permit  a  certain  quantity  of  air  to  escape  to  the 
upper  side  of  the  piston,  but  permit  none  to  return 
to  the  column  CD.  At  the  next  stroke  the  air  in 
AB  and  CD  is  again  rarefied,  and  more  water  rises 
in  DC.  This  is  repeated  until  the  water  rises  into 
the  cylinder  AB,  which  it  will  do  provided  that  the 
column  CD  be  somewhat  less  than  n/pg  in  height. 
The  piston  then  scoops  up  the  water  in  the  lower 
part  of  the  cylinder,  always  allowing  it  to  pass  to  its  upper  sur- 
face, but  never  to  return,  and  thus  at  each  upward  stroke  of  the 
pump  water  is  lifted  up  and  falls  out  at  E. 

By  the  force-pump  water  may  be  raised  to  very  great 
heights.  Fig.  122  shows  the  arrangement  of  the  valves.  The 
piston  is  solid,  and  when  it  is  pressed  down  the  valve  E  is  closed, 


346 


OF   GASES. 


[CHAP. 


"i : 


while  air  or  watei'  is  forced  through  the  valve  F  against  the 
pressure  of  air  or  water  in  the  tube  G,  which  tends  to  close  that 

valve.  In  the  Fire-engine  there  may 
be  one  or  two  such  force-pumps,  which 
drive  water  into  the  dome. 

Digression  on  Valves.  —  There  are 
three  principal  types  of  valves  in  use. 
Of  these  the  first  is  the  ordinary  and 
very  familiar  clapper  valve. 

The  second  is  the  conical  valve 
shown  in  Fig.  128.  The  pressure  of 
the  fluid  in  A  may  displace  the  valve :  a 
spring  returns  it  to  its  place  when  the 
relative  pressure  in  A  has  become  suffi- 
ciently diminished  to  permit  it  to  do  so. 
The  third  kind  is  that  shown  in  section  on  Fig.  124.  The 
piston  AB  is  furnished  with  a  cap  of  indiarubber,  which  is 
slightly  smaller  than  the  tube  in  which  the  piston  moves.  In 
the  direction  A  to  B  the  piston  can  be  freely  moved  through  the 
liquid;  but  if  the  piston  be  moved  in  the  contrary  direction, 
the  indiarubber  cap  flies  open,  and  it  exactly  and  equably  fits 


**B 

== 

—  : 

=. 

Fig.  123. 


Tig.  124. 


the  tube  so  that  no  water  can  pass  it.  A  pouch  is  formed :  the 
greater  the  pressure  within  the  pouch,  the  closer  the  apposition 
between  the  indiarubber  and  the  walls  of  the  tube,  and  the 
better  the  action  of  the  valve. 

A  somewhat  similar  form  of  valve  is  found  in  the  heart.  The  semilunar 
valves  (pulmonary  and  aortic)  consist  of  pouches  attached  to  the  walls  of  the 
vessel ;  they  lie  loosely  against  the  walls  and  allow  the  liquid  to  flow  past 
them  as  it  issues  from  the  heart;  but  when  a  backward  impulse  is  given  to 
the  blood,  or  the  valves  are  pushed  forward  against  the  blood,  they  are  caught 
by  the  liquid,  the  pouches  are  distended,  they  touch  one  another  and  com- 
pletely block  up  the  lumen  of  the  tube. 

The  other  valves  of  the  heart  are  clapper  valves,  attached  to  the  walls 
of  the  cavity  of  the  heart,  two  or  three  in  each  situation,  together  attached  to 
a  complete  circumference,  acting  together,  slightly  overlapping  one  another, 
and  completely  closing  the  lumen  of  the  tube,  and  provided  with  tendinous 
and  muscular  arrangements  which  prevent  their  being  driven  too  far  towards 
the  auricle  when  they  are  impelled  backwards  by  a  predominant  ventricular 
pressure. 


xii.]  ATMOSPHERIC   PRESSURE.  347 

Measurement  of  Atmospheric  Pressure.  —  The  atmos- 
pheric pressure  n  per  unit  of  surface  may  be  easily  calculated  if 
H,  the  height  of  the  barometric  column,  be  known,  for  n  =  Hpg. 
The  habit  of  stating  the  atmospheric  pressure  in  terms  of  the 
barometric  height,  —  as  thus,  "a  pressure  of  30  inches  of  mer- 
cury,"-- is  general,  and  if  clearly  understood  is  unobjectionable. 

The  height  of  the  barometric  column  of  mercury  is  subject 
to  corrections  for  capillarity  and  for  temperature ;  the  latter 
involve  the  consideration  of  the  less  density  of  warm  mercury, 
and  of  the  expansion  of  the  glass  of  the  tube,  which  expansion 
involves  an  alteration  in  the  correction  for  capillarity. 

The  aneroid  barometer  is  essentially  a  hollow  box  of 
elastic  metal  in  which  there  is  rarefied  air.  Any  given  amount 
of  external  pressure  produces  a  corresponding  amount  of  com- 
pression of  this  box ;  a  multiplying  arrangement  causes  a  lever 
to  indicate,  by  its  position  in  reference  to  the  face  of  the  dial, 
the  amount  of  this  compression.  Careful  preliminary  gradua- 
tion enables  the  absolute  amount  of  external  pressure  corre- 
sponding to  each  indication  of  the  instrument  to  be  recorded. 

The  pressure  n  =  Hpg,  =  (say,  when  H  is  found  to  be  76  cm. 
Hg)  1,013663  dynes  per  sq.  cm.,  is  the  same  pressure  as  would 
be  exerted  by  a  uniform  atmosphere  throughout  which  g 
was  uniform,  whose  uniform  density  was  0-0012932,  and  its  uni- 
form height  H=  n/pg  =  1,013663  -=-(0-0012932  x  981)  =  799,022 
cm.,  or  7990-2  metres.  If  a  barometer  on  the  floor  stand  at  76 
cm.,  the  same  barometer  raised  to  the  height  of  1  metre  should 
stand  at  a  height  of  76  cm.,  less  '095  mm.,  a  perceptible  dimi- 
nution. 

The  pressure  does  not  diminish  regularly  with  the  height, 
as  it  would  in  an  ocean  of  incompressible  fluid.  The  lower 
strata  of  the  air  are  compressed,  and  therefore,  to  set  up  a  given 
difference  of  pressure,  a  shorter  vertical  ascent  among  them  is 
sufficient  than  is  necessary  among  the  higher  strata. 

Each  stratum  differs  from  the  one  below  it  in  two  respects  : 
—  (1)  it  has  fewer  strata  above  it ;  (2)  it  is  therefore  less  com- 
pressed, and  for  equal  mass  has  greater  volume.  If  we  imagine 
the  whole  atmosphere  to  be  divided  into  7990-2  strata,  the  lowest 
of  them  all,  which  bears  the  weight  of  7989-2  strata,  will  be  1 
metre  thick ;  the  next,  which  bears  the  superincumbent  weight 

of  7988-82  strata,  will  have  a  thickness  of  1  metre  xx798^'2 ;  the 

''7988*2 

next  stratum  will  have  a  thickness  greater  than  this  in  the  ratio 


348  OF  GASES.  [CHAP. 

7988-2  .     .„  ,  /-   7989-2\   7988-2  7989-2 

;  and  ^ 


7QQQ.9 

nth  layer  will  be  ""  —  metres  thick. 

Altitudes  as  indicated  by  the  Barometer.  —  If  h  be  the  vertical 
height  between  two  stations,  H  the  height  of  the  mercury-barometer  at 
the  lower  station  observed  at  temperature  t°  C.,  and  H'  the  height  of  the 
barometer  at  the  higher  station  at  the  temperature  t  ',  A  being  the  latitude  ; 
then 

x  =  18393  •  (1  +  -002837  cos  X)  -log  jLfl  -f  2 


ti      \  1000  / 

(Laplace's  Formula.) 

Variations  in  the  barometric  pressure  occur  from  moment 
to  moment  as  the  atmospheric  ocean  is  disturbed  by  currents, 
driven  in  whirlpools,  varied  in  thickness  by  superficial  waves,  or 
locally  varied  in  its  superincumbent  mass  by  expansion  (due  to 
heat)  and  lateral  overflow.  When  any  spot  has  a  low  pressure, 
there  is  a  tendency  for  the  surrounding  air  to  rush  in  from  all 
sides  towards  that  spot,  the  centre  of  depression;  the  greater 
the  difference  of  pressure  between  two  places  —  i.e.,  the  steeper 
the  barometric  gradient  —  the  greater  will  be  the  tendency  to 
an  inflow  of  air  towards  the  centre  of  depression.  This  tendency 
is  so  modified  by  the  rotation  of  the  earth  from  west  to  east  (in 
a  direction  opposed  to  the  apparent  movement  of  the  sun)  that 
the  flow  does  not  take  place  directly  towards  the  centre,  but 
round  it  in  a  circular  storm  or  cyclone,  whose  direction  is 
in  the  northern  hemisphere  opposed  to,  in  the  southern  the  same 
as,  that  of  the  hands  of  a  watch  (Dove's  Law  of  Storms).  The 
wind  whirls  round  the  centre  and  also  towards  it  ;  air  ascends 
in  the  centre  :  it  expands  and  becomes  cooled  ;  moisture  con- 
denses ;  rain  falls.  "  Put  your  back  to  the  wind,  and  the  barom- 
eter is  lower  towards  your  left  hand  (in  the  northern  hemi- 
sphere)." —  (Buys-Ballot.) 

Correction  for  pressure.  —  Variations  in  the  barometric 
pressure  render  it  necessary  in  measuring  quantities  of  gas  by 
volume  to  make  a  correction  for  pressure,  and  to  reduce  the  gas 
to  standard  pressure  —  i.e.,  to  state  what  the  volume  would 
have  been  had  the  atmospheric  pressure  at  the  time  of  measure- 
ment been  76  cm.  of  mercury.  Boyle's  law  teaches  us  that  the 
volume  varies  inversely  as  the  pressure.  If  therefore  the  pres- 
sure on  gas  measured  as  x  cub.  cm.  at  76-1  cm.  had  been,  not 
76-1  but  76-0  cm.,  the  volume  of  that  gas  would  have  been 
greater  under  the  less  pressure  in  the  ratio  of  76-1  to  76-0. 


xii.]  ATMOSPHERIC   PRESSURE.  349 

The  general  rule  is,  that  a  volume  of  gas  measured  at  a  pres- 
sure Hem.  of  mercury  must  be  multiplied  by  H/1Q  in  order  to 
reduce  it  to  the  standard  atmospheric  pressure. 

Standard  Atmospheric  Pressure. — In  many  modern  books,  instead 
of  a  pressure  of  76  cm.  mercury,  or  1033-296  cm.  water,  or  1,013663-376 
dynes  per  sq.  cm.,  the  standard  atmospheric  pressure  is  taken  as  1,000000 
dynes,  or  one  megadyiie,  per  sq.  cm.  This  is  75  cm.  Hg  at  l°-8  C.,  or  1000 
cm.  of  3  %  KC1  solution  at  8°  C.  (density  =  1-0937). 

Gases  passed  into  the  Torricellian  Vacuum.  — If  a  bubble 
of  gas  be  passed  into  the  Torricellian  vacuum,  it  will  expand  so 
as  to  fill  it;  further,  it  will  exert  pressure  011  the  top  of  the 
column  of  mercury  —  it  will  therefore  depress  that  column ;  the 
extent  to  which  it  depresses  the  column  measures  the  pressure 
which  it  exerts  upon  the  mercury :  conversely,  that  depression 
measures  the  pressure  of  the  mercury  upon  it,  and  therefore 
indicates  the  pressure  under  which  it  itself  assumes  its  actual 
volume. 

Let  a  barometer  tube  whose  cross  area  is  £  sq.  in.,  and  whose  free  inter- 
nal height  is  34  inches,  have  standing  in  it  a  column  of  30  inches  of  mer- 
cury. Pass  a  cubic  inch  of  air  (measured  under  a  pressure  of  30  inches) 
through  the  mercury  into  the  four-inch-long  Torricellian  vacuum.  It  would 
exactly  fill  that  vacuum,  exerting  a  pressure  of  30  inches  on  the  top  of  the 
mercury.  This  is  impossible.  The  gas  expands ;  it  depresses  the  mercury 
through  x  inches ;  it  is  then  subjected  to  a  pressure  of  x  inches  of  mercury 
as  compared  with  the  atmospheric  pressure  of  30  inches  under  which  it  was 
measured.  Its  volume  is  now  accordingly  increased  to  1  cub.  in.  x  30/a:. 
The  length  of  tube  occupied  by  this  volume  is  (30/x)  x  4  =  120 /x  inches  ;  of 
these,  4  inches  were  already  taken  up  by  the  vacuum.  The  actual  depression 
is  therefore  (120 /x)  —  4 ;  but  this  depression  is  x  itself.  Hence  x  =  (120 /x) 
—  4 ;  or  x  =  9-1455 ;  and  the  mercury  will  stand  in  the  tube  at  a  height  of 
20-8545  inches. 


CHAPTER   XIII. 

HEAT. 

Heat  is  a  form  of  Energy.  It  would,  perhaps,  indeed  be  more 
correct  to  say  that  we  designate  under  the  one  name  Heat  two 
totally  distinct  forms  of  Energy.  The  one  of  these  is  the 
energy  of  a  wave-motion  in  the  Ether,  passing  from  a  hot  body 
to  surrounding  objects  across  the  intervening  space,  as  from  the 
sun  to  our  earth,  or  from  a  hot  fire  to  the  colder  objects  upon 
which  it  shines :  this  we  call  Radiant  Heat.  The  other  form 
is  that  of  a  confused  oscillatory  disturbance  of  the  particles  of  a 
body :  in  virtue  of  this  molecular  movement  a  body  may  appear 
to  our  cutaneous  sense  of  heat  (a  sense  quite  distinct  from  that 
of  touch)  to  be  more  or  less  hot  or  warm ;  or  in  the  converse 
case  it  may,  on  account  of  the  small  amount  of  this  movement, 
appear  to  be  relatively  cool  or  cold.  The  latter  form  of  heat 
may  be  called  Sensible  Heat,  or  Heat  simply,  and  of  it  we 
shall  proceed  to  treat  in  this  chapter.  It  is  the  only  form  of 
heat  for  the  perception  of  which  we  have  special  sense-organs. 
We  do  not  directly  perceive  the  undulations  of  radiant  heat  by 
our  senses:  when  the  sun  shines  on  us  heat-waves  strike  the 
skin,  throwing  it  into  vibrations,  and  the  sensible  heat  of  the 
skin,  not  the  radiant  heat  of  space,  affects  the  appropriate  nerve- 
ends.  When  we  touch  a  hot  body  it  communicates  its  oscillations 
to  the  nervous  system :  when  we  approach  a  hot  body  we  become 
indirectly  sensible  of  the  radiant  undulations  into  which  it  is 
throwing  the  surrounding  ether.  Thus  we  may  state  that  our 
sense  of  heat  is  our  power  of  perception  of  the  confusedly- 
vibrating  condition  of  a  body ;  and  that  the  more  pronounced 
this  condition  of  agitation,  the  hotter  will  a  body  appear.  A 
hotter  body  may  be  readily  supposed  —  and  rightly  so  if  we  con- 
fine our  attention  to  bodies  formed  of  the  same  substance  —  to 
have  in  it  a  greater  amount  of  Heat  than  a  colder  one.  And  a 
hotter  body  can  become  cold,  a  colder  body  can  become  warm : 
heat  can  be  supplied  to  bodies,  or  they  can  be  deprived  of  it ; 

350 


CHAP,  xiii.]  NATURE  OF  HEAT.  351 

heat  can  be  gained  or  lost  by  material  bodies.  The  primitive 
interpretation  of  this  was  that  Heat  was  a  substance,  a  fluid,  the 
so-called  Caloric,  invisible,  imponderable  ;  that  a  piece  of  hot 
iron  was  a  kind  of  temporary  union  of  cold  iron  with  this  subtle 
imponderable  fluid.  When  a  piece  of  metal  was  rubbed  it  became 
warm:  the  reason  assigned  was  that  Caloric  was  squeezed  out  of 
it,  like  water  out  of  a  sponge.  But  this  material  theory  of  heat 
became  untenable  when  it  was  shown  that  there  was  absolutely 
no  limit  to  the  amount  of  sensible  heat  which  might  be  so  pro- 
duced by  the  friction  of  a  trifling  amount  of  metal ;  the  amount 
of  water  that  might  be  boiled,  for  example,  by  heat  produced 
in  this  way  depended  only  on  the  mechanical  power  available 
(Rumford).  The  heat  evolved  by  friction  —  as,  for  instance,  in 
metal  boring  or  turning  —  is  practically  limitless.  Even  two 
masses  of  ice,  caused  to  rub  against  one  another,  melt  (Davy) 

—  a  fact  which  leads  the  material  theory  of  heat  into  helpless 
confusion.     Water  was  admitted  to  be  ice  plus  caloric  ;  if,  then, 
ice  with  its  caloric  rubbed  or  squeezed  out  of  it  and  lost  —  that 
is  to  say,  ice  minus  caloric  —  become  water,  how  can  the  theory 
stand?    Plainly  Heat  is  not  material:  and  the  only  alternative 
appears  to  be  that  it  is  the  Energy  imparted  to  the  system.    It  is 
equal  to  the  work  done  upon  the  system  ;  and  we  find  that  Heat 
and  the  other  forms  of  Energy  are  reciprocally  convertible. 

When  a  body  is  sensibly  hot  its  particles  are  in  an  active  state  of  motion. 
The  particles  strike  one  another  and  rebound ;  the  more  rapidly  they  do 
so,  the  greater  is  the  mean  velocity  of  the  particles,  and  the  greater  is  the 
kinetic  energy  of  the  whole  mass ;  but  it  is  impossible  that  the  energy  of 
the  molecules  should  be  entirely  due  to  such  a  movement  of  Translation. 
They  are  not  material  points,  and  they  have  —  if  not  in  solids  or  in  liquids, 
yet  certainly  in  gases  —  six  degrees  of  freedom ;  when  they  strike  each  other 
they  not  only  rebound  but  they  also  spin ;  to  the  energy  of  translation  must 
be  added  one  of  Rotation.  Further,  the  molecules  are  made  up  of  atoms: 
atoms  are  not  stationary  in  the  molecule,  but  may  be  so  violently  agitated 
as  to  leave  it  altogether,  and  thus  to  give  rise  to  the  phenomena  of  chemical 
decomposition  by  heat;  part  of  the  energy  of  a  heated  body  is  due  to 
intra-molecular  or  Atomic  Oscillations.  Lastly,  the  ether  entan- 
gled in  a  molecule  is  also  s.et  in  vibration,  and  absorbs  some  energy,  which 
appears  as  kinetic  energy  of  Ether-Vibrations.  The  sum  of  these  is 
found,  by  the  agreement  of  experimental  results  with  calculations  based  on 
the  hypothesis  that  such  is  the  law,  to  be  proportional  on  the  average 

—  an  average  not  perceptibly  departed  from  for  any  appreciable  interval  of 
time  —  to  the  kinetic  energy  of  translation  alone. 

Heat  is  not  Motion,  for  it  is  neither  Change  of  Position,  nor  yet 
Momentum;  it  is  the  Energy  of  Motion.  Double  the  quantity, of  molecular 
Motion,  and  you  quadruple  the  molecular  kinetic  Energy,  tha't  is,  the  Heat. 

Heat  is  not  liberated  by  Pressure  alone:  there  must  be  yielding  to 


352  HEAT.  [CHAP. 

the  pressure  :  then  the  work  done,  Fa,  or  the  equivalent  Energy,  has  a  deter- 
minate value,  measurable  in  ergs. 

The  convertibility  or  identity  of  Heat  with  Energy  is  inde- 
pendent of  the  inner  mechanism  of  the  moving  molecules  which 
possess  it ;  and  it  is  confirmed  by  instances  from  all  sides. 

The  Energy  of  work  which  is  apparently  wasted  in  friction 
becomes  Heat:  the  heating  of  a  locomotive  brake,  the  ignition 
of  a  lucifer  match,  the  heat  evolved  during  the  mechanical 
operations  of  metal  boring  or  turning,  the  heat  found  in  a  body 
which  has  received  a  sudden  blow  or  a  sudden  distortion,  or 
suddenly  yielded  to  pressure,  —  all  these  prove  the  proposition. 

If  work  be  done  in  driving  a  paddle  in  water,  no  work  being 
done  other  than  that  of  churning  the  water,  when  the  opera- 
tion is  over  the  work  appears  to  have  been  wasted  and  to  have 
disappeared ;  but  the  energy  is  not  destroyed ;  it  exists  in  the 
water  in  the  form  of  heat.  If  772-55  *  foot-pounds  of  work 
(measured  at  sea-level  and  latitude  of  Greenwich,  Joule)  be 
expended  in  churning  a  pound  of  water,  the  temperature  of 
that  water  will  be  raised  by  1°  F.,  from  60°  F.  to  61°  F. ;  a  simi- 
lar rise  of  1°  C.  in  a  kilogramme  of  water  will  be  effected  by 
the  expenditure  of  423-985  kilogrammetres  or  41,593,000000 
ergs  of  work;  that  is,  of  41,593000  ergs  per  gramme.  Hence 
the  water  at  the  base  of  Niagara  Falls  ought  (setting  aside  the 
effect  of  evaporation  and  of  cooling  or  heating  by  the  air)  to 
be  about  ^°  F.  higher  in  temperature  than  at  the  top,  for  the 
vertical  fall  is  161  feet.  Hence  also  the  sailor's  maxim  that 
the  sea  is  warmed  by  a  storm. 

When  in  a  steam-engine  at  work  the  steam  at  its  entrance 
to  the  cylinder  from  the  boiler  is  compared  with  that  which  goes 
to  the  condenser,  it  is  found  that  the  latter  is  colder.  The  dif- 
ference of  heat  is  found  to  be  equivalent  to  the  work  which  the 
engine  has  done ;  and  if  the  engine  do  no  work,  then  the  energy 
which  has  not  been  converted  into  work  remains  as  heat  in  the 
out-going  steam,  and  the  engine  may  become  heated  (Him). 

When  a  quantity  of  gas  or  of  liquid  is  forced  through  a 
tube,  as  in  Fig.  109,  the  potential  energy  of  the  system  before 
the  flow  is  started  is  greater  than  the  kinetic  energy  of  the  out- 
flowing stream.  If  the  resistance  be  so  great  that  the  velocity 

*  The  investigations  of  Rowland  and  Griffiths  have  shown  that  this  number  is  too 
low,  and  that  it  should  be  above  778,  or  possibly  as  high  as  779.  If  we  took  it  as 
778.5,  we  would  have  to  replace  the  number  41,593000,  used  in  this  volume,  by 
41,914000. 


xiii.]  HEAT  A  FOEM   OF  ENERGY.  353 

of  outflow  is  practically  null,  the  whole  of  the  work  done  on 
the  fluid  is  spent  in  heating  it.*  The  work  done  is  equivalent 
to  the  heat  produced. 

We  are  now  able  to  state  the  First  Law  of  Thermo- 
dynamics. Heat,  being  a  form  of  Energy,  can  be  measured 
in  ergs,  in  foot-poundals,  or  in  foot-pounds. 

This  law  is  usually  stated  in  a  somewhat  different  form.  An 
arbitrary  unit  of  heat  is  chosen,  and  designated  a  calorie  :  this  is 
the  amount  of  heat  which  is  required  to  raise  the  temperature  of 
one  gramme  of  water  from  0°  C.  to  1°  C.  This  quantity  of  heat 
is  found  to  be  41,593000  ergs.  This  last  number,  41,593000 
ergs,  is  the  "Mechanical  Equivalent  of  Heat,"  or  "Joule's 
Equivalent " :  f  it  should  perhaps  be  called  the  Dynamical 
Value  of  the  Conventional  Unit  of  heat,  the  calorie.  The  first 
law  is,  then,  that  one  calorie  (^ca)  is  equal  to  41,593000  ergs. 

Another  unit  of  heat  has  been  proposed,  the  Electromagnetic  Unit,  the 
Joule,  or  10,000000  ergs ;  this  is  the  amount  of  heat  developed  in  one  second 
in  an  electrical  circuit  or  wire  whose  resistance  is  one  Ohm  when  a  current 
passes  whose  intensity  is  one  Ampere.  (See  p.  647.) 

Heat  is  energy,  and  it  is  the  lowest  form  of  energy.  It 
may  be  said  to  have  no  organisation,  but  to  depend  on  undi- 
rected and  blind  activity  of  molecules,  which  dash  hither  and 
thither.  When  in  any  action  energy  is  liberated  which  is  not 
guided  by  the  environment  into  any  specialised  form,  it  mani- 
fests itself  as  heat ;  and  when  energy  is  spent  in  doing  work, 
the  equivalent  of  which  appears  in  no  other  form,  it  then 
appears  as  heat.  This  statement  is  widely  applicable  and 
important. 

Work  done  upon  a  dynamoelectric  machine  whose  circuit  is  complete 
appears  in  the  first  place  as  the  energy  of  an  electric  current :  if  no  exterior 
work  be  done,  the  system  as  a  whole  becomes  heated. 

A  voltaic  cell  can  do  exterior  work :  if  it  do  none,  the  current  being 
allowed  to  circulate  uselessly,  the  whole  of  the  energy  liberated  during  the 
chemical  combination  appears  as  heat  in  the  circuit. 

*  This  must  be  done  at  a  pressure  corresponding  to  a  certain  definite  head  H  of 
the  same  fluid.  The  fluid  is  found  to  rise  in  temperature  by  x°  C. ;  a  head  of  H/x 
cm.  would  cause  it  to  rise  by  1°  C. ;  a  vertical  free  fall  of  H/x  cm.  would  cause  it,  if 
abruptly  stopped,  to  rise  in  temperature  by  1°  C. ;  the  amount  of  energy  correspond- 
ing to  such  a  fall  would  be  (H/x)  •  my  ergs ;  this  energy  in  the  form  of  heat  (H/x)  •  mg 
ergs,  would  heat  a  mass  m  of  the  fluid  through  1°  C. ;  (H/x)  •  g  ergs  would  heat  one 
gramme  of  the  fluid  through  1°  C. ;  (H/x)  •  (g/<r)  ergs  (<r  being  the  specific  heat  of  the 
fluid,  p.  365)  would  heat  one  gramme  of  water  through  1°  C. 

t  Joule's  Equivalent  in  its  original  form  was  a  number  (772)  whirih  denoted  the 
number  of  foot-pounds  of  work  found  to  be  equivalent  to  the  heat  necessary  to  raise 
1  Ib.  of  water  through  1°  F. 

2A 


354  HEAT.  [CHAP. 

Heat  being  a  form  of  energy,  many  propositions  relating  to 
it  are  merely  special  cases  of  propositions  relating  to  energy. 

If  a  certain  number  of  bodies  be  arranged  in  a  system  A 
whose  potential  energy  —  depending  on  the  arrangement  of  the 
bodies  in  the  system  —  is  PA;  if  the  same  bodies  can  be  arranged 
in  other  systems  B,  C,  D,  whose  respective  potential  energies 
(less  than  that  of  the  former)  are  PB,  Pc>  PD>  etc.;  then  the 
transformation  of  the  more  highly-stressed  system  A  into  a  less- 
stressed  system  B,  if  this  be  brought  about  by  a  rearrangement 
of  its  constituent  bodies,  involves  a  liberation  of  energy  equal 
to  PA  —  PB.  If  in  this  case  the  system  A  be  converted  into  the 
system  B  without  doing  any  exterior  work,  the  whole  of  the 
energy  liberated  appears  in  the  form  of  heat ;  and,  numerically 
expressed,  the  heat  thus  liberated  is  equal  to  the  work  W  which 
would  have  had  to  be  done  upon  the  system  B  in  order  to  con- 
vert it  into  the  system  A,  if  that  converse  operation  had  been 
effected ;  that  is,  the  heat  so  liberated  is  equal  to  W  =  PA  —  PB. 

A  gramme  of  hydrogen  and  eight  grammes  of  oxygen  form  a  system 
(system  A)  which  after  explosion  may  be  converted  into  nine  grammes  of 
water-vapour  of  the  same  volume  (system  B)  at  a  temperature  of  136°-5  C. 
The  former  is  converted  into  the  latter  without  doing  any  exterior  work. 
Much  energy  is  liberated  in  the  form  of  heat,  and  though  the  absolute  values 
of  PA  and  PB  are  unknown,  their  difference  is  found  (see  Calorimetry)  to  be 
an  amount  of  energy  equivalent  to  28,580  ca. 

If  the  products  be  cooled  down  to  steam  at  100°  C.,  the  total  amount  of 
heat  liberated  is  equal  to  28,738  ca,  or  1,195300,000000  ergs;  if  to  water  at 
0°  C.,  it  is  equal  to  34,462  ca,  or  1,433380,000000  ergs,  or  1,433380  megergs. 
The  potential  energy  which  such  a  mixture  loses  when  its  particles  clash 
together  and  combine  is  the  energy  of  chemical  separation.  A  mixture  of 
explosive  gases  may  be  made  to  yield  up  some  of  this  energy  in  the  form 
of  work,  as  in  the  modern  gas-engine ;  if  no  work  be  done,  and  if  there  be 
no  other  transformation,  the  whole  of  it  must  appear  in  the  form  of  heat. 

Chemical  combination  is  thus  often  attended  with  the  evolution  of  heat. 
One  gramme  of  carbon  burned  in  oxygen  yields  8,080  ca  or  336071,520000 
ergs ;  1  gramme  of  carbonic  oxide  yields  2,403  ca  (2,431  ca,  Andrews) ; 
1  gramme  of  marsh-gas,  13,063  ca;  1  gramme  of  dry  albumen,  4,998  ca ; 
urea,  2,206  ca;  fat,  9,096  ca;  starch,  3,901-2  ca,  or  162263,000000  ergs  per 
gramme  (Frankland). 

When  copper  or  antimony  is  dropped  into  chlorine  it  takes  fire,  and  a 
chloride  is  formed  :  heat  is  evolved. 

In  some  instances  the  converse  is  true  ;  work  has  to  be  done  upon  sepa- 
rate elements  in  order  to  force  them  directly  or  indirectly  to  combine  :  and 
when  their  compound  decomposes,  heat  is  evolved.  Carbon  and  sulphur 
will  only  combine  when  they  are  kept  hot  by  an  external  source  of  heat : 
they  must  be  forced  to  combine :  and  when  CS2  is  violently  shaken,  as  by 
the  explosion  of  a  percussion  cap,  the  carbon  and  the  sulphur  fall  apart, 
evolving  heat.  Nitrous  oxide  (N2O)  evolves  heat  when  it  is  decomposed 


xiii.]  HEAT  A  FORM  OF  ENERGY.  355 

into  nitrogen  and  oxygen ;  and  hydrogen  dioxide  (H2O2)  evolves  heat 
when  it  is  decomposed  by  contact  with  platinum. 

The  heat  liberated  or  absorbed  measures  the  work  done  by  chemical 
action.  Where  there  is  none,  there  is  no  chemical  change.  Thus  hydro- 
chloric acid  gas  and  ammonia  gas  do  not  change  in  temperature  when  mixed 
hot :  there  is  no  combination :  it  is  only  on  cooling  that  they  combine  and 
liberate  heat.  Similarly  in  many  other  cases,  where  dissociation  takes  place 
on  heating  or  on  solution  in  a  liquid. 

The  Chemical  Forces  themselves  are  not  measured  by  the  total  Heat 
liberated  or  absorbed.  The  reactions  may  be  rapid  or  slow,  and  they  do  not 
necessarily  take  a  course  which  will  liberate  the  maximum  amount  of  heat. 

A  change  from  the  condition  B  to  the  condition  A  (which 
possesses  more  potential  energy)  cannot  be  effected  unless  there 
be  energy  added  ab  externo,  or  else  unless  some  of  the  kinetic 
energy  of  the  body,  if  it  have  any,  assume  the  potential  form ; 
in  the  latter  case  the  body  may  lose  sensible  heat,  and  may 
become  cold. 

When  a  chemical  decomposition  is  effected  by  heat,  if  heat  had  been 
evolved  during  the  formation  of  the  compound,  heat  must  be  continuously 
supplied  to  do  the  work  of  decomposition.  The  heat  supplied  has  the  effect 
of  throwing  the  molecule  into  such  agitation  that  the  mutual  affinity  of 
the  atoms  cannot  retain  them  in  union.  This  is  the  process  of  Dissocia- 
tion or  Thermolysis.  At  moderately-high  temperatures  the  atoms 
reunite  with  others  which  they  encounter;  at  very  high  temperatures  (from 
2300°  to  3000°  C.  in  the  case  of  oxygen  and  hydrogen)  no  such  reunion  is 
possible,  and  the  decomposition  is  complete.  Thus  the  proportion  of  decom- 
posed to  apparently  undecomposed  material  varies  with  the  temperature. 
The  process  is  favoured  by  one  or  more  of  the  resultants  of  dissociation 
being  gaseous.  After  dissociation,  the  separated  elements  contain  potential 
energy  equal  to  the  heat  expended  upon  them ;  and  upon  cooling  they  may 
recombine  with  the  evolution  of  this  energy  in  the  form  of  heat,  which  is 
gradually  lost. 

Dissociation  is  easy  at  low  pressures.  Hence  at  low  pressures  the  com- 
bustion of  a  candle  is  incomplete  and  its  flame  is  smoky. 

Dissociation  of  the  products  of  combustion  is  also  exceedingly  facilitated 
by  the  presence  of  solid  surfaces  (Sir  C.  W.  Siemens). 

In  general,  every  Change  in  the  State  or  Condition  of  a  body 
or  a  system  of  bodies  is  associated*  with  a  Change  in  the  Intrin- 
sic Potential  Energy  of  the  body  or  the  system  ;  and  this  change 
is  accompanied  and  manifested  either  by  the  liberation  of 
Energy  in  some  form,  useful  or  useless  —  e.g.,  work  or  heat  — 
or  else  by  the  disappearance  of  Energy  which  is  spent  in 
producing  the  change  of  state,  and  is  either  taken  in  ab  externo 

*  This  general  conclusion  is  subject  to  the  qualification  that  the  change  of  state 
or  condition  must  be  a  real  one,  not  one  which  consists  in  a  mere  replacement  of  the 
particles  occupying  a  given  position  by  others  physically  similar,  or  by  a  mere  change 
of  the  direction  in  which  similar  parts  of  the  substance  lie. 


356  HEAT-  [CHAP. 

or  is  transferred  from  the  kinetic  energy  already  possessed  by 
the  body,  as  is  shown  in  the  ordinary  case  by  that  body  becom- 
ing cold. 

Thus,  if  a  quantity  of  air  in  a  cylinder  be  suddenly  compressed  by  means 
of  exterior  work  done  upon  it,  it  becomes  hot :  if  the  piston  be  allowed  to 
return,  the  air  cools  down  to  its  former  temperature ;  but  if  it  be  kept  com- 
pressed until  it  has  assumed  the  temperature  of  surrounding  objects,  and  if 
it  be  then  allowed  to  drive  the  piston  out  against  atmospheric  pressure,  it 
becomes  very  cold,  for  it  obtains  the  energy  required  to  do  the  work  of 
driving  out  the  piston  at  the  expense  of  its  own  heat. 

If  the  system  A  disengage  x  units  of  energy  (as  heat,  or  in 
any  other  form)  on  being  let  down  to  the  condition  B,  and  if 
the  same  system  A  disengage  y  units  when  it  acquires  the  con- 
dition C,  then  the  system  B,  on  being  let  down  to  the  condition 
C,  will  disengage  energy  =  y  —  x.  Conversely,  if  B  and  C 
respectively  require  energy  x  and  y  to  enable  them  to  become 
converted  into  the  system  A,  the  system  C  requires  energy 
=  y  —  x  in  order  to  enable  it  to  become  the  system  B. 

The  relative  amounts  of  chemical  energy  in  organic  compounds  may  be 
estimated  by  finding  the  amount  of  heat  which  they  evolve  when  they  are 
burned  so  as  to  form  carbonic  anhydride,  water  (and  nitrogen). 

Oil  of  lemons,  turpentine,  and  terbene,  which  have  the  same  chemical 
constitution,  seem  to  have  a  different  intramolecular  arrangement,  for  on 
combustion  they  evolve  different  amounts  of  heat.  This  shows  that  the 
potential  energy  of  the  molecules  is  different  in  each  case. 

What  is  the  intrinsic  energy  of  Acetic  Acid  ?  60  grammes  of  acetic  acid 
are  found  (Berthelot)  to  disengage  on  combustion  210,000  ca  of  heat : 
3500  ca  or  145,576  megergs  per  gramme.  The  total  intrinsic  potential 
energy  of  acetic  acid  we  do  not  know;  the  number  given  indicates  the  total 
amount  available  on  combustion  with  oxygen.  Its  elements,  —  24  grammes 
of  carbon,  4  of  hydrogen  (and  32  of  oxygen),  —  yield  on  combustion  332,000 
ca.  The  difference  between  the  "  Combustion-equivalent "  of  the  60 
grammes  of  acetic  acid  and  that  of  the  same  weight  of.  its  component  ele- 
ments —  that  is,  122,000  ca  —  is  the  total  amount  of  energy  lost  by  these 
substances  when  they  pass  through  the  changes  (whatever  be  the  number, 
the  nature,  or  the  order  of  these)  in  which  they  pass  from  the  state  of  free 
elements  to  that  of  acetic  acid. 

Amorphous  sulphur  kept  in  a  solution  of  sulphuretted  hydrogen  becomes 
octohedral  sulphur  with  absorption  of  heat. 

When  zinc  is  dissolved  in  sulphuric  acid,  a  certain  amount  of  energy  is 
liberated  and  heat  is  evolved :  when  zinc  is  amalgamated  with  mercury,  it 
becomes  cold  unless  heat  be  supplied :  when  amalgamated  zinc  is  dissolved 
in  sulphuric  acid  it  evolves  more  heat  than  unamalga-mated  zinc  does,  and 
that  by  an  amount  exactly  equal  to  the  heat  absorbed  during  amalgamation. 

The  absolute  amount  of  energy  liberated  or  absorbed  during 
any  change  of  state  is  independent  of  the  rate  at  which  the 
change  is  effected. 


xiii.]  HEAT  A  FORM  OF  ENERGY.  357 

A  slow  change  of  state  (as  in  the  processes  of  decay  or  of  the  oxidation 
of  the  tissues  of  an  animal)  evolves  the  same  amount  of  heat  as  a  rapid 
change ;  the  temperature  in  the  former  case  is  lower  than  in  the  latter, 
because  the  lapse  of  time  allows  a  more  equable  distribution  of  the  heat. 
Thus,  1  gramme  of  hydrogen  and  8  grammes  of  oxygen  will  evolve  enough 
heat  to  raise  34,462  grammes  of  water  in  temperature  by  1°  C. ,  or  344,620 
grammes  by  y^0  C. :  this  it  will  do  whether  the  combination  be  explosive  or 
gradual,  as  when  the  gases  are  induced  slowly  to  combine  by  the  presence  of 
rolled  platinum.  The  final  condition  of  the  products  must  be  the  same 
in  both  cases :  if  this  be  not  borne  in  mind,  the  amounts  of  energy  evolved 
during  combination  will  appear  to  differ  in  the  two  cases  by  an  amount 
equal  to  the  work  which  would  have  to  be  done  in  order  to  convert  the  one 
final  state  into  the  other. 

When  a  system  of  bodies  passes  from  one  state  to  another  it 
is  a  matter  of  indifference  what  the  intermediate  changes  have 
been,  so  far  as  concerns  the  absolute  amount  of  energy  liberated 
or  absorbed;  the  system  A  may  have  assumed  the  conditions 
C,  D,  etc.,  and  that  in  any  order;  but  the  amount  of  energy 
liberated  depends  only  on  the  initial  state  A  as  compared  with 
the  final  state  B. 

If  it  had  been  otherwise,  the  perpetual  motion  might  be  realised  ;  for  it 
might  be  possible  to  effect  a  change  from  A  to  B  by  one  series  of  transfor- 
mations, and  to  effect  the  reverse  operation  by  another  series,  such  that  the 
one  series  of  changes  would  evolve  more  energy  than  the  converse  one  con- 
sumed, and  the  result  would  be  a  repeated  restoration  of  the  status  quo,  asso- 
ciated with  a  perpetual  supply  of  energy,  available  for  useful  work,  and 
created  out  of  nothing. 

The  energy  absorbed  by  a  system  during  a  given  change  of 
state  is  exactly  equal  to  that  which  is  liberated  when  the  change 
is  reversed. 

It  is  assumed  in  this  statement  that  no  exterior  work  is  done  through 
the  instrumentality  of  the  change  of  state. 

The  potential  energy  of  every  system  of  bodies  always  tends 
to  diminish  as  far  as  possible.  Every  system  which  possesses 
potential  energy  thus  tends  to  lose  it ;  its  potential  energy  tends 
to  become  kinetic,  and,  if  it  assume  no  other  form,  to  take  the 
unspecialised  form  of  heat.  In  any  system  which  undergoes 
spontaneous  transformation,  the  transformation  generally  tends, 
unless  prevented,  to  take  such  a  course  that  the  heat  evolved 
by  it  shall  be  a  maximum.  This  is,  however,  only  a  tendency : 
and  in  many  chemical  reactions  the  heat  evolved  is  not  the 
maximum  possible. 

In  many  cases  a  single  change  of  state  may  be  analysed  into 
several  others.  The  heat-value  of  the  total  change'  is  equal  to 
the  sum  of  the  heat-values  of  the  separate  component  changes. 


358  HEAT.  [CHAP. 

Thus  when  a  piece  of  sodium  is  put  into  water  the  following  changes 
occur  simultaneously:  —  (1)  decomposition  of  water  into  free  atoms  of 
hydrogen  and  oxygen ;  (2)  coalescence  of  atoms  of  hydrogen  to  form  mole- 
cules ;  (3)  reduction  of  hydrogen  to  the  gaseous  state  ;  (4)  exterior  work  done 
by  the  hydrogen  escaping  against  atmospheric  pressure ;  (5)  combination  of 
sodium  with  oxygen  and  hydrogen  atoms  to  form  sodium  hydrate ;  (6)  solu- 
tion of  sodium  hydrate  in  water.  Each  of  these  changes  has  its  own  heat- 
value,  positive  or  negative,  according  as  it  involves  the  evolution  or  the 
absorption  of  a  certain  amount  of  energy.  On  the  whole,  potential  energy 
is  lost  and  heat  is  liberated. 

The  combustion  of  1  grm.  H  with  8  grms.  oxygen  yields  34,462  ca  heat. 
The  same  quantity  of  the  same  elements  combining  in  the  nascent  state 
yields  54,623  ca.  Hence  the  heat  evolved  during  the  combustion  of  one 
gramme  of  hydrogen  is  the  resultant  of  an  absorption  of  energy  (20,161  ca) 
due  to  the  break-up  of  the  gaseous  molecules  into  atoms,  and  an  evolution 
(54,623  ca)  due  to  the  combination  of  these  atoms  in  the  formation  of  water- 
molecules  and  condensation  into  liquid  water  at  0°  C.  The  balance  of  the 
account  shows  energy  to  be  liberated  as  heat. 

When  a  gas  is  dissolved  in  water  there  are  two  effects  :  —  («)  liquefac- 
tion of  gas  with  evolution  of  heat;  (b)  satisfaction  of  chemical  affinity 
between  the  water  and  the  gas,  with  the  evolution  of  still  more  heat.  When 
KFT3-gas  is  dissolved  in  water,  there  is  no  evolution  of  heat  corresponding  to 
any  union  of  NH3  and  H2O  to  form  NH4HO. 

When  a  solid  is  dissolved  in  water  the  liquefaction  of  the  solid  causes 
the  absorption  of  heat  (as  in  freezing  mixtures),  while  the  satisfaction  of 
mutual  chemical  affinity  causes  its  evolution.  When  glacial  acetic  acid  is 
dissolved  in  water,  the  absorption  of  heat  caused  by  imparting  greater  fluidity 
to  the  acetic  acid  overpowers  the  evolution  of  heat  due  to  chemical  union. 

When  two  or  more  changes  of  state  occur  concurrently,  it 
may  be  that  some  of  these  changes  are  accompanied  by  the  liber- 
ation, some  by  the  absorption  of  heat,  and  that  these  changes 
exactly  compensate  each  other;  the  result  being  that  on  the 
whole  there  is  neither  absorption  nor  liberation  of  heat. 

For  example,  a  gas,  while  expanding  (from  whatever  cause), 
against  the  atmospheric  pressure,  does  work  in  lifting  the  atmos- 
phere ;  if  it  increase  in  volume  alone,  without  undergoing  any 
change  in  its  temperature,  energy  must  be  supplied  to  it  in 
order  to  enable  it  to  do  this  work ;  if  it  diminish  in  temperature 
without  suffering  any  change  in  its  volume,  it  must  necessarily 
lose  heat ;  if,  on  the  other  hand,  it  undergo  both  these  changes 
—  increase  in  volume  and  diminution  of  temperature  — concur- 
rently, it  is  possible  that  these  two  changes  may  be  so  adjusted 
that  the  body,  while  it  undergoes  the  double  change,  neither  loses 
heat  nor  acquires  energy  from  without.  Such  expansion  is 
called  (Rankine)  adiabatic  expansion  —  expansion  during  which 
the  substance  neither  gains  nor  loses  heat  by  conduction  or  radi- 
ation to  or  from  surrounding  objects,  and  in  the  course  of  which, 


xiii.]  ADIABATIC   CHANGE   OF   VOLUME.  359 

as  it  expands,  it  cools  down  by  reason  of  its  expenditure  of 
energy.  This  is  a  kind  of  operation  which  could  only  be  per- 
fectly realised  in  practice  if  the  expansion  were  infinitely  rapid ; 
but  any  gas  suddenly  expanded  is  thus  chilled.  Conversely, 
adiabatic  contraction  of  volume  is  associated  with  increase  of 
temperature. 

We  have  hitherto  regarded  any  change  of  state,  simple  or 
complex,  as  a  possible  antecedent  cause  of  the  liberation  or  of 
the  disappearance  of  heat.  We  shall  now  change  our  stand- 
point, and  consider  the  effects  (including  change  of  state  or  of 
condition)  produced  by  the  increase  of  heat  in  a  body  or  by  its 
withdrawal. 

EFFECTS  OF  HEAT. 

The  principal  effects  of  an  increase  of  heat  in  a  body  may 
be  the  following:  — 

A.  Internal  Work. 

a.  Increase  of  the  kinetic  energy  of  the  molecules  of 

the  body  —  an  increase  of  the  sensible  heat  of 
the  body  ;  i.e.  an  increase  of  temperature. 

b.  Inter  molecular   work  —  work    done   by  or  against 

molecular  forces — change  of  volume,  change  of 
cohesion,  change  of  elasticity,  etc. 

c.  Intramolecular  work — work  done  within  each  sev- 

eral   molecule  —  production   of   intramolecular 
vibrations,  rotations,  deformations. 

d.  Chemical  work,  intermolecular  and  intramolecular. 

B.  External  Work.  —  Work  done  by  or  on  a  body  as  it 

expands  or  diminishes  in  bulk. 

These  effects  are  not  necessarily  all  produced  by  the  action  of 
heat  upon  any  substance. 

There  may,  as  in  the  following  example,  be  no  external 
work  done  when  a  body  is  heated ;  the  whole  energy  imparted 
to  the  body  being  spent  upon  the  internal  accumulation  of 
energy  in  the  form  of  heat.  Water  at  3°-4  C.  if  heated  to  4°-4  C. 
first  contracts  and  then  returns  to  its  original  dimensions.  On 
the  whole  there  is  in  this  case  no  external  work  done.  Neither 
is  there  any  work  done  in  giving  the  particles  a  new  position  in 
opposition  to  the  intermolecular  forces,*  nor  is  there  any  chem- 

*  This  statement  is  only  approximately  true,  for  there  are  physical  differences  — 
of  viscosity  and  the  like  —  between  water  at  3°'4  C.  and  water  at  4^4  C.  The  tem- 
perature of  the  maximum  density  is  lowered  0°'0177  C.  per  atmo.  pressure  (Tait). 


360  HEAT.  [CHAP. 

ical  effect.  The  whole  heat  imparted  may  thus  be  held  to  be 
spent  in  raising  the  temperature  by  1°  C. 

When  a  bar  of  iron  is  heated  in  a  vacuum  there  are  two 
effects  :  (1)  increase  of  temperature  ;  (2)  expansion  of  the  iron, 
which  represents  work  done  against  the  molecular  forces.  When 
the  same  bar  is  heated  in  air,  there  is  added  a  third  effect,  viz., 
the  thrusting  aside  of  the  surrounding  air  by  the  expanding  bar, 
in  consequence  of  which  exterior  work  is  done  during  expansion. 
In  a  bar  of  iron  the  exterior  work  done  in  this  way  is  very  small, 
and  the  interior  work  done  predominates  so  largely  that  the 
exterior  work  may  for  many  purposes  be  neglected. 

When  a  mass  of  gas  is  heated,  the  work  which  is  done  in 
expanding  the  gas  itself  is  appreciably  null,  for  this  is  one  of 
the  characteristics  of  gases ;  if  there  be  any  work  done  during 
the  expansion,  it  is  all  exterior.  The  effects  in  this  case  are 
two :  (1)  the  increase  of  temperature  ;  (2)  exterior  work  done 
in  overcoming  the  exterior  (atmospheric  or  other)  pressure. 

When  water  above  3°-9  C.  is  heated  it  expands.  The  effects 
are  —  (1)  increase  of  temperature  ;  (2)  work  done  in  separat- 
ing the  molecules;  (3)  a  small  amount  of  work  done  against 
the  external  pressure. 

When  water  between  0°  and  3°-9  C.  is  heated  it  contracts. 
The  effects  are  —  (1)  increase  of  temperature  ;  (2)  ihtermolec- 
ular  work  ;  (3)  a  small  amount  of  work  done  by  the  external 
pressure. 

When  a  piece  of  caoutchouc  is  heated  it  contracts ;  when  pulled,  it 
expands  and  assumes  the  dimensions  proper  to  a  lower  temperature,  inter- 
molecular  energy  is  set  free,  and  the  caoutchouc  becomes  warm.  A  piece 
of  metal  suddenly  extended  becomes  cool. 

When  ice  at  0°  C.  is  heated,  the  whole  energy  imparted  to  it 
is  expended  in  producing  the  following  results  :  —  (1)  Fusion, 
with  contraction  of  volume  (intermolecular  work  —  work  spent 
in  producing  a  new  arrangement  of  the  molecules)  ;  (2)  A 
slight  amount  of  work  done  by  the  exterior  pressure  on  the 
body.  The  latter  may  be  for  most  purposes  neglected ;  if  we 
do  so  wre  may  say  that  all  the  heat  supplied  to  the  ice  is  spent 
in  doing  the  interior  work  of  liquefaction,  and  that  none  of  it 
is  spent  in  producing  any  increase  of  temperature.  When,  there- 
fore, a  piece  of  ice  is  heated  it  melts,  but  it  does  not  rise  in 
temperature  until  it  has  been  wholly  melted.  The  water 
produced  has  a  temperature  of  0°  C.,  and  it  does  not  begin  to 
rise  in  temperature  until  the  ice  has  entirely  disappeared :  when 


xiii.]  EFFECTS   OF  HEAT.  361 

this  has  occurred,  the  continued  action  of  heat  causes  the  water 
to  rise  in  temperature. 

A  gramme  of  ice  at  0°  C.  absorbs  (Bunsen)  80-025  ca  of  heat,  and 
becomes  a  gramme  of  water  at  0°  C.  Conversely,  a  gramme  of  water  at 
0°  C.  must  continue  to  lose  heat  until  it  has  parted  with  80-025  ca  before  it 
can  become  a  gramme  of  ice  at  0°  C. ;  whence  we  observe  that  water  does 
not  freeze  throughout  at  the  instant  of  the  thermometer's  touching  the 
freezing  point. 

It  was  obvious  that  a  gramme  of  water  differed  from  one  of  ice  in 
somehow  possessing  80-025  ca  of  heat ;  but  this  was  not  sensible  to  the  ther- 
mometer ;  hence  the  heat  so  possessed  by  the  water  was  said  to  be  hidden 
or  Latent  Heat.  We  now  know  that  it  is  not  Heat  of  any  kind ;  it  is 
latent  or  potential  Energy ;  work  must  be  done  against  molecular  forces  in 
order  to  convert  ice  into  water :  water  somehow  differs  from  ice  at  the  same 
temperature  in  possessing  more  potential  energy. 

Direct  increase  of  the  kinetic  energy  of  the  particles  of 
a  heated  eras  is  demonstrated  by  the  Radiometer. 

O  •/ 

If  a  surface  be  heated,  a  molecule  of  gas  striking  against  it 
is  heated ;  it  leaves  the  hot  surface  with  a  velocity  greater  than 
that  with  which  it  had  approached  it.  If  the  surface  be  fixed, 
the  gas  in  front  of  it  is  driven  away  from  it  by  the  bombardment 
of  the  molecules  which  have  touched  the  hot  surface,  and  on 
their  return  strike  their  fellow-molecules ;  in  front  of  the  hot 
surface  the  gas  is  therefore  under  a  greater  pressure  than  it 
would  have  been  had  the  surface  been  cold.  If  the  hot  surface 
be  not  fixed,  this  increase  of  pressure  has — reaction  being 
equal  and  contrary  to  action  —  a  tendency  to  drive  that  surface 
backwards.  This  tends  to  knock  dust  away  from  a  hot  surface. 

If  the  hot  surface  be  the  front  aspect  of  a  disc,  the  back  of 
which  is  by  some  means  kept  colder  than  the  front,  and  if  this 
disc.be  suspended  in  a  gas,  the  heat  of  the  front  surface  increases 
the  pressure  towards  the  front,  and  the  gas  flows  round  to  the 
back  of  the  disc.  Thereafter  the  disc  is  struck  on  the  hotter 
surface  by  fewer  molecules  with  greater  velocities,  on  the  colder 
surface  by  a  greater  number  of  molecules  with  lesser  velocities ; 
there  is  thus  compensation ;  the  result  is  that  the  disc  is  equally 
pressed  upon  in  front  and  on  the  back  ;  it  does  not  move. 

Let  us  now  suppose  that  the  particles  recoiling  from  the 
heated  surface  do  not  meet  other  molecules,  but  impinge  on  the 
walls  of  the  vessel.  A  layer  of  particles  in  such  a  condition  is 
called  a  Crookes's  layer. 

This  will  occur  in  two  cases  —  (1)  when  the  gas  is  so  rarefied  that  the 
mean  free  path  of  the  molecules  exceeds  the  distance  between  the  hot  sur- 
face and  the  walls  of  the  vessel ;  (2)  when,  whatever  the  density  of  the  gas, 


362  HEAT.  [CHAP. 

the  opposite  wall  is  so  near  the  hot  surface  that  the  distance  between  them 
is  less  than  the  actual  mean  free  path  of  the  molecules.  These  conditions, 
which  are  essentially  identical,  may  concur  ;  there  may  be  both  rarefaction 
of  the  gas  and  approximation  of  the  opposed  surfaces. 

In  such  a  case  there  is  no  flow  of  gas  from  the  hotter  sur- 
face towards  the  colder  one  :  each  molecule  which  strikes  the 
hotter  surface  and  rebounds  with  a  greater  speed  adds  indepen- 
dently to  the  recoil  which  the  hotter  surface  suffers,  and  if  the 
hotter  surface  be  movable,  it  is  driven  backwards.  If  it  be  not 
movable,  the  particles  which  rebound  from  it  strike  the  opposite 
wall  of  the  containing  vessel,  and  that  wall  has  a  tendency  to 
move  forward. 

There  is  yet  another  case :  if  the  rarefaction  of  the  gas 
be  extreme,  the  particles  which  strike  the  heated  surface  are 
few  in  number  or  none  at  all,  there  is  little  or  no  recoil,  and 
there  is  no  movement  set  up  when  the  rarefaction  is  carried 
too  far. 

The  disc  of  which  we  speak  —  a  disc  of  which  one  face  is 
kept  hotter  than  the  other  —  may  be  a  disc  covered  on  the  one 
side  with  some  heat-absorbent  material  such  as  lampblack,  the 
other  face  being  whitened.  When  radiant  heat  or  light  falls 
upon  the  disc,  even  in  an  equally-lighted  field,  the  blackened 
side  becomes  hotter.  If  such  a  disc  be  suspended  vertically  by 
two  threads,  it  will  diverge  slightly  from  the  perpendicular. 
Such  a  disc  may  be  attached  to  the  end  of  a  counterpoised  rod, 
the  whole  being  suspended  by  two  threads :  the  effect  of  heat 
or  light  is  to  twist  the  suspending  threads  to  a  certain  extent. 
If  the  suspensory  arrangement  be  replaced  by  a  pivoting  one, 
we  have  the  Radiometer.  A  globe  of  glass,  in  which  a  vacuum 
is  made,  carries  a  vertical  needle  axially  fixed,  on  the  summit 
of  which  is  poised  a  rotating  vane  consisting  of  light  rods,  to  the 
extremities  of  which  discs  are  affixed,  each  similarly  blackened 
on  one  side.  Such  an  instrument  placed  in  light,  even  in  a  uni- 
formly-lighted field,  has  the  black  sides  of  its  discs  more  heated 
than  the  unblackened  sides,  and  if  the  radiance  be  of  sufficient 
energy  the  vane  rotates.  Moonlight  is  too  weak  to  produce 
this  effect :  a  candle  will  make  a  sensitive  radiometer  rotate ; 
a  paraffin  lamp  without  a  globe  will  at  close  quarters  make  the 
vane  fly  round  so  fast  as  to  be  invisible. 

If  a  radiometer  be  floated  in  water,  and  if  the  vane  be  so 
constructed  —  one  of  its  spokes  being  a  magnet  —  that  a  power- 
ful magnet  in  the  neighbourhood  can  hold  it  motionless,  when 


xiii.].  RADIOMETER.  363 

the  radiometer  is  exposed  to  light  the  bulb  itself  will  rotate  in 
the  water  in  which  it  floats. 

The  radiometer  is  a  machine  in  which  heat  (generally 
derived  from  the  transformation  of  light  into  heat)  is  directly 
converted  into  the  energy  of  work. 

The  less  the  distance  between  the  discs  and  the  walls  of 
the  bulb,  the  greater  will  be  the  effect,  and  the  faster  will  the 
vane  rotate,  provided  that  the  rarefaction  be  less  complete  than 
that  which  gives  the  greatest  effect.  Too  complete  a  rarefaction 
is  not  an  advantage,  for  it  leaves  an  insufficient  supply  of  work- 
ing molecules. 

When  the  distance  between  the  disc  and  the  opposite  wall 
is  excessively  small,  the  vacuum  need  not  be  very  good ;  indeed 
the  effect  of  repulsion  may  be  made  manifest  even  in  the  open 
air. 

When  a  drop  of  water  is  placed  upon  a  very  hot  iron  it 
assumes  the  so-called  Spheroidal  State ;  it  does  not  wet  the 
hot  iron,  but  gathers  itself  into  a  drop,  which  rapidly  evapo- 
rates and  alters  the  local  conditions  of  its  surface-tension  so  as 
to  present  an  appearance  of  varying  scroll-work  on  its  surface, 
while  the  drop  oscillates  so  as  to  present  the  form  of  rosette  and 
other  patterns,  these  being  due  to  the  formation  of  nodes  and 
vibrating  loops.  The  drop  may  be  of  very  considerable  dimen- 
sions—  several  ounces  in  weight.  It  does  not  touch  the  iron; 
there  is  an  intervening  layer  of  aqueous  vapour  on  which  it 
floats ;  through  the  space  intervening  between  the  drop  and 
the  hot  solid  the  light  of  a  candle  may  be  seen.  This  layer  of 
aqueous  vapour  is  a  "  Crookes's  layer ; "  particles  strike  the 
heated  surface,  rebound,  and  strike  the  liquid,  thus  maintaining 
a  clear  space  between  the  metal  and  the  drop.  Ether  and  small 
drops  of  bromine  float  in  the  same  way  on  the  surface  of  hot 
water.  A  lump  of  carbonate  of  ammonia,  thrown  into  a  red-hot 
platinum  crucible,  assumes  the  spheroidal  state  superficially,  but 
does  not  melt.  The  hand  can  be  safely  immersed  in  melted 
metal  if  it  be  not  too  dry,  and  if  the  immersion  be  effected  with 
a  certain  degree  of  prompt  deliberation ;  a  Crookes's  layer  of 
water-vapour  intervenes  between  the  hand  and  the  metal. 

When  liquid  sulphurous  acid  is  dropped  into  a  white-hot 
platinum  crucible  it  sinks  greatly  in  temperature,  on  account 
of  its  rapid  evaporation  and  its  slow  reception  of  heat  across 
the  Crookes's  layer ;  if  a  little  water  be  added  to  it,  the  water 
freezes.  Ice  can  thus  be  produced  in  a  white-hot  platinum 


364  HEAT.  [CHAP. 

crucible.  A  similar  Crookes's  layer  is  formed  if  a  quantity 
of  solid  carbonic  dioxide  be  lightly  placed  on  the  tongue ;  the 
extreme  cold  (-80°  C.)  is  not  felt. 

When  the  hot  solid-body  cools  down,  the  Crookes's  layer 
disappears,  the  liquid  suddenly  comes  in  contact  with  the  solid 
still  relatively  hot,  and  the  liquid  explodes  in  vapour.  This 
occurs  in  the  case  of  water  and  iron  at  about  180°  C. 

Melted  copper  can  be  cast  under  water  in  a  canvas  mould ;  and,  singu- 
larly, it  often  remains  fluid  so  long  and  cools  down  so  far  in  that  condition 
that  there  is  no  explosion. 

Increase  of  Temperature.  —  We  have  freely  made  use  of 
the  term  Temperature  because  it  is  a  term  in  common  use,  and 
not  likely,  so  far  as  we  have  used  it,  to  lead  to  ambiguity.  We 
have  still  to  defer  the  consideration  of  thermometry ;  but  we 
must  now  consider  increase  of  temperature  as  directly  due  to 
increase  of  the  molecular  kinetic  energy  of  a  body.  When 
we  double  the  Molecular  Kinetic  Energy  of  a  body 
we  double  its  Temperature. 

Observe  that  it  is  not  asserted  that  we  double  the  temperature  when  we 
double  the  total  energy  of  a  body :  some  may  disappear  in  doing  work,  and 
take  the  form  of  the  so-called  latent  heat. 

This  implies  that  there  must  be  some  point  of  Absolute  Zero 
of  Temperature,  independent  of  the  conventions  of  Fahrenheit, 
Celsius,  and  others,  afterwards  to  be  explained  —  a  point  of 
Absolute  Cold,  beyond  which  no  cooling  is  conceivable. 

We  have  already  seen  that  in  a  perfect  gas  —  one  in  which 
there  is  no  complication  due  to  intermolecular  forces  —  the 
pressure  is  proportional  to  the  molecular  kinetic  energy  of  a 
given  mass,  occupying  a  given  volume ;  the  temperature  is,  or 
may  by  definition  be  held  to  be,  also  proportional  to  this  kinetic 
energy;  it  follows  that  the  Temperature  is  Proportional 
to  the  Pressure  when  the  volume  occupied  by  a  given  mass 
remains  unchanged.  It  is  found  that  in  all  gases  the  pressure 
diminishes  by  about  ^-g-  for  each  Centigrade  degree  of  cooling, 
the  temperature  of  0°  C.  being  the  starting-point,  and  the  vol- 
ume being  maintained  constant.  If  a  gas  could  be  cooled  down 
in  this  way  to  -  273°  C.  (a  feat  unachieved,  -  225°  C.  being 
the  lowest  yet  reached),  it  would  have  no  pressure  and  there- 
fore no  temperature,  for  it  would  have  no  kinetic  energy, 
no  heat.  The  Absolute  Zero  of  temperature  is  therefore 
-  273°  C.  (or  more  accurately  -273°-72  C.),  and  the  Absolute 
Temperature  of  a  body  whose  temperature,  as  measured  by 


xin.]  CHANGE   OF    TEMPERATURE.  365 

the  Centigrade  thermometer  (see  p.  402),  is  t°  C.,  is  r°  Abs. 
=  (273  -f  t)°  Abs. ;  thus  the  boiling  point  of  water,  100°  C.,  is 
373°  Abs. 

The  true  C.G.S.  unit  of  temperature  would  be  the  rise  of  temperature 
produced  by  adding  one  erg  to  the  molecular  kinetic  energy  of  one  gramme 
of  a  perfect  gas  (p.  369)  of  unit  specific  heat,  as  defined  below;  but  this  is 
an  impracticable  unit;  and  the  conventional  "unit  of  temperature,"  the 
Centigrade-thermometer  degree,  or  °  C.,  in  terms  of  which  temperatures  are 
measured,  is  equal  to  41,593000  such  true  C.G.S.  units. 

Specific  Heat.  —  The  heat-energy  of  a  molecule  of  hydro- 
gen is  equal  to  that  of  a  molecule  of  oxygen  at  the  same  tem- 
perature ;  but  the  latter  weighs  sixteen  times  as  much  as  the 
former,  and  a  mass  of  hydrogen  contains  sixteen  times  as  many 
molecules  as  an  equal  mass  of  oxygen  under  similar  physical 
conditions.  Hence  a  given  mass  of  hydrogen  at  a  given  tem- 
perature possesses  sixteen  times  as  much  heat-energy  as  an 
equal  mass  of  oxygen  at  the  same  temperature. 

To  produce  a  given  rise  in  the  temperature  of  a  mass  of 
hydrogen  we  must  supply  sixteen  times  as  much  heat  as  we 
would  find  necessary  to  produce  an  equal  rise  in  temperature 
in  an  equal  mass  of  oxygen  ;  the  capacity  for  heat  —  the  Ther- 
mal Capacity  —  of  hydrogen  is  sixteen  times  that  of  oxygen. 

The  Thermal  Capacity  of  a  Substance  is  the  number  of  units, 
that  is,  the  number  of  ergs  of  Heat-energy  with  which  a  gramme 
of  that  substance  must  be  supplied  in  order  to  raise  its  temper- 
ature by  one  "  unit  of  temperature,"  that  is,  by  1°  C. 

The  Specific  Heat  of  a  substance  is  the  ratio  between  its 
thermal  capacity  and  that  of  water:  and,  since  one  gramme  of 
water  requires  1  ca  to  raise  its  temperature  1°  C.,  the  Specific 
Heat  of  a  substance  is  also,  numerically,  the  Number  of  calo- 
ries of  Heat  required  to  raise  the  temperature  of  one  gramme 
of  that  substance  by  1°  C. 

If  the  specific  heat  of  a  substance  be  <r,  its  thermal  capacity 
will  be  41,593000o-  ergs  per  gramme  (or  41,593000/30-  ergs  per 
cub.  cm.). 

The  ratio  between  the  Specific  Heats  of  two  substances  is 
the  same  as  that  between  their  Thermal  Capacities. 

The  « thermal  capacity  of  a  body,'  such  as  the  earth  or  a  lump  of  copper, 
is  ma-  co,  or  41,593000  mo-  ergs,  per  °  C. 

In  general,  the  lighter  the  molecules  of  which  a  substance 
is  made  up,  the  more  numerous  must  they  be  in  argiven  mass, 
and  the  higher  the  thermal  capacity  of  the  substance,  i.e.  the 


366  HEAT.  [CHAP. 

more  heat  must  be  expended  upon  it  in  producing  a  given  rise 
of  temperature. 

The  law  here  indicated — that  the  specific  heat  of  an  element 
varies  inversely  as  its  atomic  weight  —  is  based  on  the  assump- 
tion that  a  mass  of  a  heated  substance  behaves  like  a  group  of 
isolated  molecules  which  have  no  action  on  one  another.  It  is 
not  surprising,  accordingly,  to  find,  when  heat  supplied  to  a 
body  is  spent  not  only  in  raising  the  temperature  of  a  body,  but 
also  in  doing  internal  and  external  work,  that  the  law  is  only 
approximately  obeyed.  Still,  the  approximate  obedience  is  suffi- 
ciently striking  to  have  caused  Dulong  and  Petit  to  enounce  it 
as  a  law,  and  as  such  it  bears  their  name.  It  has  been  utilised 
as  one  means  among  others  of  ascertaining  the  atomic  weight  of 
different  elements. 

1   " 

For  the  formula  "  sp.  heat  <x we  may  substitute  sp. 

at.  wt. 

heat  = — ;    or   sp.   heat  x  at.   wt.  =  const.      This    constant 

at.  wt. 

product,  which  bears  the  name  of  Atomic  Heat,  is  about  6-4; 
the  metals,  phosphorus,  sulphur,  may  be  said  to  form  a  group 
in  which  it  varies  from  5-86  to  6-93.  Divergences  from  the 
average  value  are  most  marked  in  the  case  of  elements  whose 
atomic  weight  is  below  30.  In  the  case  of  carbon,  silicon,  and 
boron  at  ordinary  temperatures  the  product  is  small,  being  1-8 
to  2-8,  3-9  to  4-2,  and  2-5  to  2-7  respectively;  but  at  higher 
temperatures  the  specific  heat  of  these  substances  increases  so 
that  the  product  rises  to  about  5'5. 

Hydrogen  possesses  a  higher  thermal  capacity  than  water;  its  sp. 
heat  at  constant  volume  (p.  367)  is  2411,  and  that  at  constant  pressure 
is  3-409  calories  per  gramme. 

The  Molecular  Heat  of  a  compound  —  i.e.  the  product 
of  its  sp.  heat  into  its  molecular  weight  —  is  approximately 
equal  to  the  sum  of  the  atomic  heats  of  its  component 
elements,  reckoned  with  reference  to  each  component  atom  sepa- 
rately. This  rule  applies  with  tolerable  accuracy  to  gaseous 
compounds  formed  without  condensation  ;  solid  and  liquid  com- 
pounds, and  even  gaseous  compounds  whose  formation  from 
their  elements  is  accompanied  by  condensation,  —  compounds 
which  in  some  sense  approximate  to  the  liquid  state,  —  depart 
from  it  in  a  marked  degree. 

This  product  —  the  atomic  heat  of  elements,  the  molecular 
heat  of  compounds  —  has  the  following  physical  meaning.  Of 


xiii.]  MOLECULAR   AND   ATOMIC   HEAT.  367 

any  substance  whose  atomic  or'  molecular  weight  we  know,  we 
may  take  a  number  of  grammes  numerically  equal  to  the  atomic 
or  molecular  weight ;  for  example,  35-5  grammes  of  chlorine, 
16  grammes  of  marsh  gas ;  we  may  call  such  a  quantity  the 
gramme-atom  or  the  gramme-molecule  of  the  sub- 
stance. The  Atomic  Heat  or  the  Molecular  Heat  of  a  substance 
is  the  number  of  calories  of  heat  necessary  to  raise  the  tempera- 
ture of  a  gramme-atom  or  of  a  gramme-molecule  of  the  substance 
through  1°  C.  The  atomic  heats  of  elementary  substances 
are  approximately  the  same  —  another  form  of  Dulong  and 
Petit's  law ;  so  are  the  molecular  heats  of  substances  of  similar 
composition. 

The  specific  heat  of  a  substance  determines  the  temperature 
which  it  will  assume  when  a  definite  quantity  of  heat  is  supplied 
to  it  or  liberated  in  it. 

Thus  when  1  gramme  of  hydrogen  and  8  of  oxygen  are  exploded 
together,  but  are  not  allowed  to  expand  in  volume,  28,580  ca  of  heat  are 
liberated.  If  we  could  assume  the  action  to  be  instantaneous,  we  might 
assume  that  none  of  the  heat  is  lost.  The  28,580  ca  would  then  be  divided 
among  9  grammes  of  water-vapour  whose  sp.  heat  at  constant  volume  is 
0-37;  the  temperature  attained  would  be  gfg  =  8883°  C.  above  the  tem- 
perature (136°'5  C.)  proper  to  a  volume  equal  to  the  original  volume  of 
the  mixture.  This  case  is  instructive  as  showing  the  influence  of  disso- 
ciation ;  for  when  a  temperature  of  3000°  C.  is  actually  attained,  further 
combination  becomes  impossible  and  the  action  is  arrested,  but  not  wholly, 
for  it  is  gradually  completed  pari  passu  with  the  loss  of  heat  by  conduction 
or  by  radiation.  If,  however,  the  exploding  mixture  be  allowed  to  expand, 
doing  external  work,  the  temperature  of  3000°  may  never  be  attained.  The 
possible  tempe.rature  is  also  lowered  by  a  progressive  increase  in  the  specific 
heat  of  the  products  as  the  temperature  rises. 

Where  a  substance  while  being  heated  is  not  allowed  to 
expand,  there  is  probably  no  internal  work  done  ;  neither  is  there 
any  external  work  done ;  all  the  heat  supplied  is  applied  in 
raising  the  temperature.  The  thermal  capacity  in  this  case  is 
specially  known  as  c  the  thermal  capacity  at  constant  volume. 
If,  however,  the  substance  be  allowed  to  expand  while  it  is  being 
heated,  an  external  pressure  being  maintained,  both  external  and 
internal  work  are  done,  and  in  order  to  effect  a  given  increase  of 
temperature  more  heat-energy  is  required  than  in  the  former 
case.  The  thermal  capacity,  k,  of  any  particular  gas  under  a 
constant  pressure  is  therefore  greater  than  the  thermal  capac- 
ity, e,  at  constant  volume ;  and  it  is  found,  in  the  case  of  air,  to 
exceed  it  in  the  ratio  of  14058  : 1,  whatever  be  fhe  external 
pressure,  so  long  as  that  is  maintained  constant. 


368  HEAT.  [CHAP. 

That  k  (in  ergs  per  gramme)  does  not  depend  upon  p,  follows  from 
the  equations  kmr  =  cmr  +  pij  (p.  370,  note),  and  joij  =  I&WT  (p.  370). 

The  ratio,  =  k/c,  of  the  thermal  capacity  of  a  gas  under  any  constant  ex- 
ternal pressure,  to  that  at  constant  volume,  may  be  found  in  two  ways. 

I.  Let  a  gas  suddenly  exchange  its  pressure  p,  its  density  p,  and  its 
temperature  T°,  on  the  Absolute  (centigrade-degree)  Scale,  p.  364,  for  others 
Pi">  P/»  T/°>  tne  ratio  of  its  thermal  capacities  being  =  k/c ;  then  from  the 
adiabatic  equation  p/p,  =  (p/pt)k/c  (see  p.  395,  footnote),  and  the  equations 
p  =  2&-/DT  and/),  =  3&-  p,Tt  (see  page  370),  we  find  that,  for  any  given  kind 
of  gas,  k/c  ={log  (p/p,)  +  log  (p/p,}}  (i)  ;  or,  since  p/p,  =  pr/p^,,  k/c  = 
log  (pT/p,T,)-5-  log  (p/p,)  =  (log  (T/T,)  -i-  log  (p/p/)}  +  1  (ii)  \  or  again,  since 
(  P/P,}  =  (P/P,Y/C=  (»pT/»PyT,  •  T,/T)*/«  =  (pr./p.T)*/',  k/c  =  {log(;«-,//>,T) 
•*•  l°g  (P/P/)}  (iii)-  Hence  if  two  of  the  changes  />  to  ;?y,  p  to  p,,  r  to  rt,  can 
be  found,  the  value  of  k/c  may  be  calculated,  on  the  assumption  that  the 
gas  is  a  perfect  gas.  The  experimental  adiabatism  necessary  is  very  diffi- 
cult to  ensure;  yet  Rontgen  has  performed  the  following  series  of  operations 
upon  known  quantities  of  air  and  determined  the  value  k/c  =  14053. 

1.  Air  in  a  reservoir  at  a  pressure  p  dynes  per  sq.  cm.,  exceeding  the 
atmospheric,  at  density  p,  and  temperature  T°  Abs. 

2.  Open  a  stopcock  :  air  rushes  out  of  the  reservoir  until  the  pressure  p 
falls  to  n,  the  atmospheric  pressure,  per  sq.  cm. 

3.  Immediately  close  the  stopcock.     The  air  within  the  reservoir  is  at 
pressure  n,  but  has  been  cooled  by  doing  external  work  during  expansion  ; 
it  soon  comes  to  the  same  temperature  as  surrounding  objects  —  that  is, 
again  T°  Abs. :  it  now  has  the  pressure  pf  and  the  density  p,,  which  can  be 
found  at  leisure,  and  the  above  formulae  applied. 

II.  From  the  velocity  of  sound.  This  is,  in  air,  33,200  cm.  per  sec. 
Newton's  law  of  the  velocity  of  propagation  of  waves  is  that  v  =  Vft/p. 
The  coefficient  of  elasticity  ft  is  equal  numerically  to  the  pressure  p  in  a  gas 
if  the  temperature  be  constant;  .-.  ft  =  Vp/p  =  Vn/p  if  the  pressure  be  the 
atmospheric.  For  air  p  =  0-0012932  grms.  per  cub.  cm.  at  0°  C.  and  76  cm. 
bar.  pr.  at  Paris ;  n  =  1,013663  dynes  per  sq.  cm.  Hence,  according  to 

Newton's  law,  v  =  y '°j^^  =  27,997  cm.  per  sec. ;  but  in  fact  it  is  found 

to  be  33,200  cm.  There  is  here  to  all  appearance  a  material  divergence 
from  Newton's  law ;  but  it  is  explained  when  we  observe  that  the  assumption 
that  the  temperature  is  constant  is  unfounded ;  that  a  travelling  wave  of 
sound  subjects  the  air  to  adiabatic  compression  —  adiabatic*  because  the 
heat  has  not  time  to  become  diffused ;  that  the  elasticity  of  air  so  com- 
pressed is  greater  than  that  of  air  maintained  at  a  constant  temperature ; 
that  the  ratio  of  these  two  elasticities  of  a  gas  is  otherwise  known  (p.  324) 
to  be  the  same  as  the  ratio  of  their  specific  heats  at  constant  pressure  and 
at  constant  volume ;  and  therefore  that  the  coefficient  of  elasticity  in  the 
formula  should  have  been,  not  ft  the  elasticity  at  const,  temp.,  but  ft'  the 
elasticity  under  adiabatic  compression,  =  k/c  x  ft.  Whence  v  =  Vft'/p  = 
Vkf^tTp  I  33,200  =  27,997  ^Jkjc  ;  k/c  =  1-40622.  .; 
The  mean  value  of  k/c  is  thus  1-4058,  for  air. 

*If  the  heat  produced  had  time  to  become  diffused,  or  if,  as  might  be  the  case  in 
excessively  slow  vibrations  or  rare  gases,  the  gas  had  time  to  flow  round  the 
vibrating  object,  so  that  it  could  not  become  compressed  and  evolve  heat,  the  speed  of 
propagation  would  tend  to  approximate  to  the  value 


xm.]  SPECIFIC    HEAT.  369 

Differences  in  the  ratio  between  the  specific  heats.  —  There  is  a 
wide  range  of  difference  between  the  ratios  observed  in  different  gases  ;  k/c 
=  1-66  in  mercury-vapour  (Kundt)  ;  k/c  =  1-03  in  oil-of-turpentine  vapour. 
The  reason  of  this  is  the  following :  —  Heat  communicated  to  a  gas  at  a 
constant  external  pressure  does  at  least  three  things  ;  (1)  it  is  expended  in 
doing  external  work ;  (2)  it  increases  the  molecular-translational  kinetic 
energy;  (3)  it  increases  the  rotational  and  other  intramolecular  kinetic 
energy.  The  first  of  these  is  limited  ;  it  does  not  depend  on  the  nature  of 
the  gas  undergoing  a  given  expansion  ;  it  is  equal  to  the  constant  external 
pressure/)  resisted,  in  to  the  increase  of  volume,  b;  i-e.  it  is/>b.  If  we  take  a 
gramme  of  hydrogen  at  0°  C.  and  76  cm.  bar.  pr.,  we  find  that  it  occupies 
11,164-5  cub.  cm.  On  being  heated  through  1°  C.,  it  expands  by  ^  of  its 
volume;  hence  fc  —  (11,164-5  -r-  273)  cub.  cm.  The  atmospheric  pressure,  at 
76  cm.  of  mercury,  is  1,013663  dynes  per  sq.  cm.  Accordingly,  pit  = 
(11,164.5  -5-  273  x  1,013663)  =  41,454000  ergs  =  very  nearly  1  ca  per  °  C.,  per 
gramme  of  hydrogen. 

The  second  term,  the  increment  of  molecular-translational  kinetic 
energy,  cannot  exceed  this  more  than  50  per  cent ;  for  (p.  249)  p'o  =  f  of 
the  whole  of  this  kinetic  energy ;  and  similarly,  pij  =  f  the  increment  of 
this  translational  energy,  which  will  therefore  be,  very  nearly,  I1  ca  per 
0  C.,  per  gramme  of  hydrogen. 

If  hydrogen  were  a  substance  whose  molecules  did  not  rotate  or  vibrate, 
li  ca  per  degree  C.  would  be  an  amount  of  energy  sufficient  to  impart  the 
necessary  increment  of  translational  energy  to  1  gramme  of  hydrogen 
imprisoned  within  any  given  space ;  and  the  specific  heat  of  hydrogen  at 
constant  volume  would  be  1|.  If  the  one  gramme  of  hydrogen  wrere 
allowed  to  expand  under  constant  atmospheric  pressure,  another  ca  per 
degree  C.  would  be  required  in  order  to  thrust  away  the  air,  and  the  specific 
heat  under  any  constant  external  pressure  would  be  2|.  The  ratio  k/c 
would  then  be  2£  -s- 1|,  or  1-66.  But  the  specific  heat  under  a  constant 
external  pressure  is,  in  hydrogen,  more  than  2-5  ca  per  gramme ;  it  is 
3-409  :  that  at  constant  volume  is  more  than  1-5,  being  2-411.  The  excess  is 
due  to  intra-  and  inter-molecular  work ;  and  this  work,  which  differs  very 
much  from  one  substance  to  another,  but  which  is  approximately  propor- 
tional in  any  one  substance  to  the  translational  kinetic  energy,  so  long  at 
least  as  the  specific  heat  remains  constant,  causes  the  ratio  k/c  to  be  not 
1-66,  but  3-409  -  2-411  =  1-414. 

An  equal  bulk  of  oxygen,  11,164-5  cub.  cm.,  which  ought  to  take  up  the 
same  amount  of  heat  as  the  same  volume  of  hydrogen,  takes  up  3-480  and 
2-4816  ca  respectively;  k/c  —  1-402.  An  equal  bulk  of  benzol-vapour  takes 
up  14-64  and  13-65  ca  respectively,  and  k/c  =  1-073 :  ether-vapour,  17-75  and 
16-76  ca,  and  k/c  =  1-058  :  turpentine,  34-4  and  33-4  ca,  and  k/c  =  1-03.  In 
the  last  example,  any  heat  which  is  communicated  to  the  vapour  is  used  to 
the  extent  of  more  than  nineteen-twentieths  in  doing  work  upon  the  com- 
plex molecules  themselves,  or  in  altering  their  mutual  relations.  In  sharp 
contrast  to  this  we  have  the  vapour  of  mercury,  whose  low  specific  heat  and 
ratio  k/c  =  1-66  point  towards  extreme  simplicity  of  the  molecule,  which  is, 
on  chemical  grounds,  otherwise  believed  to  be  monatomic. 

In  a  perfect  gas  —  one  whose  molecules  did  not  act  upon 
one  another  —  the  thermal  capacity  at  constant  vorlume  would 
be  quite  independent  of  the  temperature  or  of  the  pressure. 

2s 


370  HEAT.  [CHAP. 

In  air  the  specific  heat  is  sensibly,  though  not  perfectly,  constant  at  all 
temperatures  between  —  30°  C.  and  +  225°  C.,  and  at  pressures  from  1  to  10 
atmospheres.  We  shall  see  that  this  justifies  us  in  relying  upon  the  indi- 
cations of  the  air  thermometer.  In  carbonic  acid  it  increases  with  the 
temperature,  becoming  about  doubled  at  2000°  C. 

In  a  perfect  gas  the  pressure  at  constant  volume  and  the 
volume  under  constant  pressure  would  both  vary  directly  as  the 
Absolute  temperature. 

The  general  law  is,  that  job  oc  mr,  or  job  =  2&mr,  or,  whatever  may  be  the 
volume,  p  —  2&  •  pr,  where  B  is  a  Constant  ;  this  constant  is  numerically 
equal*  to  the  Difference  between  the  two  Thermal  Capacities  of  the 
particular  gas,  at  constant  external  pressure  and  at  constant  volume  respec- 
tively, and  measured  in  ergs  per  gramme. 

In  the  case  of  air,  p  =  3&  •  pr  =  (k  —  c)  •  pr  =  04058  cpr,  where  c  is  the 
Thermal  Capacity  at  constant  volume,  measured  in  ergs  per  gramme.  Hence 
c  =  (p  -H  0-4058pr)  ergs  per  gramme  ;  or  the  Specific  Heat  at  constant  volume 
-  {(p  +  0-4058pr)-f-  41,593000}  ca  per  gramme. 

When  p  =  u  =  1,013663  dynes  per  sq.  cm.,  the  density  p  of  atmospheric 
air  is  (1  •*•  773-2833)  at  0°  C.  or  273°-72  Abs.  Then  the  thermal  capacity 
at  const,  vol.,  c  =  {1,013663  -  (04058  -773-2833  x  273-72)}  ergs,  and  the 
specific  heat  of  air  at  const,  vol.  =  0-1696  ca,  per  °C.  per  gramme.  The 
observed  value  of  the  specific  heat  at  const,  vol.  is  0-1684  ca  per  °C.  per 
gramme,  which  corresponds  to  41,731000  ergs  per  ca. 

When  a  gas  is  compressed  it  becomes  heated  —  that  is,  pro- 
vided that  external  pressure  have  produced  the  compression,  and 
added  energy  to  the  gas  by  doing  work  upon  it. 

When  a  gas  is  allowed  to  expand  it  becomes  cool  —  that  is, 
provided  it  expand  against  external  pressure  and  sacrifice  energy 
by  doing  external  work. 

The  Work  done  upon  or  by  the  gas  appears  or  is  lost  as 
Haat.  The  rise  of  temperature  may  be  calculated,  on  the 
express  assumption  that  there  is  no  internal  work  done  —  an 
assumption  approximately  but  not  perfectly  true  (Joule)  —  by 

*  Assume  a  given  mass  of  gas  (m  grammes)  to  be  maintained  under  a  constant 
external  pressure  p  ;  then,  Heat  imparted  in  raising  its  temperature  r  through  the 
small  increment  f  is  k  •  mr  ergs  ;  and  this  is  equal  to  c  •  m  •  f  (mere  heating,  as  if  at 
constant  volume)+pii  (external  work  done)  :  k-  m  •  r  =  c  •  mr+pfo.  But  p  =  &  •  mr/ij  ; 
hence  k-  mr  =  c-mf  +  ia-  rnr  •  ij/fo  ;  or  (k  —  c)  •  r  /r  =  la  .  ft/ft.  But,  at  constant 
pressure,  changes  of  volume  are,  in  perfect  gases,  proportional  to  changes  of  absolute 
temperature;  hence  f/T  =  {j/jj.  Therefore  (k  —  c)  =  3R,  the  Thermodyuamic  Con- 
stant. 

It  is  assumed  in  the  above  that  no  internal  work  is  done  during  the  expansion  of 
a  gas,  or  that  the  Latent  Heat  of  Expansion,  L  (p.  377)  ,  =  0.  If  the  gas  be  not  perfect, 
or  if  L  otherwise  not  =  0,  the  heat  supplied  to  mass  m  is  kmt  =  craf 


It  may  be  explained  that  as  on  p.  324,  so  in  this  chapter,  r,p,  ij,  etc.,  are  small 
actual  increments,  not  increments  per  unit  of  time.  For  these  symbols  the  student 
may  substitute  ST,  dp,  5b,  etc.  ;  and  he  may  read  them  as  'the  change  of  tempera- 
ature,'  etc.,  positive  or  negative  as  the  case  may  be. 


xiii.]  SPECIFIC   HEAT.  371 

dividing  the  whole  work  done  on  the  gas  (measured  in  terms 
of  calories  of  heat)  by  the  mass  and  by  the  specific  heat  of  the 
gas  :  or  the  work  measured  in  ergs,  by  the  mass  and  by  the 
thermal  capacity. 

.  Saturated  vapour  behaves  in  this  regard  in  a  peculiar  manner.  If  work 
be  done  upon  saturated  steam  at  any  temperature  below  789°-8  Abs. 
(51G°-8  C.),  the  heat  evolved  causes  the  vapour  to  become  a  superheated 
vapour,  and  heat  must  be  parted  with  in  order  to  allow  the  steam  to  remain 
saturated.  Conversely,  if  saturated  steam  below  516°-8  C.  be  allowed  to 
expand,  doing  external  work  while  no  heat  is  supplied  to  it,  it  loses  energy, 
loses  latent  heat,  and  is  partly  condensed  ;  and  it  does  not  fall  in  tempera- 
ture during  expansion  as  'much  as  it  would  do  if  it  were  a  perfect  gas 
expanding  to  the  same  extent,  for  the  liquefaction  of  the  vapour  liberates 
heat.  Thus  an  expanding  saturated  vapour,  such  as  steam,  liberates  more 
energy  and  can  do  more  work  than  an  expanding  gas.  Above  5160>8  C.  a 
sudden  adiabatic  expansion  of  saturated  steam  would,  on  the  other  hand, 
produce  evaporation  of  water  in  contact  with  it  ;  and  compression  would  pro- 
duce condensation. 

The  above  facts  are  comprised  in  the  equation  —  k  being  the  thermal 
capacity  and  a-  the  specific  heat  of  saturated  steam  (caused  to  rise  in  tempera- 
ture as  saturated  steam,  without  superheating)  at  constant  external  pressure, 
T  the  Abs.  temperature  — 

k  =  (42,136000  -  33'280'000000)  ergs  per  gramme  ; 
33,280,000000 


___      m  per  gramme. 

Below  789°-8  Abs.  <r  is  negative;  above  that  temperature  it  is  positive. 

The  vapour  of  bisulphide  of  carbon  acts  at  ordinary  temperatures  like 
that  of  water  below  516°-8  C.  ;  that  of  ether,  on  the  other  hand,  is  rendered 
cloudy  by  compression  even  at  ordinary  temperatures. 

The  specific  heat  of  substances  is  not  perfectly  constant 
at  all  temperatures  :  whence  the  necessity  of  the  qualification 
"  from  0°  to  1°  C."  This  want  of  constancy  is,  among  gases, 
most  remarkable  in  those  which  are  most  condensible  ;  but 
among  solids  and  liquids  the  variations  of  specific  heat  are  still 
more  remarkable,  and  indicate  differences  in  the  amount  of 
internal  work  associated  with  changes  of  temperature  at  differ- 
ent temperatures,  this  internal  work  being  done  in  effecting 
changes  in  the  density,  'the  intermolecular  stresses,  the  allotropic 
form,  and  so  on. 

The  thermal  capacity  of  a  body  may  be  expressed  by  the  fraction  — 

Increment  of  Heat  supplied  to  unit-mass 
Increment  of  Temperature  produced 

where  both  the  increments  are  very  small  :  if  an  amount  of  heat,  8H  ergs, 
produce  a  change  of  temperature  St,  the  thermal  capacity  is  8H/&  /  and  this 


372  HEAT.  [CHAP. 

is  one  of  the  Specific  thermal  capacities  of  a  body,  of  which  six  may  be 
distinguished. 

1.  Specific  thermal  capacity  per  unit-increase  of  temperature  at  constant 

pressure  ;  the  amount  of  heat  required  to  raise  the  temperature  of 
unit-mass  by  1°  C.,  the  pressure  being  constant,  whatever  may  be 
its  amount.  This  is  called  the  Thermal  Capacity  at  Con- 
stant Pressure,  &  ergs  per  gramme. 

2.  Specific  thermal  capacity  per  unit-increase  of   pressure  effected,  the 

temperature  being  constant.     This  has  no  special  name. 

3.  Specific  thermal  capacity  per  unit-increase  of  pressure  effected  by  heat, 

at  constant  volume.     This  has  no  other  name. 

4.  Specific  thermal  capacity  per  unit-increase  of  volume,  the  pressure 

being  constant. 

•  5.   Specific  thermal  capacity  per  unit-increase  of  temperature  at  constant 

volume  (the  Thermal  Capacity  at  Constant  Volume,  c). 
6.    Specific  thermal  capacity  per  unit  expansion  per  gramme,  the  tempera- 
ture being  constant.    This  is  called  the  Latent  Heat  of  Expan- 
sion, L;  and  it  is  usually  specified  as  a  certain  number  of  calories, 
not  of  ergs,  per  gramme.     We  adhere,  however,  to  ergs. 

Under  No.  5,  heat  =  m  •  c  ergs,  supplied  to  a  mass  m,  produces  a  unit-increase 
of  temperature ;  a  rise  of  temperature  f  is  produced  by  heat  =  m  •  cr  ergs. 
Under  No.  6,  in  the  same  way,  heat  =  m  -  Lfc/ij  ergs  produces  a  proportionate 
expansion  tr/ij.  We  commit  no  sensible  error  if  we  suppose  that  when  the 
temperature  and  volume  both  vary,  the  amount  of  heat  which  must  be 
supplied  to  a  mass  m  of  any  substance  is  found  by  simple  addition,  and 
is  equal,  if  there  be  any  external  work,  pij  ergs,  done  during  expansion,  to 
{m(cT  +  Lb/ij)  +  pb}  ergs. 

Internal  Work.  —  If  any  substance  were  a  perfect  gas,  heat 
imparted  to  it  would  to  no  extent  be  spent  in  doing  internal 
work  against  intermolecular  or  intramolecular  forces. 

In  that  case  the  latent  heat  of  expansion,  L,  =  0. 

There  is,  however,  no  such  perfect  gas,  as  we  shall  now 
show. 

If  our  physical  gases  were  perfect  gases  we  would  find  — 

1.  That  the  amount  of  heat  evolved  on  compressing  a  gas 
would  be  exactly  equal  (when  measured  in  ergs)  to  the  work 
done  in  compressing  the  gas. 

If  the  original  pressure  and  volume  be  jt>0,  ij0,  and  the  new  volume  ij7, 
the  work  done  is  jt?0ij0  log  (b7/ij3)  =  p  t>3  log  (p0/p/)  :  a  conclusion  deduced, 
on  the  assumption  that  the  heat  evolved  is  withdrawn  at  once,  so  that  there 
is  no  rise  in  temperature,  from  the  two  equations  —  • 

/•&, 
(1)  work  done  =  I  pclti',  and  (2)  j»0ij0  =p,'o/  (Boyle's  Law). 

*\ 

2.  That  when  a  gas  expands,  doing  external  work,  the  gas 
loses  energy ;  and  that  a  perfect  gas  would  in  this  way  lose  heat 


XIII.] 


PERFECT   GASES. 


373 


exactly  equal  in  amount  to  the  external  work  done,  and  would 
accordingly  sink  in  temperature.     *   . 

A  vessel,  A,  of  compressed  air  (Fig.  125)  is  provided  with  an  exit  tube 


Fig.126. 


furnished  with  a  stopcock :  the  extremity  of  the  exit  tube  dips  under  water 
in  a  bell  jar  B.  The  stopcock  is  opened  ;  air  flows  out ;  it  replaces  the  water 
in  the  bell  jar :  in  so  doing  it  forces  water  down  against  the 
atmospheric  pressure  :  it  thus  does  work  ;  the  air  remaining 
in  A  becomes  cold  (Joule). 

A  similar  vessel  of  compressed  air  (Fig.  126) ;  the  extrem- 
ity of  the  exit  tube  communicates  with  the  open  air.  The 
stopcock  is  opened;  air  flows  out;  it  thrusts  aside  the  air 
immediately  surrounding  the  orifice ;  the  air  within  A  thus 
does  work  against  the  atmospheric  pressure  :  the  air  remain- 
ing in  A  becomes  cold.* 

3.  That  if  a  stream  of  a  perfect  gas  were 
checked,  the  whole  kinetic  energy  lost  by  the  gas 
would  appear  as  heat  in  it. 

The  heating  effect  of  checking  a  stream  of  gas  may  be  readily  shown 
(Yerdet)  by  pinching  a  rapidly-issuing  jet  of  air  between  the  finger  and 
thumb,  or  by  partly  blocking  it  with  the  finger-tip. 

*  The  expansion  here  is  nearly  adiabatic ;  let  us  assume  it  to  be  perfectly  so,  and 
the  air  to  be  a  perfect  gas,  so  that  pp.  /p,p  =  T/T,,  from  the  equations  p  =  2Spr  and 
p  =&O(T(.  The  law  of  adiabatic  expansion  is  (p.  393)  p/p.  =(0/P,)k/c;  whence  (p, //>) 
=-(p //>,)-«/*.  Then  T/Tl={(p/lSip)  -=-  (pl/Kp,)}=pp,/plp=(p/p,)(p/pl)-c/*  = 
(p/p,)(*-cV*.  Let  the  initial  and  final  pressures,  p  and  p.,  be  6  atmos.  and  1  atmo., 
so  that  p/p,  =6;  let  the  initial  and  final  temperatures  be  0°  C.  (r=  273°  Abs.)  and  the 
unknown  r°  Abs.;  and  for  air,  &/c  =  1-4058.  From  these  data  we  find  273/r, 

=  G1^  =  (>°-2886,  or  log  (273 /T,)  =  0'2886  log  6  =  0' 16252  =  log  1-4533;  whence  273/r( 
=  1-4533,  or  T.  =  273  +  1-4533  =  187°'8  Abs. ;  this  is  -86°  C.,  the  temperature  of  the 
residual  air  in  the  vessel  A:  not  that  of  the  escaping  air,  which  reconverts  some  "of 
its  kinetic  energy  into  heat  by  friction. 

It  may  be  noted  here  that  if  instead  of  expansion  we  have  compression,  the  cal- 
culation is  of  precisely  the  same  kind  as  the  preceding ;  if,  for  example,  the  initial 
and  final  pressures  be  1  atmo.  and  6  atmos.,  so  that  p/p,  =  1,  instead  of  6  as  above, 
the  result  is  that  the  final  temperature  is  617°  C. 


374 


HEAT. 


[CHAP. 


In  the  same  way  a  jet  of  high-pressure  steam,  when  liberated  into  the 
free  air,  suddenly  expands  and  partly  condenses  into  scalding  droplets  ;  then 
a  little  way  farther  on,  by  reason  of  friction  against  the  air  and  of  intermo- 
lecular  friction,  it  is  deprived  of  its  momentum,  and  is  heated  so  far  as  to 
become  superheated  or  gaseous  steam,  in  which  condition  it  will  rapidly 
dry  (and  even  cool)  any  moist  surface  on  which  it  plays ;  still  farther  on,  it 
again  becomes  opaque,  and  is  then  scalding  steam. 

If  the  vessel  A  of  Figs.  125  and  126  be  connected  with  another  in  which 
a  vacuum  has  been  produced,  the  air  in  A  loses  energy  and  is  cooled.  The 
part  of  the  gas  which  first  arrives  in  B  is  heated  by  compression  exercised 
by  the  part  which  arrives  afterwards ;  the  latter  is  also  heated  by  having  its 
motion  checked :  the  temperature  in  B  thus  becomes  higher  than  the  origi- 
nal temperature.  The  result  is  as  if,  of  the  particles  in  A,  those  possessed 
of  the  higher  translatory  velocities  had  escaped  into  B  (Natanson). 

4.  That  expansion  of  a  perfect  gas  would  not,  if  no  exter- 
nal work  were  done,  affect  its  mean  temperature  :  for,  no  inter- 
nal and  no  external  work  being  done,  the  amount  of  kinetic 
energy  possessed  by  the  gas  would  remain  unaltered,  and  the 
mean  temperature  would  be  unchanged.  There  is  no  gas  whose 
mean  temperature  remains  unaffected  under  such  circumstances ; 
therefore  there  is  no  perfect  gas. 

The  apparatus  of  Fig.  127  being  immersed  in  a  large  vessel  of  water,  the 
stopcock  is  opened  ;  the  air  in  A  is  cooled,  that  in  B  is  warmed ;  the  amount 

of  heat-energy  gained  by 
B  is  equal  to  that  lost  by 
A :  the  water  surround- 
ing A  and  B  (which  must 
be  stirred)  is  not  on  the 
whole  perceptibly  cooled 
or  warmed.  This  experi- 
ment, made  by  Joule,  was 
believed  to  show  that  air 
did  behave  approximately 
as  a  perfect  gas ;  for  the 
temperature  of  the  water, 
and  therefore  the  average 
temperature  of  the  whole  gas  in  A  and  B,  remained  unchanged  after  open- 
ing the  stopcock. 

The  objection  to  this  experiment  is,  that  a  rise  or  fall  of  temperature  in 
the  gas,  even  though  by  no  means  insignificant,  would  under  such  circum- 
stances be  imperceptible.  The  mass  of  water  surrounding  the  vessels  A  and 
B  cannot  be  made  much  less  than  about  7  kilogrammes :  the  specific  heat 
of  water  is  high,  that  of  air  is  low ;  and,  besides,  it  is  desirable  that  the 

experiment  be  continuous,  and  that 

Fig.128.  the  effects,  if  there  be  any,  be  accu- 
mulated. 

Hence  a  new  form  of  the  exper- 
iment was  devised  by  Joule   and 
Thomson  (Lord  Kelvin).     A  tube 
obstructed  by  a  diaphragm  with  a  narrow  orifice  takes  the  place  of  the 


Fig. 127. 


xm.]  NO   PERFECT   GASES.  375 

vessels  A  and  B.  Air  is  forced  from  A,  Fig.  128,  towards  B.  The  pres- 
sure within  A  is  greater  than  that  within  B ;  the  gas  which  passes  into  B 
ultimately  becomes  simply  the  same  gas  with  a  larger  volume :  it  cannot 
become  cooler  by  reason  merely  of  its  thrusting  the  exterior  air  at  C  out  of 
the  tube,  for  it  simply  acts  as  a  buffer  between  the  air  in  A  and  the  exterior 
air  at  C,  and  the  exterior  work  which  it  does  is  equal  to  that  done  upon  it. 
If  the  air  were  a  perfect  gas,  the  temperature  at  B  would  be  the  same  as 
that  at  A.  It,  is  found  in  such  apparatus  to  vary  from  spot  to  spot  on  account 
of  eddies ;  these  must  be  got  rid  of.  This  is  done  by  substituting  for  the 
diaphragm  with  the  single  opening  a  porous  plug  of  graphite  or  of  cotton 
wool.  It  is  then  found  that  air  is  not  a  perfect  gas;  the  temperature 
in  B  is  a  little  lower  than  that  in  A.  Energy  has  been  consumed  in  doing 
internal  work — probably  in  separating  the  particles  of  the  gas  —  to  the 
extent,  when  the  pressures  in  A  and  B  both  differ  little  from  the  atmospheric 
pressure,  of  about  ¥^7  of  the  whole  work  spent  upon  the  gas  in  forcing  it 
through  the  plug.  The  proportion  of  the  total  energy  spent  in  doing  internal 
work  varies  from  substance  to  substance,  and  from  condition  to  condition. 
In  carbonic  dioxide,  at  a  pressure  varying  little  either  in  A  or  B  from  the 
atmospheric  pressure,  it  amounts  to  about  ^5  in  air  at  a  pressure  in  A  of 
19  atmospheres,  it  amounts  to  as  much  as  ^3  of  the  whole. 

In  the  case  of  hydrogen,  curiously,  there  is  a  slight  increase  of  tempera- 
ture :  the  expanded  gas  has  more  kinetic  energy  than  the  unexpanded  gas : 
energy  is  liberated  when  hydrogen  expands ;  its  particles  seem  to  repel  one 
another. 

At  equal  temperatures,  therefore,  compressed  air  contains  less  intrinsic 
inter  molecular  potential  energy  than  an  equal  mass  of  rarer  air;  compressed 
hydrogen  the  reverse. 

High-pressure  steam,  treated  after  the  fashion  of  Fig.  127,  becomes 
superheated  or  gaseous  steam. 

Van  der  Waals's  Law.  —  Van  der  Waals  has  been  able  to  get  the 
departures  from  Boyle's  law,  presented  by  various  gases,  approximately 
dealt  with  by  using,  in  the  formula  job  =  2&r  (where  the  quantity  of  gas 
referred  to  is  one  gramme),  instead  of  the  observed  pressure  p  a  quantity 
(p  +  a/b2),  and  instead  of  the  observed  volume  b  a  quantity  (b  —  b).  Then 
the  equation  job  =  I&T  becomes  (p  +  «/b2)  •  (b  —  ft)  =  I&T.  The  former  of 
these  terms  is  the  theoretical  value  of  the  pressure,  that  is,  the  sum  of  the 
observed  pressure  p  and  a  mutual  attraction  which  varies  directly  as  the 
square  of  the  density,  and  is  most  observable  when  the  gas  is  approaching 
liquefaction ;  in  the  latter,  &  is  4  times  the  volume  (per  gramme-mass) 
occupied  by  the  molecules  themselves. 

We  shall  now  write  these  quantities  in  the  more  general  form  (p  +  ap2) 
and  {(1/p)  —  b}  or  (1  —  pb)/p.  Then,  corresponding  to  the  equation 
P  =  E  •  p  •  T  (p.  370),  we  have  (p  +  ap2)  =  &  •  p  (1  -  fy)-1  •  T.  When  multi- 
plied out,  this  becomes  ab'- p3  —  a- p2  +  (bp  -f  I£T)P  —  p  =  0,  or  p3  —  p2/b  + 
P  (P/a  +  3&T/ab)  —  p/ab  =  0.  This  is  a  "  cubic  equation  "  in  p.  Now, 
every  cubic  equation  has  three  roots,  of  which  the  whole  may  be  real,  or 
one  may  be  real  and  two  imaginary,  not  corresponding  to  any  physical 
reality,  or  there  may  be  three  equal  roots.  Which  of  these  results  corre- 
sponds to  any  particular  case  depends  on  the  actual  coefficients  in  the  equa- 
tion. When  there  are  three  equal  roots  in  the  equation  xz  —  ax2  +  fix  —  y  =  0, 
we  have  x  =  a/3  =  V/J/3  =  \/y.  Hence  there  are  three  coincident  and 
equal  solutions  of  the  cubic  equation  above,  when  p  =  l/3&  = 


376  HEAT.  [CHAP. 

=  Z/pi  ab.  By  elimination  among  these  equations  we  find  that  p  =  a/2762, 
and  T  =  8a/27i&6:  and  when  these  are  the  values  of  p  and  T,  p  has  only  one 
value,  1/36. 

The  other  relations  between  p,  T,  and  p  may  be  represented  by  a  dia- 
gram, Fig.  128a,  in  which  ordinates  represent  pressures,  and  abscissae  the 
reciprocals  of  the  density,  or  the  relative  volumes  of  a  given  mass  of  gas, 
while  the  different  curves  correspond  to  different  fixed  values  of  T.     It  will 
be  seen  that  the  curve  marked  0  is  the  first  curve,  going  upwards  from  a  low 
to  a  high  temperature,  in  which  it  ceases  to  be  possible  to  find  three  values 
of  ij  =  1/p  for  some  given  pressure  p,  with  a  mini- 
mum  and  a  maximum  lying  between  these.     The 
point  P  in  that  curve  is  the  point  at  which  the 
three  values  above-mentioned  (T  =  8a/273&6,  p  = 
a/2762,  and  p  =  1/36)  concur:  and  at  that  point, 
r  is  6,  the  Critical  Temperature,  p  is  Co  the  Crit- 
ical Pressure,  and  p  is  the  Critical  Density,  the 
reciprocal  of  <j>   the    Critical  Volume   per   unit- 
mass  of  the  gas.     ]f  the  temperature  be  above  0 
there  is  only  one  real  value  of  p  for  any  value  of 
p ;  if  it  be  below  0,  there  will,  within  certain  limits 
Of  value  of  p,  be  three  real  values  of  p  for  each 
value  of  p. 

In  the  curve  T,,  at  pressures  below  OA,  the  substance  is  a  gas :  at  pres- 
sures above  OB,  it  is  a  liquid.  At  pressures  between  OA  and  OB,  however, 
if  the  curve  corresponding  to  the  cubic  equation  were  completely  verified, 
the  condition  of  the  material  would  be  unstable.  At  the  volume  Oc,  it 
would  be  increasing  in  volume  with  increasing  pressure.  What  does  happen 
is,  that  between  A  and  B  the  curve  of  unstable  condition  is  replaced  by 
a  line  representing  nearly  uniform  pressures;  the  substance  is  partly  gas, 
partly  liquid.  Now  increase  the  temperature ;  the  range  AB  diminishes, 
and  the  volume  is  also  somewhat  higher.  Continue  raising  the  tempera- 
ture ;  the  range  AB  disappears,  and  at  the  Critical  Temperature  0  the 
volume  of  the  compressed  gas,  the  inaccessible  volume  in  the  unstable  con- 
dition, and  the  volume  of  the  liquid  at  that  temperature  and  pressure,  all 
come  to  coincide. 

At  temperatures  above  the  Critical,  the  swerve  in  the  curve  diminishes, 
and  the  higher  the  temperature,  the  more  nearly  does  the  curve  coincide 
with  the  rectangular  hyperbola  of  a  perfect  gas.  At  temperatures  above  the 
Critical  Temperature,  there  is  thus  no  condensation,  no  separation  of  liquid 
from  a  compressed  gas  as  the  pressure  rises,  and  the  higher  the  temperature 
the  more  nearly  does  the  condition  of  the  substance  approximate  at  all  pres- 
sures to  that  of  a  perfect  gas.  At  temperatures  below  0,  on  the  other  hand, 
the  lower  the  temperature,  the  smaller  is  the  pressure  required  to  condense 
the  gas  into  a  liquid. 

Wroblewski  calculated  from  the  behaviour  of  hydrogen,  which  gave  data 
for  the  constant  numerical  terms  a,  6,  and  2&  in  the  equation,  that  the  criti- 
cal temperature  and  pressure  of  hydrogen  are  respectively  32°-6  Abs.,  or 
—  240°4  C.,  and  13-3  atmospheres.  For  nitrogen  the  critical  temperature 
is  -146°  C. ;  for  oxygen  -118°-8  C.  (Olszewski). 

It  will  be  understood  that  what  we  have  called  the  swerve  in  the  curve 
will  cause  the  appearance  of  anomalies.  As  the  pressure  increases,  the  vol- 
ume first  diminishes  somewhat  more  rapidly,  then  less  rapidly  than  it  would 


xiii.]  VAN  DER   WAALS'    LAW.  377 

have  done  in  a  perfect  gas :  so  that  there  is  always  some  point  at  which  the 
observed  pressure  p  and  the  observed  density  p  are  such  that  p/p,  =  p*Q  per 
gramme,  is  a  minimum.  If  now  we  compare  different  gases  at  the  same 
temperature  and  pressure,  these  anomalies  seem  to  be  inexplicably  different ; 
but  if  we  compare  the  different  gases  at  temperatures  and  pressures  which 
are  equal  multiples  of  their  respective  critical  temperatures  and  pressures,  it 
is  found  that  they  all  behave  similarly.  The  curves  are  then  approxi- 
mately identical  for  them  all.  For  example,  if  any  gas  be  heated  to  a  tem- 
perature T°  Abs.  =  14  times  its  critical  temperature,  p/p  is  always  a  minimum 
when  the  pressure  p  =  3d) ;  but  if  the  temperature  rise  to  r,0  Abs.  =  30,  p/p  is  a 
minimum  when  the  pressure  =  d> :  and  if  it  rise  to,  say,  56  or  60,  p/p  is  a  mini- 
mum only  when  the  pressure  is  exceedingly  small.  In  hydrogen,  therefore, 
at  0°  C.  =  273°  Abs.,  =  about  84  x  0,  the  gas  is  more  compressible  at  all  high 
pressures  than  Boyle's  Law  would  indicate ;  but  at  —183°  C.,  the  tempera- 
ture of  boiling  liquid  oxygen,  which  is  about  30  on  the  Absolute  scale,  it  has 
been  observed  by  Wroblewski  that  the  compressibility  of  hydrogen  begins 
to  diminish  at  14  atmospheres'  pressure,  and  the  gas  behaves  as  ordinary 
gases  do  at  ordinary  temperatures.  In  carbonic  acid,  on  the  other  hand, 
whose  critical  temperature  0  is  30°-92  C.  =  304°  Abs.,  and  whose  critical 
pressure  at  =  73  atmospheres,  the  minimum  compressibility  would  be  found 
at  about  219  atmospheres  and  153°  C.  (=426°  Abs.  =  14  x  304),  or  again, 
at  73  atmospheres  and  639°  C.  (=912°  Abs.  =  3  x  304°  Abs.) ;  beyond  which 
limits  carbonic  acid  would  behave  like  hydrogen  at  ordinary  temperatures 
and  pressures. 

In  gases  the  amount  of  heat  which  disappears  during  expan- 
sion in  doing  internal  work  is  generally  small  in  proportion  to 
the  external  work  done  against  the  atmospheric  pressure :  in 
solids  and  liquids  the  internal  work  done  is  relatively  much 
greater. 

When  a  substance  is  heated  and  rises  in  temperature  with- 
out being  allowed  to  expand,  so  much  heat  is  absorbed  during 
a  given  rise  of  temperature;  when  expansion  is  permitted,  an 
additional  supply  of  heat  is  required.  The  Latent  Heat  of 
Expansion,  L,  of  a  substance  may  thus  be  found  by  difference. 

When  temperature  varies,  the  volume  being  constant,  the  heat,  H,  sup- 
plied to  a  mass  m  is  equal  to  m-cr  ergs ;  when  the  pressure  is  kept  constant 
and  expansion  is  allowed,  the  heat  supplied,  H;,  is  equal  to  ra-  (cf  +  Lij/ij) 
+  j0b;  whence  L  =  (H/-H  -joi)to/mb.  H,  is  equal  to  m-kr,  where  k  is  the 
thermal  capacity  at  constant  pressure,  whence  L,  in  ergs,  =  {(k  —  c)  rb/b 
—  p/p},  per  gramme.  In  a  perfect  gas  L  =  0. 

Latent  heat  of  expansion  is  difficult  to  measure.  To  ascer- 
tain that  of  water  between  0°  C.  and  100°  C.,  for  instance,  it 
would  be  necessary  to  compare  the  amounts  of  heat  required 
to  heat  a  certain  mass  of  water  from  0°  to  100°  C.  when  it  is  free 
to  expand  and  when  it  is  prevented  from  expanding:  but  the 
latter  investigation  would  require  the  application  of  a  pressure 
of  8772  atmospheres. 


378  HEAT.  [CHAP. 

In  the  same  way  wrought-iron  heated  through  15i°  Fahr.  exercises  a 
pressure  of  1  ton  per  square  inch. 

The  Latent  Heat  of  Expansion  of  a  substance  is,  as  a  numer- 
ical coefficient,  the  amount  of  heat  (usually  reckoned  in  calories) 
required  to  effect  unit-expansion  in  a  gramme  of  that  substance 
—  that  is,  to  double  its  volume  —  and  which  disappears  in  doing 
that  work,  without  affecting  the  temperature. 

The  work  of  expansion  is,  however,  associated  with  that  of 
raising  the  temperature  ;  only  in  idea  can  we  form  an  abstract 
conception  of  the  amount  of  heat  required  to  effect  a  certain 
expansion  while  the  temperature  is  supposed  to  remain  un- 
changed. Temperature  and  volume  vary  simultaneously,  and 
the  physical  constant  known  as  the  coefficient  of  expansion 
states  numerically  the  relation  between  these  associated  effects 
of  heat. 

A  substance  whose  volume  is  b0  at  a  temperature  TO  assumes  a  volume 
b;  at  the  temperature  T,;  the  change  of  temperature  is  rt  —  TO;  the 


tionate  change  of  volume  is  (b/  —  ba)/b0  ;  the  quotient  (ij,  —  b0)/iJ0(r/  —  TO) 
is  the  coefficient  of  expansion.     If  rl  —  TO  —  1°,  the  coefficient  of  expansion 

fe(&,-&0)/toc. 

The  coefficient  of  expansion  of  any  substance  is  the  ratio 
between  the  increase  of  volume  which  it  undergoes  when  its 
temperature  is  raised  by  1°  C.,  and  its  original  volume. 


If  a  cube  of  volume  ij0  assume  volume  b,,  its  side  v^  becomes  v^  ;  its 

coefficient  of  linear  expansion  is  therefore  -  '-$-=  —  9?  or  approximately 

vb0 

J(by  —  fc0)/b0.  Thus,  if  a  body  measuring  a  cubic  foot,  on  being  heated  1°> 
assume  a  volume  of  1-0003  cub.  ft.,  the  side  of  the  cube  (1  foot)  has  become 
nearly  1-0001  linear  foot. 

Since  we  have  in  general  to  deal  with  expansions  propor- 
tionately very  small,  we  may  say  that  the  coefficient  of  linear 
expansion  —  the  proportionate  increase  in  length,  breadth,  or 
thickness  per  degree  centigrade  —  is  equal  to  one-third  the  coef- 
ficient of  cubical  expansion. 

If  L  be  the  coefficient  of  linear  expansion  of  a  body  whose  length  at  TO 
is  /0,  the  length  of  the  body  at  the  temperature  T,  is  I,  =  10  +  (L  •  10  •  (T,  —  TO)) 
or  Z0(l  +  L  (r,  —  TO)).  In  this  equation  there  are  five  terms,  any  four  of 
which  being  known,  the  fifth  can  be  found. 

In  some  cases  —  many  crystalline  bodies  —  the  coefficients  of 
linear  dilatation  are  not  equal  in  all  directions.  Crystals  have 
three  axes,  in  the  directions  of  which  the  coefficients  of  expan- 
sion (L^  Ln,  Lnt")  are  not  always  equal  to  one  another  ;  thus  the 


xiii.]  EXPANSION.  379 

angles  of  crystals  are  often  modified  by  changes  of  temperature. 
Substances  belonging  to  the  regular  system  have  the  coefficients 
equal  in  the  three  axial  directions,  and  they  preserve  similarity 
of  figure  when  heated ;  dimetric  crystals  have  two  axial  coeffi- 
cients equal,  the  third  different;  trimetric  crystals  have  all 
three  coefficients  unequal.  In  general  the  cubical  coefficient 
=  (i,  +  Llt  +  Lln). 

Take  plates  of  gypsum,  cut  parallel  to  the  prismatic  axis :  cement  them 
together  so  that  the  direction  of  the  axis  of  one  plate  forms  a  right  angle 
with  that  of  the  other.  Heat  until  the  cement  is  melted ;  allow  to  cool.  The 
unequal  contraction  in  cooling  will  warp  the  whole  (Fresnel).  In  the  case 
of  this  substance  a  contraction  in  one  direction  is  associated  with  expansion 
in  two  others. 

Indiarubber  and  iodide  of  lead,  iodide  of  lead  and  silver 
(Pb  I2,  Ag  I),  iodide  of  silver  up  to  156°*5  C.,  and  garnets,  as 
well  as  water  between  0°  and  3°-9  C.,  contract  when  heated : 
their  coefficient  is  negative.  In  some  substances  (zinc  and  iron) 
the  coefficient  of  expansion  slowly  alters  with  lapse  of  time. 

When  a  hollow  body  such  as  a  flask  or  thermometer-bulb  is 
heated,  it  expands  almost  exactly  as  if  it  were  solid :  a  glass 
tube  expands  as  if  it  were  a  glass  rod.  It  follows  that  when  a 
hollow  body  is  heated,  its  internal  cavity  increases  in  volume 
in  the  same  proportion  as  it  would  have  done  if  it  had  been 
occupied  by  a  solid  the  same  as  that  which  surrounds  it. 

Examples  of  Expansion  by  Heat.  —  Bodies  which,  when 
cold,  exactly  fill  certain  apertures,  when  they  are  warmed  will 
not  enter  these.  Railway  rails  are  not  laid  in  exact  contact ; 
allowance  must  be  made  for  their  summer  expansion  and  winter 
contraction.  In  designing  lattice-girders  for  bridges,  the  same 
necessity  must  be  taken  into  account.  Railway-distance  signals 
are  controlled  by  rods,  which  differ  considerably  in  length  at 
night  and  by  day;  provision  must  be  made  for  tightening  them 
up  or  the  reverse.  If  the  neck  of  a  stoppered  bottle  clasp  the 
stopper  too  tightly,  it  may  be  loosened  by  causing  the  neck  to 
expand  while  the  stopper  does  not  do  so  ;  this  may  be  effected 
by  winding  a  string  rou  ad  the  neck  and  pulling  it  backwards 
and  forwards  so  as  to  produce  heat  by  friction ;  the  neck  is 
heated  before  the  stopper  itself  is  affected.  Glass  suddenly 
heated  expands  superficially  while  the  inside  is  still  cool :  under 
the  stress  set  up  the  glass  may  break ;  hence  the  thinner  a  flask, 
the  less  risk  there  is  of  its  cracking  when  it  is  heated.  A  cart- 
wheel tire  is  fitted  on  when  it  is  hot ;  when  it  cools  down  it 


380  HEAT.  [CHAP. 

contracts  and  holds  the  rim,  spokes,  and  hub  firmly  together :  if 
it  be  originally  too  small  it  may  break  itself  by  its  own  con- 
tractile tension.  The  lead  on  a  roof  expands  by  day  and  con- 
tracts at  night;  gravity  aids  the  one  and  checks  the  other 
tendency  ;  the  lead  creeps  down.  The  same  theory  has  been 
applied  to  glacier  movement. 

Applications  of  Expansion. —  The  Compensation-pendulum 
is  a  pendulum  constant  in  length,  whatever  be  the  temperature. 
A  simple  bar  of  metal  would,  by  its  variations  in  length,  produce 
oscillations  irregularly  unequal,  the  clock  going  slow  in  sum- 
mer, fast  in  winter.  In  order  to  correct  this,  the  bob  of  the 
pendulum  is  suspended  from  a  framework  of  bars  of  iron  and 
brass,  so  arranged  that  expansion  of  the  bars  of  iron  tends  to 
depress  the  bob :  that  of  the  bars  of  brass  tends  to  raise  it ;  by 
proper  adjustment  of  the  lengths  of  the  bars  these  effects  com- 
pensate one  another. 

The  bob  itself  is  sometimes  made  of  a  tube  containing 
quicksilver :  the  expansion  of  the  suspending  bar  tends  to  lower 
the  centre  of  gravity  of  the  pendulum:  that  of  the  mercury 
tends  to  raise  it ;  a  proper  adjustment  of  the  quantity  of  mer- 
cury in  the  bob  produces  sensibly  accurate  compensation. 

Sometimes  the  rod  of  a  pendulum  bears  a  transverse  bar, 
which  is  loaded  at  each  end  with  heavy  masses.  This  transverse 
bar  consists  of  strips  of  different  metals ;  in  weather  warmer 
than  the  average  the  lower  strips  expand  most,  distort  the  bar, 
raise  the  heavy  masses,  and  thus  raise  the  centre  of  gravity  of 
the  whole  pendulum:  in  colder  weather  the  reverse  effect  is 
obtained,  for  the  lower  strips  contract  most.  These  effects  may 
be  adjusted  so  as  to  neutralise  the  effect  of  the  lengthening  or 
shortening  of  the  pendulum  itself. 

Measurement  of  Coefficients  of  Expansion. — In  solids 
the  coefficient  of  linear  expansion  is  found  by  direct  observation. 
A  bar  is  heated  to  a  known  temperature  ;  its  original  length  and 
temperature  are  known.  The  elongation  of  the  bar  may  be 
measured  by  a  traversing  bar  with  micrometer,  or  by  the  method 
of  Fig.  5,  or  by  the  expansion  of  the  bar  in  a  tube  pushing  out 
a  piece  of  porcelain,  which  can  move  outwards  but  cannot  return. 
The  first-mentioned  method  is  by  far  the  least  liable  to  error, 
especially  when  the  distance  between  two  distinctive  points  on 
the  bar  is  observed  at  two  given  temperatures. 

1  ~  °  . is  the  coefficient  of  linear  dilatation. 


xiii.  1  EXPANSION.  381 

The  coefficient  of  cubical  expansion  may  be  found  by  multi- 
plying the  coefficient  of  linear  expansion  by  3 ;  or,  better,  by 
finding  the  different  specific  densities  of  the  solid  at  different 
temperatures. 

The  mass  (  =  weight/*/)  remaining  the  same,  p  and  pt  being  the  densi- 
ties, the  volumes  b0  (  =  m/p)  and  I0J(=  m/p^)  are  easily  found;  and  (T/—  TO) 

being  the  difference  of  temperatures,  (  — •  —      — ) ,  the  coefficient  of 

\     b0        Tt  —  TO  / 
cubical  expansion,  can  be  computed. 

If  a  solid  be  heated  in  a  flask  with  a  narrow  orifice  and 
completely  filled  with  mercury,  the  mercury  expands,  the  flask 
expands,  and  so  does  the  solid  immersed  in  it.  The  absolute 
expansion  of  the  mercury  is  previously  known,  that  of  the  glass 
vessel  must  be  known,  and  the  amount  of  mercury  which  would 
fall  out  of  the  flask  if  the  flask  were  completely  filled  with  mer- 
cury and  heated  to  the  same  degree  is  already  known;  when  the 
solid  is  immersed  in  the  mercury,  a  different  quantity  of  mer- 
cury escapes  from  the  flask  when  heated ;  the  difference  is  due 
to  the  difference  of  dilatation  between  mercury  and  the  immersed 
solid:  the  coefficient  of  expansion  of  the  immersed  solid  can 
thus  be  calculated. 

In  liquids  the  expansion  may  be  found  by  observation  of 
the  apparent  increase  of  bulk  undergone  by  a  liquid  contained 
in  a  flask.  The  width  of  the  neck  may  be  ascertained  by  the 
addition  of  known  quantities  of  mercury :  an  apparent  rise  of  the 
liquid  in  the  neck  may  be  interpreted  as  corresponding  to  so 
many  cubic  cm.  apparent  increase  of  bulk.  But  it  is  important 
to  bear  in  mind  that  the  cavity  of  the  flask  also  expands,  and 
that  the  real  expansion  of  the  liquid  is  the  sum  of  the  expansion 
of  the  cavity  of  the  flask,  and  the  apparent  expansion  of  the 
liquid  in  the  neck.  If  the  liquid  have  the  same  cubical  coeffi- 
cient as  the  glass,  there  will  be  neither  a  rise  nor  a  fall  in  the 
neck ;  if  it  have  a  less  rate  of  expansion  than  glass  it  will  sink 
in  the  neck,  and  will  then  apparently  contract;  only  when 
it  has  a  greater  coefficient  of  expansion  than  the  glass  will  it 
rise  in  the  neck,  and  thus  under  such  circumstances  manifestly 
appear  to  expand. 

When  a  thin  glass  flask  filled  with  water  is  suddenly  heated  it  expands 
before  the  water  contained  in  it  has  had  time  to  become  heated,  and  the 
liquid  in  the  first  place  appears  to  shrink  into  the  flask.  Then  the  liquid 
becomes  heated  and  rises  in  the  neck  of  the  flask. 

The  expansion  of  a  liquid  may  also  be  found  by  observing 


382 


HEAT. 


[CHAP. 


its  density  at  different  temperatures.  This  may  be  done  by 
means  of  separate  observations.  It  may  also  be  done  by  the 
observation  of  the  simultaneous  heights  of  a  hotter  and  a  colder 
column  of  the  same  liquid,  which  balance  one  another  in  a 
U-tube.  The  heights  are  reciprocally  proportional  to  the  den- 
sities,  and  thus  it  is  easy  to  find  the 
coefficient  of  expansion  per  degree  cen- 
tigrade. Fig.  129  shows  that  each  limb 
of  the  U-tube  is  maintained  at  a  con- 
stant temperature  by  surrounding  baths 
(of  water,  mercury,  oil,  etc.)  whose  tem- 
peratures are  known.  The  heights  of 
the  columns  may  be  measured  by  means 
of  a  cathetometer.  The  absolute  expan- 
sion of  mercury  is  by  this  method  found 
to  be  per  degree  centigrade  (between 
—  36°  C.  and  100°  C.)  almost  exactly  ^F  of  its  total  bulk  at 


0°  C.  ;  above  100°  C.  it  increases  rapidly  with  the  temperature. 
The  total  amount  of  expansion  is  thus  not  exactly  proportional 
to  the  rise  of  temperature. 

In  gases  the  coefficient  of  expansion  is  nearly  uniform, 
about  2^F  ^or  everj  degree  centigrade.  Not  quite  uniform  ;  for 
all  gases  are  not  necessarily  in  the  same  physical  condition  merely 
because  they  are  at  the  same  temperature,  for  some  may  be  near, 
others  far  from,  their  point  of  condensation  ;  and  the  volume  of 
gases  is  not  exactly  proportional  to  their  absolute  temperature. 

The  coefficient  of  expansion  in  gases  may  be  determined  by 
direct  observation,  the  volume  being  allowed  to  vary,  while 
the  pressure  is  maintained  constant  during  a  given  change  of 
temperature;  or  inferentially,  by  observation  of  the  increase 
in  pressure  exercised  by  a  gas  when  its  volume  is  kept  constant 
during  a  given  change  of  temperature,  coupled  with  the  assump- 
tion that  Boyle's  law  is  perfectly  obeyed,  and  that  the  volume 
and  the  pressure  bear  an  exact  inverse  ratio  to  one  another. 
The  latter  method,  as  we  shall  see,  is  more  valuable  in  ther- 
mometry  than  in  the  determination  of  the  actual  coefficient  of 
expansion  of  a  gas. 

If  we  assume  Boyle's  law  and  Charles's  law  to  be  both  true,  we  have  the 
equation  p'o/r  =  const.  If  the  same  quantity  of  gas  change  in  pressure, 
volume,  or  temperature,  again  p^/r^the  same  const.  Hence  p<o/T=pllQl/ri. 
This  enables  us  to  solve,  to  a  first  approximation,  such  problems  as  the 
following  :  — 


xiii.]  EXPANSION.      >  383 


Fifteen  litres  of  air  at  0°  C.  and  761  mm.  bar.  pr.  afe  heated  to  10°  C. 
while  the  barometer  sinks  to  759  mm. ;  what  volume  does  the  air  assume  ? 

pb  _  pfr, .  761  x  15  _  759  x  b, 
T  ~  ry  '       273  283 

Whence  ir  =  |  —  x  — -  x  15  )  litres. 


,  =  gx  §  x  15)  li 


Again,  15  litres  of  air  at  0°  C.  (273°  Abs.)  and  762  mm.  Hg  pressure  are, 
when  they  are  heated  to  an  unknown  temperature  and  exposed  to  a  pressure 
of  1000  mm.  Hg,  doubled  in  volume  :  what  is  the  unknown  temperature? 

pb  _  pX  .  762  x  15  _  1000  x  30  . 
r  ~~  T,  '       273  T/ 

Whence  T,  =  ^  .  ~  -  273  =  716°-5  Abs.  =  443°-5  C. 

We  may  combine  with  these  equations  the  two  following  propositions  :  — 

1.  The  specific  density  of  a  gas  is  numerically  equal  to  half  its  molec- 
ular weight. 

2.  One  gramme  of  hydrogen  measures  11-1645  litres  at  0°  C.  and  760 
mm.  bar.  pr. 

Problem. 

Fourteen  litres  of  carbonic  acid  are  measured  at  10°  C.  and  759  mm. 
pressure  :  what  is  their  mass  ? 

First  reduce  the  14  litres  to  the  volume  which  they  wrould  occupy  at 
0°  C.  and  760  mm.  bar.  pr.  —  i.e., 


1          44 
Each  litre  of  carbonic  acid  at  0°  C.  and  760  mm.  weighs  -  x  — 

grammes.     The  whole  weighs 

x  2-Ii!  x  75!»  x  _t_  x  «)  grammes. 
283     760      11-1645      2  /  & 

Problem. 

What  bulk  is  occupied  by  20  grammes  of  ammonia  gas  at  15°  C.  and 
740  mm.  bar.  pr.? 

One  gramme  of  hydrogen  occupies  at  0°  C.  and  760  mm.  a  bulk  of 
11-1645  litres;  at  15°  C.  and  740  mm.  it  would  have  a  volume  of 
(11-1645  x  f|-f  x  ||§)  litres  ;  but  ammonia  gas  has  a  sp.  density  =  -y  ; 
hence  20  grammes  of  ammonia  occupy  a  bulk 


It  may  be  left  to  the  student  as  an  exercise,  to  find  what  corrections 
should  be  applied,  to  reduce  the  apparent  weight  of  a  substance  weighed  in 
air  at  a  given  temperature  to  the  real  weight  at  a  standard  temperature,  say 
0°  C.,  the  coefficients  of  expansion  of  air,  of  the  counterpoising  weights,  and 
of  the  substance  weighed,  being  supposed  to  be  known. 


384  HEAT.  [CHAP. 

Fusion.  —  Heat  sometimes  operates  liquefaction  of  solid 
bodies.  The  temperatures  at  which,  fusion  is  effected  differ 
widely:  the  fusing  point  of  solid  alcohol  (  — 130°-5  C.),  that  of 
mercury  (-40°  C.),  and  that  of  platinum  (about  1775°  C.) 
which  can  only  be  fused  by  the  oxyhydrogen  blowpipe  or  the 
electric  arc,  may  be  taken  as  examples. 

Fusion  upon  heating,  and  solidification  upon  cooling,  occur 
normally  at  the  same  temperature ;  Melting  points  and  Freezing 
points  are  the  same,  except  in  cases  of  over  cooling,  in  which 
the  temperature  may  fall  below  the  freezing  point,  and  in  which 
solidification  may  be  made  to  start  by  dropping  in  a  piece  of 
the  solid.  During  fusion  or  during  freezing  or  solidification, 
when  this  has  once  begun,  the  temperature  remains  the  same 
until  the  process  is  complete.  Energy  is  being  absorbed  or 
liberated  in  the  form  of  Heat.  See  Latent  Heat,  p.  361. 

In  general  there  is  expansion  during  fusion ;  in  such  event 
there  may  be  a  small  amount  of  work  done  against  external 
pressure.  If  the  external  pressure  be  increased,  the  amount  of 
heat-energy  that  must  be  supplied,  in  order  to  effect  this  external 
work  in  addition  to  the  internal  work  of  fusion,  is  proportion- 
ately increased.  The  temperature  of  fusion  is  thus  in  most 
cases  raised  by  increase  of  pressure.  In  the  cases  of  water, 
antimony,  cast-iron,  and  many  rocks,  the  freezing  point  is  low- 
ered by  pressure,  because  these  substances  expand  when  they 
freeze.  Tables  of  melting  points  therefore  denote  the  melting 
points  of  substances  at  the  atmospheric  pressure. 

We  may  here  state  the  reasoning  by  which  it  was  predicted*  that  an 
increase  of  pressure  would  be  found  to  lower  the  melting  point  of  ice; 
though  some  of  the  steps  will  not  be  understood  until  after  we  have  con- 
Fig  130.  sidered  Carnot's  cycle  of  operations  and  his 
"perfect  engine." 

A  cylinder  with  a  square  base,  1  cm.  square, 
contains  one  gramme  of  water  —  i.e.  1  cub.  cm.  S 
is  a  source  of  heat  at  0°  C.  (which  must  be  situated 
within  a  sufficient  space  entirely  devoid  of  air). 
R  is  a  refrigerator  situated  within  a  region  where 
the  atmospheric  or  other  pressure  is  equal  to  p 
dynes  per  sq.  cm. ;  it  is  maintained  at  a  constant 
temperature  —t°  C.,  very  slightly  below  0°  C. 

1.  The  cylinder  is  kept  upon  the  source  S 
until  the  water  assumes  the  temperature  0°  C. 
We  now  have  1  cub.  cm.  water  at  0°  C. 


*  By  Prof.  James  Thomson ;  experimentally  confirmed  by  his  brother,  Prof.  Sir 
William  Thomson,  now  Lord  Kelvin. 


xin.]  MELTING  POINT   OF   ICE.  385 

2.  Put  the  cylinder  on  the  refrigerator  R ;  keep  it  there  until  the  water 
is  wholly  frozen  to  ice  at  0°  C.  We  now  have  1-0908  cub.  cm.  ice  at  0°  C. 
(the  sp.  density  of  ice  being  -91674,  Bunsen). 

Work  has  been  done  during  expansion;  the  piston  has  been  thrust 
upwards  through  -0908  cm.  against  an  external  pressure  p  dynes  per  sq.  cm. ; 
the  work  done  by  the  expanding  substance  is  0-0908  p  ergs. 

Put  the  cylinder  again  on  the  source :  the  temperature  of  the  source  is 
supposed  to  be  by  an  infinitely  small  amount  higher  than  that  of  the  ice.  In 
course  of  time  the  ice  melts ;  now  we  again  have  1  cub.  cm.  of  water  at  0°  C. 
(while  no  work  has  been  done  upon  the  melting  ice  by  any  exterior  pressure). 
The  melting  ice  has  had  heat  imparted  to  it  equal  to  the  latent  heat  of 
fusion  of  1  cub.  cm.  of  water  —  that  is,  80-025  ca  =  3,328,480000  ergs. 
This  amount  of  heat  has  been  absorbed  from  the  source  at  0°  C. ;  heat  has 
been  lost  to  the  refrigerator  at  — 1°  C.  The  piston  returns  to  its  normal 
position,  as  we  have  seen,  and  the  whole  contrivance,  perfectly  imaginary, 
will  act  as  a  "  perfect  engine,"  with  ice  or  water  as  its  working  substance, 
provided  that  t  has  a  certain  value  to  be  deduced  from  the  equation  — 

Work  done     _  difference  of  temp,  between  source  and  refrigerator  _    t 
Heat  absorbed  absol.  temperature  of  source  273 

0-0908  y?  t 

3,328,480000  ~  273* 

273  x  0-0908 
3,328,480000  P' 

When  the  external  pressure  p  changes  by  n  =  1,013663  dynes  per  sq.  cm. 
— that  is,  when  it  changes  by  an  amount  equal  to  one  atmosphere —  t  changes 
by  -0074°  C.  This  means  that  such  an  engine  is  reversible,  and  its  operation 
is  theoretically  perfect,  when  the  freezing  operation  is  conducted  at  a  tem- 
perature lower  than  0°  C.  by  an  amount  equal  to  -0074°  C.  for  every  addi- 
tional atmosphere-pressure  suffered  by  the  freezing  water.  If  the  freezing- 
could  occur  at  a  higher  temperature  than  this,  there  would  be  production  of 
work  by  the  expanding  ice,  accompanied  by  a  withdrawal  of  heat  from  the 
source  insufficient  to  account  for  the  work,  and  the  perpetual  motion  would 
become  possible. 

When  a  piece  of  ice  is  placed  in  contact  with  another,  both  being  at  0°  C., 
a  very  slight  pressure  will,  by  lowering  the  melting  point,  cause  a  certain 
quantity  of  ice  at  the  point  of  contact  to  melt.  When  the  pressure  is  relieved, 
the  mass  solidifies  and  becomes  continuous  ice. 

Ice  is  not  without  plasticity  at  temperatures  not  far  from  0°  C.,  and  can 
slowly  flow  down  a  slope  of  1  in  4  under  a  pressure  equal  to  the  weight  of 
300  feet  of  ice-cliff  (Moseley  and  Browne);  but  at  temperatures  between 
0°  C.  and  about  —  }°  C.  it  can  be  driven  through  narrow  passages  by  the 
above  process  of  Re  gel  at  ion,  for  when  crushed  the  fragments  are  relieved 
of  pressure  and  reunite,  again  to  be  crushed  and  forced  onwards.  To  the 
small  plasticity  of  ice  and  to  the  process  of  crushing  or  regelation,  as  well  as 
to  creeping  (p.  380),  is  to  be  mainly  ascribed  the  flow  of  glaciers. 

Sometimes  the  fusion-point  of  a  mixture  is  below  that  of  its 
ingredients.  A  mixture  of  common  salt  with  about  2J  parts  of 
crushed  ice  melts  at  about  —18°  C.  or  0°  Fahr. :  above  this  tem- 

2c 


386  HEAT.  [CHAP. 

perature  it  is  liquid  ;  and  when  ice  and  salt  are  mixed,  the  result 
is  very  cold  liquid  brine. 

When  the  pavements  in  snowy  weather  are  cleared  by  means  of  salt,  the 
brine  thus  formed  being  at  a  temperature  of  0°  Fahr.,  or  at  "  thirty-two 
degrees  of  frost,"  penetrates  the  shoe-leather  and  chills  the  feet  of  pedes- 
trians, while  it  refuses  to  dry,  the  salt  being  hygroscopic  — that  is,  having  a 
great  affinity  for  water. 

This  example  is  a  particular  case  of  a  general  proposition, 
that  a  solution  of  a  solid  in  a  liquid  has  a  lower  freezing  point 
than  the  pure  liquid  itself. 

The  extent  to  which  the  freezing  point  of  a  liquid  is  lowered  by  dissolv- 
ing a  substance  in  it  varies  directly  as  the  number  of  molecules  dissolved, 
inversely  as  the  number  of  molecules  in  the  solvent  liquid,  and  directly  as 
a  constant  which  depends  upon  the  nature  of  the  solvent.  For  example,  if 
n  molecules  be  dissolved  in  10000  molecules  of  water,  the  freezing  point  will 
fall  by  0-0063n°C.  This  simple  relation  is  most  nearly  adhered  to  in  the 
most  dilute  solutions;  certain  abnormalities  are  observed  however,  which 
are  interpreted  as  showing  either  (1)  that  there  is  coalescence  of  molecules 
of  the  substance  dissolved,  which  fall  asunder  on  increasing  the  dilution, 
particularly  with  water,  which  is  somehow  unfavourable  to  molecular  coales- 
cence or  polymerisation  ;  or  (2)  that  there  is  a  break-up  or  dissociation  of 
the  molecules.  This  latter  particularly  occurs  in  aqueous  solutions  of  elec- 
trolytes (p.  590)  or  acids,  bases,  and  salts,  in  which  the  ions  (p.  281) 
become  separated  in  the  solution  ;  and  each  of  the  ions  produces  its  own 
independent  effect  upon  the  freezing  point  of  the  solution. 

Sublimation.  —  When  a  solid  on  being  heated  becomes  a 
vapour  without  passing  through  the  liquid  state  it  is  said  to  be 
sublimed.  Examples  of  this  are  furnished  by  arsenic  trioxide 
and  pentasulphide,  metallic  arsenic,  and  some  metallic  chlorides, 
as  well  as  by  many  organic  substances. 

Sometimes  the  word  sublimation  simply  means  the  distillation  of  a 
solid,  as  in  the  case  of  sulphur,  bichloride  of  mercury,  or  benzoic  acid,  all 
of  which  melt  before  vaporising. 

Sulphide  of  zinc  and  sulphide  of  cadmium  are  not  volatile  when  pure  ; 
but  when  mixed  with  traces  of  metallic  zinc  or  cadmium  respectively,  they 
are  very  volatile. 

Boiling  or  ebullition  is  a  rapid  process  of  reduction  of  a 
liquid  to  vapour.  Evaporation  is  thus  distinguished  from 
ebullition;  in  evaporation  particles  possessing  more  than  the 
average  kinetic  energy  fly  from  the  surface  and  mingle  with  the 
particles  of  gas  or  vapour  already  existing  in  the  neighbourhood 
of  the  surface  of  the  liquid,  and  drive  or  repel  only  a  certain 
proportion  of  them  away  from  the  surface :  in  boiling,  the 
particles  which  fly  from  the  surface  bombard  the  surrounding 
particles  so  hotly  as  to  drive  them  all  from  the  neighbourhood 


xiii.]  BOILING.  387 

of  the  surface  of  the  boiling  liquid,  and  to  take  their  place. 
Thus  the  vapour  of  a  boiling  liquid  has  to  exert  a  pressure 
which  is  just  a  little  greater  than  the  atmospheric,  or,  in 
general,  the  exterior  pressure,  whatever  that  may  happen  to  be ; 
the  vapour  of  an  evaporating  liquid  exerts  a  pressure  which  is 
only  a.  fractional  part  of  the  atmospheric  or  exterior  pressure. 

This  pressure,  just  a  little  greater  than  that  of  the  atmosphere,  may  be 
made  up  in  different  ways.  Volatile-oil  vapour,  led  into  water,  or,  equally, 
water-vapour  led  into  oil,  will  cause  boiling  of  the  water  or  the  oil  at  tem- 
peratures below  the  boiling  point  of  the  oil;  each  liquid  contributes  its 
own  quota  of  pressure  independently.  Two  liquids  partially  miscible,  in 
two  layers,  distil  at  a  constant  temperature  until  one  of  the  layers  has  dis- 
appeared, the  temperature  being  generally  higher  the  greater  the  mutual 
solubility  and  therefore  the  greater  the  diminution  of  the  joint  vapour- 
pressure  by  reason  of  the  mutual  attraction  of  the  two  liquids :  and  the 
vapour-pressure,  at  any  given  temperature,  of  a  saturated  solution  of  either 
liquid  in  the  other  is  the  same.  A  mixture  of  miscible  liquids  presents 
two  cases :  ethylic  alcohol  and  water  on  repeated  distillation  ultimately 
give  a  distillate  of  nearly  pure  alcohol ;  a  mixture  of  propylic  alcohol  and 
water  gives  a  distillate  containing  75  per  cent  of  propylic  alcohol  which 
cannot  be  separated  by  farther  distillation ;  while  if  the  liquid  distilled  be 
richer  than  this  in  propyl-alcohol,  water  and  propyl-alcohol  pass  over,  and 
propyl-alcohol  remains  in  the  retort.  In  such  cases,  there  are  differences 
in  the  action  of  the  components  of  the  mixture  on  one  another,  and  of  the 
attraction  of  the  boiling  liquid  for  the  components  of  the  mixed  vapour ; 
and  the  boiling  point  depends  upon  the  resultant  vapour-pressure. 

The  boiling  point  of  a  solution  may  differ  considerably  from  that  of  the 
solvent :  thus  a  saturated  solution  of  caustic  soda  in  water  boils  at  215°-5  C., 
and  one  of  calcium  chloride  at  179°-5  C. 

Vapour-Pressure  of  a  Solution.  —  The  pressure  of  the  vapour  of  a 
solution  is  less  than  that  of  the  solvent  alone  at  the  same  temperature,  and 
the  boiling  point  is  correspondingly  higher.  Whatever  be  the  temperature 
and  the  concentration,  and  whatever  be  the  nature  of  the  solvent  and  the 
substance  dissolved,  the  Fall  of  Vapour-Pressure  is  proportional  to  the  ratio 
of  the  number  of  molecules  of  the  substance  dissolved  to  the  total  number 
of  molecules  in  the  solution  (Raoult).  Apparent  departures  from  this  law, 
in  the  direction  of  an  excess  in  the  fall  of  vapour-pressure,  sometimes  mani- 
fest themselves,  particularly  in  solutions  of  acids,  bases,  and  salts  :  but  these 
departures  are  interpreted  as  showing  that  the  number  of  molecules  in  the 
substance  dissolved  is  altered  by  Dissociation  taking  place  when  solution 
occurs. 

Besides,  the  process  of  evaporation  is  restricted  to  the 
exterior  free  surface :  that  of  boiling  occurs  both  at  this  surface 
and  at  the  internal  surface  of  bubbles  in  the  interior  of  the 
liquid. 

A  liquid  may  be  heated  to  a  temperature  above  its  boiling 
point,  and  if  there  be  no  bubbles  formed,  no  point  "at  which  the 
action  may  preferably  start,  the  whole  liquid  may  become  over- 


388  HEAT.  [CHAP. 

stressed,  like  a  Rupert's  drop,  and  when  it  does  give  way  and 
form  vapour,  it  may  do  so  explosively.  This  kind  of  explosive 
boiling  may  be  observed  when  water  void  of  air  is  heated,  or 
when  drops  of  water  are  suspended  in  a  mixture  of  light  arid 
heavy  oils  of  the  same  specific  density  as  water  and  then  heated, 
or  when  water  is  heated  in  a  glass  vessel,  especially  if  it  have 
been  carefully  cleaned  with  sulphuric  acid.  In  the  last  case  the 
surface  of  the  vessel  is  very  uniform,  and  there  is  no  sharp  point 
or  roughness  at  which  a  bubble  may  commence :  thus  the  tem- 
perature rises  above  the  boiling  point  until  it  is  brought  down 
by  a  sudden  outburst  of  vapour,  and  bumping  ensues.  There  is 
less  of  this  in  a  smooth-metal  vessel  than  in  a  glass  one ;  still 
less  in  a  rough-metal  vessel ;  still  less  where  jagged  pieces  of 
platinum  or  stone  have  been  immersed  in  the  liquid  to  be  boiled. 
The  process  of  boiling  depends  to  a  great  degree  for  its  regu- 
larity on  the  presence  of  air-bubbles:  we  may  sometimes  see 
that  water,  when  long  boiled,  ceases  to  evolve  bubbles  and 
evaporates  only  at  the  surface,  with  an  occasional  outburst  of 
steam. 

A  bubble  of  air  or  vapour,  produced  in  the  interior  of  a  hot 
liquid,  is  increased  in  size  by  molecules  escaping  into  it  from  the 
surrounding  liquid;  if  the  temperature  of  these  molecules,  their 
energy,  their  velocity,  their  pressure,  be  such  that  they  can 
expand  the  bubble  against  the  surrounding  pressure  and  against 
the  surface-tension  within  the  bubble  itself,  the  bubble  enlarges 
and  rises.  If  we  artificially  produce  bubbles  in  the  interior  of  a 
heated  liquid,  as  when  we  electrolyse  hot  water,  the  liquid  boils 
very  rapidly  at  the  electrodes,  where  gaseous  oxygen  and  hydro- 
gen are  being  given  off. 

The  uniformity  of  the  boiling  point  is  interfered  with  by  variations  in 
the  size  of  the  bubbles,  and  therefore  in  the  inward  tension  of  their  liquid 
boundaries,  which  resist  expansion. 

The  boiling  point  at  different  pressures. — The  greater 
the  external  pressure  to  be  overcome,  the  greater  must  be  the 
energy,  and  therefore  the  greater  the  temperature,  of  the  rising 
vapour.  The  temperature  of  ebullition  and  the  external  pres- 
sure are  not  directly  proportional  to  one  another,  but  are  found 
experimentally,  and  recorded  in  tables  such  as  those  great  tables 
of  Regnault's,  to  be  found  in  his  Relation  des  Experiences. 

At  mountain  heights  the  atmospheric  pressure  is  less  and  the  boiling 
point  is  lower;  thus  at  Quito,  at  a  height  of  9540  feet,  water  boils  at 
90°-1  C. 


xiii.]  BOILING.  389 

If  a  flask  containing  water,  boiling  at  100°  C.,  be  corked  and  set  aside 
until  it  has  cooled,  say,  to  90°  C.,  and  if  the  upper  part  of  the  flask,  the  part 
containing  the  vapour  of  water,  be  suddenly  cooled  by  cold  water  dashed 
upon  it,  the  vapour  in  it  will  be  partly  condensed,  and  a  partial  vacuum  will 
be  formed :  the  water  will  find  itself  at  a  temperature  of  90°  C.  under  a 
pressure  of  less  than  525-45  mm.  of  mercury,  and  it  will  again  begin  to  boil : 
the  water  is  thus  seemingly  induced  to  boil  by  the  application  of  cold  to  the 
flask  containing  it. 

Ifacryophorus  tube,  Fig.  131,  of  which  both  bulbs  are 
half  filled  with  water,  have  one  bulb  immersed  in  a  freezing 
mixture,  the  vapour  in  the  cold  bulb  is  condensed ;  the  vapour 
in  the  tube  is  pushed  into  the  cold  bulb  by  the  uncompensated 
pressure  of  particles  rising  from  the 
liquid  in  the  warmer  bulb;  this  pro- 


cess is  continuous ;  work  is  continu- 
ously done  in  maintaining  the  flow  of 
vapour,  which  is  as  continuously  condensed ;  the  liquid  in  the 
warmer  bulb  continuously  evolves  vapour,  and  does  so  so  rapidly, 
the  pressure  being  small,  as  to  boil ;  it  continuously  does  work, 
but  receives  no  energy;  it  cools  and  ultimately  freezes,  even 
while  evaporating. 

Boiling  and  evaporation  may  thus  involve  not  only  the 
giving  of  momentum  to  particles  of  the  liquid,  but  also  external 
work  done  against  resistances ;  and  during  evaporation  there 
may  be  cooling  due  not  only  (1)  to  the  latent  heat  of  evapora- 
tion absorbed  in  producing  change  of  state,  but  also  (2)  to  the 
external  work  which  is  done  by  the  evaporating  body  —  work 
which  generally  takes  the  form  of  thrusting  aside  the  external  air. 

Examples  of  cooling  due  to  evaporation  are:  —  The  cooling  of 
the  skin  by  perspiration  or  by  a  draught  of  air,  even  though  the  air  be  warmer 
than  the  skin ;  a  dog  cooling  himself  by  panting  with  his  tongue  exposed ;  a 
porous  water-cooler  or  alcarraza,  the  evaporation  at  the  surface  of  which  cools 
the  contained  water ;  the  practice  in  some  hot  countries  of  cooling  a  room  by 
throwing  water  over  the  floor ;  the  cooling  of  air  supplied  for  the  ventilation 
of  coalpits  by  injecting  water-spray  into  it ;  the  cooling  of  the  compressed  air 
of  refrigerators  by  the  same  means ;  the  cooling  undergone  by  a  liquid  which 
is  being  rapidly  evaporated,  as,  for  example,  the  rapid  cooling  of  sulphurous 
anhydride  or  of  ammonia,  which  is  effected  in  the  course  of  the  process  of 
artificial  ice-making  by  the  rapid  evaporation  of  the  liquefied  gases  under  a 
powerful  air-pump;  the  cooling  of  a  jet  of  liquefied  carbonic  acid  when 
allowed  to  escape  into  the  air,  so  that  the  substance  is  in  part  solidified. 

Ethylene  (olefiant  gas)  may  be  liquefied  by  cold  and  pressure;  on 
being  rapidly  evaporated  under  the  air-pump  it  becomes  so  cold  that  air, 
greatly  compressed,  can  be  liquefied  by  it.  This  liquefied  air,  when 
allowed  to  evaporate  freely,  produces  temperatures  apparently  below  —  210° 
C.  (Olszewski). 


390  HEAT.  [CHAP. 

The  latent  heat  of  evaporation  of  steam  is  X  =  (33011,504000  - 
33,200000r)  ergs  per  gramme,  where  the  temperature  of  ebullition  is  r° 
Abs.  At  994°-32  Abs.  or  720°-6  C.,  A.  =  0,  and  this  temperature  is  for  steam 
the  Critical  Temperature,  beyond  which  there  is  no  change  of  state  when 
liquid  water  becomes  water-vapour. 

Saturation-pressure.  —  In  the  case  of  every  vapour  we  find 
that  for  each  particular  temperature  there  is  a  maximum  density  ; 
if  we  compress  the  vapour  beyond  this  density,  a  portion  of  it 
will  be  liquefied.  If  we  allow  it  to  expand,  then — provided 
that  the  temperature  be  kept  constant,  and  that  the  vapour  be 
kept  in  contact  with  its  own  liquid  —  a  portion  of  the  liquid 
will  be  evaporated ;  thus  the  density  is  maintained  constant  and 
the  vapour  is  kept  saturated.  Each  volatile  liquid  has  its  own 
saturation-pressure  for  each  temperature,  this  being  the 
pressure  necessary  to  bring  the  vapour  to  its  maximum  density. 

A  vapour  which  is  not  saturated  may  by  compression, 
exerted  until  the  pressure  of  the  vapour  is  equal  to  the  satura- 
tion-pressure, be  made  saturated,  and  by  further  pressure  will 
be  caused  partly  to  condense. 

The  saturation-pressure  of  any  vapour  at  any  temperature 
is  the  same  as  the  pressure  at  which  the  corresponding  liquid 
boils  at  that  temperature. 

Even  in  contact  with  ice,  water-vapour  has  a  saturation- 
pressure,  and  evaporation  will  go  on  until  this  pressure  is 
attained.  A  strong  wind  blowing  over  a  snowfield  may  remove 
much  of  the  snow  by  true  evaporation  without  liquefaction. 

Saturated  steam  in  contact  with  ice  at  t°  C.  has  a  pressure  p  —  (107-2 
+  (6255  x  1-080')}  dynes  per  sq.  cm.  (Regnault). 

As  a  general  rule  each  component  of  a  mixture  of  gases 
exercises  its  own  pressure,  and  is  not  affected  by  the  others  which 
accompany  it.  Yet  this  rule  is  not  absolute  ;  for  if  we  heat  in 
a  flask  a  certain  quantity  of  air  alone,  we  find  that  it  exerts  a 
certain  pressure;  a  certain  quantity  of  water-vapour  introduced 
alone  into  a  vacuum  would  exert  a  certain  pressure ;  but  when 
both  the  water-vapour  and  the  air  are  introduced  into  the  same 
vessel,  the  joint  pressure  falls  somewhat  short  of  the  sum  of  the 
several  pressures,  and  thus  it  is  shown  that  there  is  an  attractive 
action  between  water  and  air. 

Vapours  at  variable  pressures  and  temperatures  generally 
obey  Boyle's  law  with  tolerable  regularity  until  the  pressure 
comes  up  to  about  -^  of  the  saturation-pressure,  and  that 
whether  they  be  alone  or  mingled  with  air  or  with  other  vapours. 


XIII.] 


VAPOUR  DENSITY. 


391 


Fig. 132. 


Measurement  of  Vapour  Density  at  different  temperatures. 

a.  By  measurement  of  the  pressure  exercised  by  the  vapour  of  liquid  at 
a  series  of  known  temperatures. 

This  is  effected  by  the  arrangement  sketched  in  Fig.  132.  The  mean 
temperature  of  boiling  is  indicated  by 
four  thermometers,  two  in  the  liquid, 
two  in  the  vapour:  the  vapour  is 
condensed  in  A  and  returned  to  the 
flask :  the  pressure  is  measured  by  a 
manometer. 

The  use  of  this  method  depends 
on  a  tacit  assumption  that  Boyle's 
law  is  obeyed  throughout  all  ranges 
of  temperature;  but  this  method  is 
not  applicable  except  at  low  tempera- 
tures and  low  pressures ;  for  at  high 
pressures  the  vapour  assumes  abnor- 
mally small  volumes  as  it  approaches 
its  saturation-pressure. 

/?.    By  measurement  of  the  volume 
occupied  by  a  known  weight  of  fluid,  or  by  measurement  of  the  weight  of 
vapour  which  can  occupy  a  known  volume. 

The  first  of  these  methods  is  that  of  Gay  Lussac.  A  tube  filled  with 
mercury  is  inverted  like  a  Torricellian  barometer  in  a  vessel  of  mercury,  and 
has  a  Torricellian  vacuum  at  its  upper  part ;  the  whole  is  immersed  in  a  bath 
of  liquid  kept  at  a  definite  temperature.  A  little  bulb  containing  a  known 
quantity  of  the  liquid  to  be  vaporised  is  passed  up  into  the  tube;  being 
heated  it  bursts;  the  vapour  occupies  a  certain  volume  of  the  tube;  the 
mercury  stands  at  a  certain  height  in  the  tube.  The  mercury  stands  at  a 
different  height  in  an  ordinary  barometer;  the  difference  of  readings  indicates 
the  pressure  exercised  by  the  vapour.  Its  weight  is  known,  its  volume,  and 
its  temperature.  A  series  of  observations  is  made  at  different  bath-tempera- 
tures. It  is  difficult  to  ensure  that  all  the  substance  is  liquefied.  Y.  and 
C.  Meyer  use  a  large  long-necked  closed  flask :  this  is  heated  until  no  more 
air  escapes ;  when  this  is  the  case,  a  little  glass  globe  containing  a  known 
quantity,  m  grammes,  of  the  liquid  to  be  tested  is  dropped  from  the  cool 
top  of  the  neck :  it  breaks  and  the  liquid  evaporates  :  the  vapour  drives  out, 
say,  to  cub.  cm.  of  air  from  the  flask :  then,  in  C.G.S.  measures,  the  vapour 
density  at  the  temperature  and  (corrected)  pressure  of  observation  is  m/to. 

The  second  method  is  that  of  Dumas.  A  bulb  with  a  long-drawn  neck 
is  filled  with  liquid  and  immersed  in  a  heated  bath.  The  liquid  in  the  bulb 
violently  rushes  out  in  the  vapourous  state  through  the  narrow  neck ;  this 
ceases  and  equilibrium  is,  set  up;  the  bulb  is  filled  with  vapour  at  the 
temperature  of  the  bath.  The  end  of  the  neck  is  then  sealed  by  a  blow- 
pipe-flame ;  the  whole  is  removed,  cooled,  weighed.  This  gives  the  weight 
of  bulb  +  vapour;  already  the  weight  of  the  bulb,  its  volume,  the  bath- 
temperature,  are  known ;  the  density  of  the  vapour  occupying  the  bulb  at 
the  temperature  of  the  bath  can  be  thus  found.  At  high  temperatures  bulbs 
of  porcelain  or  iron,  and  baths  of  mercury-vapour,  sulphur-vapour,  or  zinc- 
vapour,  may  be  used  (Deville  and  Troost). 

The  density  of  saturated  vapour.  —  Fairbairn  and  Tate  found  the 
density  of  saturated  steam  by  introducing  into  a  recipient  of  known  capacity 


392  HEAT.  [CHAP. 

and  devoid  of  air  a  known  quantity  of  water,  and  by  measuring  the  temper- 
ature at  which  the  whole  of  the  water  was  evaporated. 

The  measurement  of  the  pressure  of  un  saturated  vapour,  if  it  present 
itself  alone,  is  simply  the  measurement  of  gaseous  pressure,  and  calls  for  no 
further  remark. 

The  measurement  of  the  pressure  exercised  by  an  unsaturated  vapour 
which  forms  one  of  the  components  of  a  mixture  is  in  one  case  —  that  of 
Aqueous  Vapour  in  the  Air  —  a  matter  of  importance.  A  numerical  example 
will  illustrate  this.  If  water  be  exposed  to  a  pressure  of  9-16  mm.  of  mer- 
cury (=0-01205  atmos.),  it  will  boil  at  10°  C.;  if  water- vapour,  of  such 
density  (supposed  constant)  that  it  exerts  a  pressure  of  0-01205  atmos.,  be 
exposed  to  a  temperature  above  10°  C.,  it  will  be  unsaturated;  at  10°  C.,  it 
will  be  saturated ;  at  any  temperature  below  10°  C.,  it  will  be  in  part  con- 
densed. 10°  C.  is,  then,  the  Condensation-Temperature  for  aqueous 
vapour  of  this  pressure  of  -01205  atmos.,  just  as  the  latter  is  the  saturation- 
pressure  for  aqueous  vapour  at  a  temperature  of  10°  C.  If,  now,  we  take 
moist  air  containing  aqueous  vapour  and  air  in  the  proportion  of  0-01205 
to  0-98795,  at  the  ordinary  atmospheric  pressure  :  at  any  temperature  above 
10°  C.  it  will  not  deposit  moisture  ;  at  10°  C.  it  will  begin  to  do  so.  10°  C. 
is  the  condensing  temperature  or  Dewpoint  for  air  containing  this  propor- 
tion of  moisture.  To  other  proportions  of  moisture  other  dewpoints  cor- 
respond ;  these  can  be  found  in  any  table  of  the  boiling  points  of  water  at 
different  pressures.  Hence,  if  we  can  find  the  temperature  at  which  air 
containing  aqueous  vapour  begins  to  deposit  moisture,  we  can  by  reference 
to  such  tables  find  the  proportion  of  aqueous  vapour  in  the  air.  This  tem- 
perature is  ascertained  by  a  Hygrometer. 

The  essential  part  of  a  hygrometer  is  a  glass  —  or,  better,  a  smooth  silver 
—  surface,  which  can  be  cooled  down  until  the  moisture  of  the  air  begins  to 
deposit  as  a  film  upon  it,  and  whose  temperature  at  the  instant  of  the  dim- 
ming of  its  brightness  can  be  accurately  ascertained.  The  surface  may  be 
fashioned  into  a  bulb  :  this  bulb  may  contain  ether ;  the  bulb  may  be  cooled 
by  blowing  through  and  thus  rapidly  evaporating  the  ether  ;  the  tempera- 
ture at  the  instant  of  dimming  of  the  surface  can  be  read  off  on  a  ther- 
mometer whose  lower  end  is  dipped  in  the  evaporating  ether.  The  whole 
may  be  allowed  spontaneously  to  become  warmer ;  as  it  does  so,  the  film  dis- 
appears :  the  temperature  at  which  this  occurs  is  noted.  The  film  is  again 
caused  to  appear  and  disappear ;  by  dint  of  repetition  a  mean  point  between 
the  highest  temperature  of  appearance  of  the  film  and  the  lowest  tempera- 
ture of  its  disappearance  is  obtained,  which  is  the  Dewpoint  required. 

Another  method  for  ascertaining  the  dewpoint  —  one  for  doing  so  by  a 
single  observation — is  the  following:  —  If  a  thermometer  bulb  be  by  any 
means  kept  cool  by  evaporation  —  being  covered  with  a  wet  piece  of  linen 
which  dips  in  water,  or  the  like  —  the  bulb  is  cooled ;  the  extent  of  cooling 
depends  on  the  rapidity  of  evaporation :  the  rapidity  of  evaporation  depends 
on  the  Humidity  of  the  air  —  that  is,  on  the  ratio  between  the  amount  of 
aqueous  vapour  actually  present  in  the  air,  and  that  which,  at  the  tempera- 
ture of  the  air,  would  be  present  if  the  air  were  saturated  with  moisture. 
The  less  the  humidity  of  the  air,  the  greater  will  be  the  evaporation,  and  the 
greater  will  be  the  difference  between  the  readings  of  a  thermometer  kept 
cool  in  this  way  and  those  of  a  thermometer  subjected  to  normal  circum- 
stances. Tables  have  been  constructed  in  which,  for  each  reading  of  the 
"  dry  bulb  "  and  of  the  "  wet  bulb,"  the  corresponding  percentage  of  aqueous 
vapour  in  the  air  is  recorded. 


XIII.] 


DEW. 


393 


Dew.  —  When,  on  a  clear  night,  the  earth,  stones,  plants, 
etc.,  become  cool  by  free  radiation,  their  temperature  may  sink 
below  the  condensation-temperature  proper  to  the  particular 
percentage  of  aqueous  vapour  in  the  air.  When  the  tempera- 
ture thus  sinks  below  the  dewpoint,  the  moisture  of  the  air  is 
partly  deposited  in  the  form  of  dew ;  and  the  more  highly 
charged  with  moisture  the  air  had  become  during  the  day,  the 
earlier  and  the  heavier  is  the  deposit  of  dew  at  night. 

The  soil  immediately  underneath  the  surface  is  at  the  same 
time  warmer  than  the  air  or  the  surface  of  the  soil ;  moisture  is 
condensed  on  the  under  surface  of  cold  stones,  etc.  Much  of 
what  is  called  Dew  is,  however,  liquid  transuded  from  plants 
themselves  (J.  Aitken). 


TRANSFORMATIONS  OF  HEAT. 

Transformation  of  Work  into  Heat  may  be  effected  directly 
by  the  agency  of  friction,  or  indirectly  by  the  transformation  of 
kinetic  energy  into  the  energies  of  noise,  light,  electrical  condi- 
tion, which  are  in  their  turn  converted  into  heat.  Even  the 
conversion,  apparently  direct,  by  the  agency  of  friction  may  be 
due  in  the  first  place  to  the  generation  of  local  electrical  currents 
or  conditions,  the  energy  of  which  is  afterwards  converted  into 
heat. 

Transformation  of  Heat  into  Work.  —  From  our  previous  discus- 
sion of  the  Indicator-Diagram  we  understand  that  the  work  done  by  any 


FigM33. 


Fig.134. 


substance  during  expansion  can  be  represented  by  the  area  PP'VVP  (Fig. 
133),  where  OP,  OP'  represent  the  original  and  final  pressures,  OV  and  OV 
the  original  and  final  volumes.  The  work  is  positive,  done  by'ihe  expanding 
substance  (steam,  air,  etc.)  if  the  expansion  be  positive,  from  OV  to  OV  ; 


394 


HEAT. 


[CHAP. 


negative,  done  upon  it  if  the  expansion  be  negative,  as  from  OV  to  a  less 
value  OV. 

Where  work  is  done  both  by  and  upon  the  working  substance,  as  in  Fig. 
134,  the  negative  work  p'p"yy'p'  being  subtracted  from  the  positive  work 
PFV'VP,  there  is  left  an  area  PP'P"V"V,  which  represents  the  work  done. 
If  the  curve  PP'P"  be  complicated,  the  total  work  done  may  be  found  by 
dissecting  the  figure ;  any  complex  operation  may  be  resolved  into  a  number 
of  simple  ones,  of  which  each  produces  its  own  effect ;  the  work  done  is  found 
by  a  process  of  summation  of  positive  and  negative  areas. 

When  the  working  substance  returns  to  its  original  volume  and  pressure, 
as  in  Fig.  135,  the  shaded  area  again  indicates  the  amount  of  work  done  by 
the  working  substance,  just  as  if.  in  Fig.  134  the  line  P"V"  had  been  made 

to  coincide  with  PV.  The  work  is  positive 
if  the  change  of  pressure  and  of  volume  have 
been  effected  in  the  direction  of  the  arrows ; 
negative  if  effected  in  the  contrary  sense. 
Such  an  operation  is  a  Cycle. 

The  advantage  of  studying  the  amount 
of  work  done  by  a  working  substance  oper- 
ating in  a  cycle  is  that  we  are  riot  called 
upon  to  take  any  internal  work  into  account. 
The  body  returns  at  the  end  of  the  operation 
to  its  primitive  condition,  and  there  is  no 
balance  of  work  done  either  by  or  against 
internal  forces. 

Into  the  consideration  of  a  cycle  we  in- 
troduce an  assumption  that  it  is  possible  for 


Fig.135. 


a  working  substance  to  return  to  the  same 

condition  as  regards  pressure  and  volume  at  the  original  temperature; 
this  might  not  have  been  true  as  regards  any  actual  substance,  though  it  is 
theoretically  true  as  regards  perfect  gases ;  it  is,  however,  actually  true  as 
regards  physical  gases,  for  the  elasticity  of  gases  is  perfect. 

We  must  choose  some  particular  kind  of  cycle  for  our  ideal  operations  ; 
that  to  be  explained  is  the  one  best  adapted  for  the  study  of  the  relations 
between  work  and  heat,  and  was  devised  in  its  primitive  form  by  Sadi 
Carnot ;  it  is  hence  known  as  Carnot's  cycle. 

If  a  gas  expand  at  constant  temperature,  we  know  by  Boyle's  law 
that  the  pressure  and  the  volume  vary  inversely ;  this  law  can  be  expressed 
graphically  by  an  equilateral  hyperbola,  for  in  that  curve  xy  —  const.  The 
pressures  and  volumes  at  different  temperatures  are  represented  by  points 
on  different  hyperbolas.  Imagine  the  curves  of  Fig.  136  to  represent 
portions  of  the  hyperbolas  corresponding  to  temperatures  rf  Abs.  and 
T2°  Abs.  for  a  given  mass  of  substance.  This  substance,  at  the  temperature 
T2  and  pressure  pv  will  have  the  volume  to :  to  the  pressure  p2  at  the  same 
temperature  corresponds  volume  b2 ;  if  the  temperature  be  rl  and  pressure 
pv  the  volume  will  be  not  fo,  but  to,,  a  point  on  the  higher  hyperbola,  on  the 
line  —  the  so-called  Isothermal  line  —  corresponding  to  the  higher  temper- 
ture  T,°  Abs. 

Expansion  of  a  gas  involves  a  more  rapid  fall  of  pressure  when  it  is 
effected  adiabatically  than  when  effected  at  constant  temperature,  for  the 
gas  cools  down  :  the  Adiabatic  lines,  which  express  the  relations  between 
pressure  and  volume  when  heat  is  neither  supplied  nor  allowed  to  escape, 


XIII.] 


ADIABATIC   EQUATION. 


396 


slope  more  steeply  than  the  isothermal  lines  for  the  same  substance.  The 
equation  by  which  any  one  of  these  lines  may  be  traced  out  is  called  the 
Adiabatic  Equation,  and  it  is  p/pk/c  =  const.,  or,  for  a  given  mass  of  gas, 
j»to*/c  =  const.,*  where  k/c  is  the  ratio  of  the  two  thermal  capacities  of  the 
gas  in  question.  Fig.  137  represents  these  lines,  and  shows  the  relations 
between  the  pressures  and  volumes  of  a  substance  starting  from  conditions 
Pv  fcp  Ti»  anc*  Pv  &»  T2>  which  correspond  to  those  of  the  previous  figure. 


Fig.136. 


Fig.137. 


Let  us  now  superpose  the  two  figures  136  and  137,  and  we  obtain  Fig. 
138,  and  are  now  prepared  to  understand  Carnot's  cycle  in  its  modern  form. 
The  steps  of  Carnot's  cycle  :  — 
1.    Starting  with  our  working  substance  at  the  condition  pv  bi,  rf  Abs. 

*  From  &-?wr  =  pij  [i]  we  get,  by  differentiation,  &•  mf  =p<o  +  (op  [ii] ;  and 
also,  p.  370,  &  =  k  —  c  [iiij.  Of  any  small  element,  =  k  •  mf  ergs,  of  Heat  supplied, 
c  •  mf  ergs  would  be  consumed  in  raising  the  temperature  by  f ,  where  c  is  the  ther- 
mal capacity  at  const,  vol. ;  and  the  remainder  would  do  external  work  equal  to  p'o 
ergs ;  whence  the  heat  supplied  =  k  •  raf  =  c  •  mf  -\-pij  [iv] ;  but  this  =  0  under  adia- 
batic  conditions ;  whence  c  •  mf  -\-pif  =  0  [v].  From  equations  [v],  [ii],  and  [iii],  we 
get  k  -pij  +  c  -pij  =  0,  by  eliminating  f  and  multiplying  by  k  —  c  ,•  and  this,  on  being 
transformed  into  &ij/ij  +  c j  /p  =  0,  may  be  integrated,  and  we  then  get  pclok  =  const., 
orp/pt  =(b//b)*/c-  Then,  whatever  the  mass  m  may  be,  p/pt  =(P/Pi)k/c]  and 
pc/pk  =  const.  It  is  assumed  in  this  that  the  gas  is  perfect,  and  L  =  0. 

Otherwise. — Suppose  a  body  of  mass  m  grammes  to  possess  on  the  whole  H 
ergs  of  Heat,  at  a  temperature  T°  Abs. :  then  the  quotient  H/rar  is  the  Entropy  or 
Thermodynamic  Function  <P.  This  function^  represents  a  certain  condition 
of  the  body ;  it  increases  if  the  total  Heat  H  in  the  body  increase,  and  vice  versa ; 
and  it  cannot  change  unless  Heat  enter  or  leave  the  body,  so  long  as  the  body  is  single, 
and  is  not  a  system  of  unequally-heated  parts,  of  which  some  gain  while  others  lose 
heat.  If  the  body  be  thus  single,  and  not  such  a  system,  the  isentropic  curves  must 
be  the  same  as  the  adiabatic  curves.  Any  small  amount  of  Heat,  5H  ergs,  supplied 
to  a  given  mass  m  at  an  average  temperature  r,  would  produce  a  change  5H/rar  =  <f 
in  the  Entropy ;  but  5H  =  (c- raf +  />£):  and  therefore  fjp  =  5H/mr  =(cr/r  +  pij/wr) 

=(cf/r  +  a^/b)  ={(C/fcmT)  (pi  +  bp)  +  » •  ij/b}  =  {(c/pb)  (pij  +&/>)+(&— c)-i/b} 

=  (cp/p  +  Mj/fc) ;  which  on  being  integrated  gives  <P  =  c  logp  +  k  logfo,  whence 
pcfofc  =  e^,  and  finally,  whatever  be  the  value  of  m,  p/pi  =  (P/P/rV.  Here  e  is  the 
base  of  the  Naperian  logarithms,  and  e^  is,  for  any  determinate  value  of  <p  or 
H/WT,  a  constant  quantity. 


390 


HEAT. 


[CHAP. 


(point  A),  we  allow  it  to  expand  at  the  temperature  TI°,  this  temperature 
being  maintained   constant.      Running  through  successive  pressures  and 

volumes   represented  by  succes- 

Fig.i38.  sive  points  on   the    isothermal 

line.  TV  it  assumes  a  pressure, 
say,  />/  and  volume  to/  (point 
B). 

Work  is  done  equal  to  pjtoj 
log  (to/AO  =  the  area  ABV/ \r 
This  work  is  done  at  the  ex- 
pense of  heat-energy  supplied 
to  the  working  substance  from 
an  external  source. 

2.  Starting  from  the  con- 
dition j»/,  to/,  TJ  (point  B)  we 
allow  the  working  substance  to 
expand  adiabatically,  until  it 
assumes  the  temperature  T2°  and 

^      ^         Vg  the  corresponding  condition  p2', 

BC  is  a  part  of  the  adiabatic  line  passing  through  B  and 


to/,  T2°  (point  C). 


cutting  the  r2  isothermal  in  C. 

Work  equal  to  the  area  BCV2'V/  is  done  by  the  expanding  substance, 
but  at  the  expense  of  its  own  heat-energy,  for  no  heat  is  supplied  to  it. 

3.  The   substance   is  now  compressed  until  it  assumes  the  condition 
p2,  to2,  T2 — that  is,  until  it  runs  from  C  so  far  up  the  isothermal  line  r2  as 
to  meet  at  D  an  adiabatic  line,  which  passes  through  the  original  point  A. 
Work  is  done  equal  to  the  area  CDV2V2'  =  p2(Q2  log  (to2'/to2)  :  but  it  is  done 
upon  the  working  substance,  for  that  substance  is  compressed  :  and  heat  to  a 
corresponding  amount  is  lost  by  the  working  substance,  for  it  passes  to  sur- 
rounding objects,  and  may  be  wasted  by  conduction  and  radiation  into  all 
the  universe. 

4.  The  body,  from  which  no  more  heat  is  allowed  to  escape,  is  now  sup- 
posed to  be  still  further  compressed  until  it  has  regained  its  original  condi- 
tion pv  tor  r/3.     Work  is  done  on  the  working  substance  thus  compressed, 
but  appears  as  heat  in  the  substance,  not  as  external  work  either  positive  or 
negative,  and  the  temperature  rises,  for  no  heat  is  allowed  to  escape. 

During  the  adiabatic  expansion  in  Step  2,  the  change  of  temperature  is 
from  rl  to  T2;  and  (see  footnote,  p.  373)  in  that  case  T1/r2  =  (PI /p2ryk~c^k 
=  (b2'/to/)(*~c)/*.  Similarly,  in  Step  4,  the  temperatures  are  again  r2  and  TJ  ; 
and  rl/T2  =  (to2Ai)(*~c)/*-  Therefore  too'/to/  =  to2/toL ;  or  to2'/to2  =  D/Ar 

The  whole  energy  supplied  to  the  working  substance  from  the  source 
is  p^  log  (to/Xtoj)  ;  that  wasted  is  />2to2  log  (to2'/to2)  =jt>2b2  log  (to//toj) ;  that 
utilised  is  [(Iog(to//b0-(^1-p2to2))^(rtlog(to//to^]-^:^  of 

the  whole.  But  j»1to1  =  m  •  T&rl ;  jt?2to2  =  m  -  2&r2,  where  rl  and  T2  are  the 
respective  temperatures.  Hence  the  proportion  of  energy  utilised  is 
{(wi&Tj  —  ml&To)  -4-  m&Tj}  or  (rl  —  T2)/rl  of  the  whole. 

The  working  substance  operating  in  such  a  cycle  acts  as  a  distributor  of 
energy ;  it  divides  5H,  the  heat-energy  supplied  to  it  from  the  Source  of  heat, 
into  two  parts  :  one  part,  5'H,  passing  to  the  Condenser,  is  lost  by  conduction 
and  radiation ;  the  remainder,  W,  is  usefully  converted  into  external  Work. 

The  heat  SH  is  supplied  at  the  higher  temperature  TI  ;  the  quantity  of 


xiii.]  CARNOT'S   CYCLE.  397 

heat  S'H  is  lost  to  surrounding  objects  at  the  lower  temperature  TO;  the 
Efficiency  of  such  an  ideal  arrangement  is  the  ratio 

Heat  utilised  _  8H  -  8'H  _  W      r1-r2 
Heat  supplied  5H  8H          rl 

Thus,  so  far  as  Carnot's  cycle  is  concerned,  even  though  we  could  find  a 
working  substance  and  construct  a  machine  which  could  carry  the  cycle  out 
in  practice,  yet  there  would  be  a  great  waste  of  heat-energy,  unavoidable 
unless  we  had  a  condenser  at  a  temperature  of  absolute  zero.  If  the  tem- 
perature of  the  boiler  of  an  ideal  engine  competent  to  work  out  Carnot's 
cycle  were  120°  C.  (393°  Abs.),  and  that  of  the  condenser  0°  C.  (273°  Abs.), 

the  work  done  by  such  an  engine  could  not  exceed  — t  ~  "  ' ,  or  about  30-6 
per  cent  of  the  whole  energy  supplied  as  heat. 

The  cycle  above  considered  is  reversible  ;  each  step  in  it  can  be  retraced 
—  or  could  be  retraced  if  we  could  construct  an  engine  capable  of  working 
without  waste  of  energy  in  noise,  friction,  excessive  conduction  and  radi- 
ation of  heat,  and  the  like — work  being  done  not  by  but  upon  the  engine 
as  it  is  driven  backwards. 

The  effect  of  reversing  such  a  cycle  would  be  that  work  W  being  done 
upon  the  engine,  the  quantity  8'H  of  heat  would  be  taken  from  the  con- 
denser, and  the  quantity  8H  of  heat  would  be  communicated  to  the  source. 

Any  engine  which  operates  through  periodic  cycles  must  be  a  reciprocat- 
ing engine  :  and  in  every  reciprocating  engine  there  is  an  absolutely  neces- 
sary waste  of  energy  arising  from  the  necessity  of  restoring  the  engine  to 
its  primitive  position  in  order  that  its  piston  may  repeat  its  effective 
thrusts. 

Carnot's  ideal  "  perfect "  engine  is  one  which,  with  a  working  substance 
capable  of  returning  to  its  primitive  condition,  will  work  out  the  reversible 
cycle  above  described,  and  thus  attain  the  efficiency  above  indicated :  an 
engine  which  wastes  no  energy  otherwise  than  by  restoring  the  primitive 
condition  of  its  working  substance. 

The  perfection  of  a  perfect  engine  depends  not  on  the  nature  of  the 
working  substance,  but  on  the  reversibility  of  the  cycle  which  it  operates, 
and  the  efficiency  of  such  a  reversible  engine  depends  only  on  the  temper- 
atures between  which  it  works.  Carnot's  Principle,  as  enounced  by 
himself,  is  —  the  motive  power  of  heat  is  independent  of  the  material  agents 
employed  to  realise  it ;  its  quantity  is  determined  solely  by  the  temperatures 
between  which  the  "  transport  of  Caloric  "  *  is  effected. 

The  efficiency  ^—  =  <£  (r,  T  —  f  ),  where  f  =  TI  —  r2. 
*  "    oH 

=/(T,T)-/'(T,T)=O-,KT)T. 

The  efficiency  depends  upon  f,  the  difference  of  temperatures  between 
the  source  and  condenser,  and  upon  \j/  (T),  a  function  of  T  which  is  called 
Carnot's  function,  C. 

We   have   also  seen   that  efficiency  =  difference   of    temperatures  =  f 

temperature  of  source          T 
Hence  C  =  l/r;   and  Carnot's  function  is  the  reciprocal  of  the  Absolute 

Temperature  of  the  Source. 

f 

*  An  expression  implying,  as  in  his  day,  the  material  theory  of  heat. 


398  HEAT.  [CHAP. 

The  Efficiency  of  a  Carnot's  Reversible  Reciprocating 
Engine  is  greater  than  that  of  any  other  reciprocating  engine.  If  it 
were  possible  to  devise  a  more  efficient  reciprocating  engine  it  might  be 
employed  with  the  expenditure  of  a  certain  amount  of  heat  to  drive  a 
reversible  reciprocating  engine  backwards  ;  the  source  and  the  condenser 
of  the  Carnot's  engine  might  be  the  same  as  those  of  the  more  efficient  en- 
gine :  the  Carnot's  engine  would  be  occupied  in  restoring  to  the  source  the 
heat  taken  from  it  by  the  better  engine  ;  on  the  whole,  a  surplus  of  work 
would  during  each  cycle  be  done  by  the  conjoined  mechanism  —  a  surplus 
not  accounted  for  by  heat  lost  by  any  body  —  a  creation  of  energy. 

If  the  better  engine  were  employed  in  driving  a  larger  Carnot's  engine 
backwards,  there  might  be  no  surplus,  no  external  work  done  ;  but  a  greater 
amount  of  heat  would  be  conveyed  to  the  source  by  the  reversed  Carnot  than 
would  be  taken  from  it  by  the  more  efficient  but  smaller  engine,  and  the 
whole  heat  of  the  universe  might  be,  step  by  step,  induced  to  travel  through 
the  condenser  into  the  source  of  the  conjoined  mechanism  —  a  conclusion 
evidently  absurd. 

That  this  conclusion  is  absurd,  or  at  any  rate  contrary  to  experience,  so 
long  as  we  cannot  deal  like  Clerk  Maxwell's  Demon  (p.  52)  with  single 
molecules,  it  is  the  aim  of  the  Second  Law  of  Thermodynamics  to 
state  :  —  Heat  cannot  of  itself  pass  from  a  colder  body  to  a  hotter  one,  nor 
can  it  be  made  so  to  pass  by  any  inanimate  material  mechanism  :  and  no 
mechanism  can  be  driven  by  a  simple  cooling  of  any  material  object  below 
the  temperature  of  surrounding  objects. 

The  word  simple,  or  some  equivalent  word,  is  necessary  in  the  above 
statement  of  the  second  law  for  the  following  reason  :  —  A  quantity  of  com- 
pressed gas  can  do  external  work,  and  in  so  doing  cool  itself  below  the 
temperature  of  surrounding  objects  ;  but  its  cooling  is  not  a  simple  loss  of 
heat-energy  ;  there  is  a  concurrent  change  of  condition  of  the  gas,  a  change 
which  cannot  be  reversed  without  the  expenditure  of  heat  exceeding  in 
amount  the  heat  converted  into  work  by  the  expanding  gas. 

This  being  admitted,  we  may  reason  backwards  arid  arrive  at  the  ratio 

of  efficiency  -        -  in  a  reversible  engine  as  a  direct  corollary  of  the  prop- 

osition ;  and  the  statement  of  that  ratio  of  efficiency  in  a  reversible  recipro- 
cating engine  is  also  known  as  the  Second  Law  of  Thermodynamics. 

This  Protean  law  assumes  another  form,  apparently  different  from  but 
essentially  identical  with  both  the  preceding.  Temperature  being  assumed 
proportional  to  the  total  heat-energy,  the  amount  of  heat-energy,  SH  ergs, 
supplied  at  the  higher  temperature  rv  is  proportional  to  TT  ;  8H  =  dmr1  ; 
=  <f>.  Similarly  S'H,  the  heat  lost  to  the  condenser  at  the  lower  tem- 


perature T2,  is  8'H  =  ^wr2;  8'H/mr2=:c£.*   Hence  8H/mr1  =  8'H/mr2;  and  from 
this  we  may  not  only  derive  the  former  equation  -  ~  —  =  -i-^l^,  but  also 

Oil  T 


the  equation  SH/mrj  —  S'H/??ir2  =  0  ;  an  equation  which,  in  the  most  general 
case,  takes  a  form  applicable  to  the  most  complex  reversible  cycle,  namely, 
2(SH/mT)  =  0,  or,  when  the  mass  of  gas  referred-  to  is  a  unit-mass, 
JWH/T  =  0  (Lord  Kelvin)  ;  an  expression  very  convenient  for  mathematical 
purposes,  but  difficult  to  translate  into  words.  —  In  a  perfect,  a  reversible 

*  The  value  of  $  is  the  same  in  both  these  cases,  because  in  both  cases  the 
change  of  entropy  is  the  difference  between  the  entropies  of  the  isentropic  or  adia- 
batic  lines  AD  and  BC,  Fig.  138. 


xiii.]  SECOND   LAW   OF  THERMODYNAMICS.  399 

cycle,  the  Entropy,*  the  sum  of  the  equivalences  of  all  the  transformations, 
is  zero  (Clausius).  In  a  non-reversible  process  the  sum  of  the  transforma- 
tions is  positive,  and  since  all  processes  are  non-reversible,  the  sum  of  the 
entropies  in  the  universe  tends  to  a  maximum.  According  to  Rankine's 
mode  of  expression,  substantially  identical  with  the  preceding,  the  second 
law  is :  If  the  absolute  temperature  of  a  uniformly-hot  substance  be  divided 
into  any  number  of  equal  parts,  the  effect  of  each  of  those  parts  in  causing 
work  to  be  performed  is  equal.  This  implies  that  the  absolute  temperature 
is  proportional  to  the  total  heat-energy,  and  so  merges  into  the  preceding 
form  of  the  second  law. 

Lastly,  Carnot's  Principle  itself  is  often  called  the  Second  Law  of 
Thermodynamics. 

We  have  already  studied  the  direct  transformation  of  heat 
into  work  in  the  radiometer.  In  the  steam-engine  the  heat  of 
the  steam  may  be  in  part  converted  into  work ;  the  piston  is 
bombarded  by  the  particles  of  the  steam,  and  if  the  resistance  to 
its  onward  movement  be  not  excessive,  it  is  thrust  forward  by 
the  joint  impact  of  the  particles  which  impinge  on  it,  their  several 
components  of  motion  parallel  to  the  piston-rod  being  effective  in 
this  respect. 

Even  under  the  most  favourable  circumstances  which  can  be 
conceived,  heat  cannot  be  wholly  converted  into  work  by  any 
form  of  continuously-acting  mechanism.  The  efficiency  of  the 
ideal  perfect  engine  —  small  though  that  efficiency  be  —  is  never 
approached  in  practice ;  and  the  efficiency  of  the  human  body 
considered  as  a  machine  —  one-fifth  of  the  total  energy  supplied 
to  it  being  capable  of  utilisation  —  is  remarkable  when  we  con- 
sider the  narrow  limits  within  which  it  operates. 

Work  can  thus  be  wholly  converted  into  heat,  but  heat  can 
never  be  wholly  converted  into  work ;  whence  a  universal  ten- 
dency to  the  Degradation  of  Energy  into  Heat,  the  lowest  of 
its  forms. 

MEASUREMENT  OF  HEAT. 

Temperature  we  have  now  seen  to  be,  when  measured  from 
an  absolute  zero  —  a  zero  of  absolute  cold  —  (1)  proportional 
to  the  absolute  amount  of  molecular  kinetic  energy,  and  (2)  the 
reciprocal  of  Carnot's  function. 

What  is  meant  by  equal  degrees  of  heat  ?  Why  is  the  dif- 
ference between  0°  C.  and  1°  C.  supposed  to  be  equal  to  that 

*  Clausius,  dealing  always  with  unit-masses,  has  applied  the  term  Entropy  to  the 
expression  S(5H/r)=  £(£) ;  and  it  will  not  be  difficult  to  see  that  where  the  sum  is 
positive,  more  heat  is  given  to  the  engine  by  the  source  than  is  giyen  when  that  sum 
=  0,  the  work  done,  W,  remaining  unchanged;  but  this  excess  is  wasted  bypassing 
through  the  condenser  to  the  external  universe. 


400  HEAT.  [CHAP. 

between  100°  C.  and  101°  C.?  —  In  a  perfect  gas  equal  differ- 
ences of  temperature  correspond  to  equal  increments  of  energy. 

In  a  diagram  containing  a  system  of  adiabatic  and  isothermal  lines,  the 
isothermal  lines  must  be  so  drawn  as  to  cut  off  equal  areas  between  the 
adiabatic  lines. 

Absolute  zero  would  correspond  to  total  absence  of  molecu- 
lar kinetic  energy. 

If  we  had  a  perfect  gas  at  command  we  might  measure 
temperature  by  its  means  in  either  of  two  ways :  — 

(1)  We   might  observe  its  pressure  at  constant  volume : 
equal  increments  of  pressure  correspond  to  equal  increments  of 
temperature. 

(2)  We    might   observe   its  varying  volume   at   constant 
pressure :   the  volume  is  proportional  to  the  absolute  tempera- 
ture, and  equal  small  increments  of  volume  approximately  cor- 
respond to  equal  small  increments  of  temperature. 

The  former  is  the  more  accurate  method. 

We  have  no  perfect  gases  to  experiment  upon :  air,  etc., 
are  not  perfect  gases.  Yet  we  may  perform  either  of  the  above 
operations  on  a  quantity  of  air  confined  in  a  flask,  and  thus 
construct  an  air  thermometer.  The  former  method — that  of 
observation  of  pressure  —  is  here  doubly  preferable  to  the  lat- 
ter —  that  of  observation  of  expansion  —  because  in  the  former 
there  is  no  waste  of  energy  in  doing  either  internal  or  external 
work,  and  the  increase  of  pressure  is  appreciably  the  same  as 
that  of  a  perfect  gas.  The  indications  of  an  air  thermometer 
used  in  this  way  may  hence  be  assumed  as  an-  approximate 
standard  of  comparison. 

For  the  corrections  necessary,  see  the  table  in  Tait's  Heat,  p.  340. 

By  the  air  thermometer  we  find  that  for  a  fall  of  1°  C.  (from  1°  to 
0°  C.)  on  the  mercurial  thermometer,  the  pressure  sinks  in  the  ratio 
of  274  to  273 ;  hence  the  temperature  sinks  in  the  same  ratio,  absolute 

zero  is  —  273°  C.,  and  Carnot's  function  has  the  numerical  value  of  ^75  for 

-j  <£/  O 

a  temperature  of  0°  C.,  and  of  — ; for  a  temperature  of  t°  C. 

Two  bodies  are  said  to  be  at  different  temperatures  when 
the  one  has  a  tendency  to  lose  heat  to  the  other ;  to  have  the 
same  temperature  when  there  is  no  such  tendency :  and  bodies 
are  at  the  same  temperature  when  they  have  the  same  kinetic 
energy  per  molecule,  not  per  unit  of  weight. 

Differences  of  temperature  may  be  roughly  perceived  by 
the  hand ;  the  sense  of  temperature  can  even  be  cultivated  like 


XIII.] 


TEMPERATURE. 


401 


Fig.139. 


that  of  musical  pitch  so  as  to  arrive  at  approximate  accuracy 
without  actual  recurrence  to  a  standard  of  known  temperature. 

Any  of  the  effects  of  heat  may  be  used  for  detecting  the 
presence  of  heat  and  for  constructing  a  thermoscope.  Arbi- 
trary graduation  of  any  thermoscope  will  enable  it  to  be  used 
as  a  thermometer. 

Breguet's  metallic  thermometer  is  a  spiral  strip  composed  of 
three  metal  strips  soldered  together  by  their  broad  surfaces  :  the 
different  rates  of  expansion  cause  the  spiral  to  roll  or  unroll 
according  to  the  variations  of  temperature,  and  thereby  to  move 
a  pointer. 

The  air  thermometer,  one  of  whose  forms  is  shown  in 
Fig.  139,  is  principally  used  as  a  standard  of  reference.  AD  is 
a  manometer,  in  which  above  D  there  is 
a  Torricellian  vacuum:  H  is  an  air 
chamber,  E  an  auxiliary  cistern  of  mer- 
cury. As  far  as  the  mercury  A  is  de- 
pressed below  a  certain  mark,  so  far  is 
the  level  of  mercury  at  B  raised  by 
raising  the  mercury  cistern  E,  closing 
the  stopcock,  and  effecting  a  fine  adjust- 
ment by  means  of  the  screw  C.  The 
volume  of  the  gas  between  B  and  A  is 
thus  made  constant,  and  the  column  of 
mercury  AD  measures  the  pressure  of 
the  gas  in  H. 

The  same  thermometer  may,  by  an 
adjustment  of  the  height  of  the  column  AD,  be  used  as  a  con- 
stant-pressure-and-variable-volume  air  thermometer. 

The  air  thermometer  presents  the  disadvantage  of  being 
extremely  unwieldy  ;  but  it  has  the  advantage  that  the  compar- 
atively small  expansion  of  the  glass  produces  little  effect  in 
causing  any  difference  between  the  apparent  and  the  real  expan- 
sion of  the  air,  or  in  vitiating  the  adjustment  to  constant  volume. 

The  ordinary  -mercurial  thermometer  is  a  familiar 
object.  Its  simplest  form  would  be  that  of  a  flask  with  a  long 
neck.  If  the  neck  were  open,  the  mercury  would  be  in  danger 
of  accidental  loss  and  of  evaporation ;  the  neck  must  therefore 
be  closed.  If  it  were  simply  closed,  the  air  contained  in  the 
neck  would  at  high  temperatures  be  compressed ;  the  bulb  would 
burst;  hence  a  vacuum  must  be  produced  in  the' upper  part  of 
the  neck.  This  vacuum  is  produced  by  closing  the  tube  while 


402 


HEAT. 


[CHAP. 


mercury  is  boiling  within  it ;  on  cooling,  the  mercury  contracts 
and  retracts,  leaving  a  space  containing  only  a  certain  quantity 
of  the  vapour  of  mercury. 

As  to  the  graduation  of  the  mercurial  thermometer,  this 
might  be  effected  by  comparison  with  an  air  thermometer,  a 
troublesome  process,  resulting  in  degrees  true  but  unequal  in 
size ;  or  by  taking  advantage  (Renaldini)  of  the  fact  that  the 
"freezing  point"  of  water  —  or,  better,  the  melting  point  of  ice 
—  and  the  "  boiling  point "  of  water  —  or,  better, 
the  temperature  of  steam  at  the  pressure  of  760 
mm.  Hg  —  are  constant  temperatures,  and  may 
be  taken  as  fixed  points ;  that  tne  height  assumed 
by  the  mercurial  column  at  these  two  tempera- 
tures may  be  marked  on  the  tube  ;  and  that  the 
tube  between  these  two  marks  may  (Newton)  be 
mechanically  graduated  by  equal  division 
into  degrees  —  a  method  certainly  convenient, 
but  only  approximately  correct. 

The  boiling  point  of  water  is  estimated  by  inserting 
the  thermometer  in  an  atmosphere  of  steam  surrounded 
by  a  steam-jacket  (Fig.  140),  intended  (Berthelot)  to  check 
irregular  condensation.  The  pressure  must  be  the  stand- 
ard, 760  mm.  Hg.  The  "  freezing  point "  must  then  be  at 
once  determined  by  the  position  assumed  by  the  mercury  when  the  water 
which  trickles  off  melting  ice  flows  in  a  stream  over  the  mercury  bulb,  the 
whole  being  surrounded  by  a  jacket  of  melting  ice. 

On  the  Centigrade  thermometer  (Linnaeus)  the  "freezing 
point "  and  the  boiling  point  are  respectively  0°  and  100°  C. ; 
on  the  Fahrenheit  scale  they  are  32°  F.  and  212°  F. ;  0°  F.  being 
the  lowest  temperature  attained  by  Fahrenheit  (Phil.  Trans. 
1724)  by  means  of  a  mixture  of  ice,  water,  and  salt  or  sal- 
ammoniac. 

A  temperature  of  t°  C.  is,  accordingly,  equal  to  Qfjf  t  +  32)°  F. ;  and 
one  of  x°  F.  to  $%$(x  -  32)°  C. 

Fahrenheit  did  not  use  the  boiling  point  of  water  as  a  standard,  but 
imagined  his  zero  to  be  an  absolute  zero,  and  then  made  or  intended  to  make 
the  freezing  point  of  water  to  stand  at  one-third  between  this  absolute  cold 
and  the  temperature  of  the  human  body,  which  for  convenience  he  called  96°. 

Water  has  been  used  as  the  expanding  substance  in  thermometers ;  it  is 
objectionable  on  account  of  its  point  of  maximum  density.  Alcohol  is  used 
at  very  low  temperatures,  because  it  is  not  readily  frozen.  Mercury,  which 
is  very  advantageous  on  account  of  its  low  specific  heat  and  its  ready 
response,  was  brought  prominently  into  notice  by  the  astronomer  Halley. 

The    sensitiveness    of    thermometers  —  the    power    of 


xiii.]  THERMOMETERS.  403 

revealing  minute  variations  of  temperature — is  increased  by 
narrowing  the  tube  or  by  enlarging  the  bulb.  A  large  bulb  is, 
however,  inconvenient;  because  it  is  difficult  of  insertion 
in  apertures  —  a  fault  which  may  be  remedied  by  giving  the 
bulb  a  cylindrical  form;  because  it  may  alter  materially  the 
temperature  of  the  object  whose  temperature  is  to  be  ascertained ; 
because  it  slowly  equalises  its  temperature  with  that  of  the 
object.  A  narrow  tube  is  inconvenient  because  a  narrow  thread 
of  mercury  is  difficult  to  see  ;  this  may  be  remedied  by  using  a 
tube  of  flat  elliptical  section,  and  by  enamelling  the  back  of  it. 

The  main  causes  of  error  in  the  use  of  a  thermometer  are 
that  the  graduation  alters,  the  "zero  rises,"  or  a  thermometer 
inserted  in  melting  ice  comes  in  course  of  time  apparently  to 
indicate  a  temperature  somewhat  above  0°  C.  or  32°  F.,  this 
effect  being  probably  due  to  a  slow  yielding  of  the  bulb  to 
atmospheric  pressure ;  and  further,  that  it  is  not  always  pos- 
sible to  ensure  that  the  whole  of  the  mercury  is  at  the  same 
temperature. 

In  testing  a  thermometer  it  is  important  to  see  that  the 
"  freezing  point  "  and  the  "  boiling  point "  are  accurately  indi- 
cated by  it,  or  that  it  agrees  with  a  thermometer  in  this  respect 
correct ;  and  that  the  bore  of  the  tube  is  uniform,  so  that  a 
little  detached  portion  of  the  thread  of  mercury  may  occupy 
an  equal  length  in  all  parts  of  it. 

For  accurate  comparison  of  thermometers  they  should  be 
immersed  together  in  a  cooling  fluid  rather  than  in  one  which 
is  being  heated  (Fourier)  ;  the  temperature  indicated  by  a 
thermometer  in  a  cooling  fluid  is  always  a  little  higher  than 
that  of  the  fluid. 

For  practical  details  connected  with  testing  thermometers  see  Gscheid- 
len,  Physiologische  Methodik,  p.  76. 

For  observations  of  the  temperature  of  the  skin  it  is  well  (Colin)  not  to 
cover  the  bulb  with  flannels,  or  to  leave  the  thermometer  in  such  circum- 
stances for  too  long  a  time,  for  the  skin  assumes  the  temperature  of  the 
interior ;  rather  should  quickly-acting  thermometers  be  used.  Apply  a  ther- 
mometer quickly,  fresh  from  the  pocket  or  the  hand :  keep  it  closely  in 
contact  with  the  skin ;  avoid  blowing  on  the  bulb ;  put  a  little  cupola  of 
paper  or  cotton  over  the  bulb,  but  not  in  contact  with  it. 

Special  Forms  of  Mercury  Thermometers.  —  In  the  Maximum 
thermometer,  above  the  column  of  mercury,  a  small  bubble  of  air  is 
introduced ;  above  this  a  little  thread  of  mercury.  When  the  temperature 
rises,  the  air  is  compressed,  the  thread  is  pushed  upwards ;  when  the  tem- 
perature falls  back,  the  thread  of  mercury  does  not  return. 

The  Minimum  thermometer  is  usually  a  spirit  thermometer  with 
a  little  broad-headed  piece  of  wire  loosely  fitting  in  the  spirit.  It  is  ad- 


404  HEAT.  [CHAP. 

justed  with  its  head  touching  the  surface  of  the  thermometric  liquid. 
When  the  liquid  contracts,  surface-tension  drags  the  wire  with  it ;  when  the 
temperature  rises,  the  liquid  passes  the  wire  without  forcing  it  upwards: 
the  position  of  the  end  of  the  wire  nearest  the  free  surface  indicates  the 
lowest  level  to  which  the  surface  had  sunk,  and  therefore  the  lowest  level 
which  had  been  attained  since  the  last  observation. 

In  Met  astatic  thermometers  any  part  of  the  mercury  may  be  removed 
from  the  column  and  shaken  aside  into  an  apical  cavity,  or  restored  in 
whole  or  in  part  to  the  main  thread ;  the  thermometer,  a  very  delicate  one, 
being  thus  competent  to  read  to  very  small  fractions  of  a  degree  at  any  part 
of  the  scale  chosen  at  will.  See  Gscheidlen,  p.  84.  The  principle  of  over- 
flow—  liquid  being  caused  to  expand  and  overflow,  or  vapour  (iodine, 
mercury)  being  boiled  out  of  a  heated  flask,  what  remains  being  weighed 
when  cooled  —  is  utilised  in  the  construction  of  some  pyrometers. 

High  temperatures,  be}rond  the  reach  of  the  mercury  ther- 
mometer, may  be  measured  by  '  pyrometric '  means,  of  which 
the  chief  are  the  following :  —  (1)  readings  of  air-,  or  rather  of 
hydrogen-thermometers ;  (2)  the  dilatation  of  solids,  such  as 
porcelain  (^oVoono  its  length  per  °C.),  made  manifest  by  the 
method  of  Fig.  5 ;  (3)  the  fusion  of  masses  of  known  melting- 
point  (a  series  of  fusible  porcelains  or  of  fusible  alloys) ;  (4)  vari- 
ations of  viscosity  of  air  and  consequent  variations  in  the  flow 
of  air  through  apertures,  at  different  temperatures ;  (5)  the 
temperature  attained  by  a  uniform  stream  of  water  directed,  at  a 
known  speed,  through  the  hot  region ;  (6)  exposure  of  a  known 
quantity  of  a  substance  (such  as  iron  or  platinum),  of  known 
specific  heat,  to  the  temperature  in  question,  then  dropping  it 
into  a  known  quantity  of  water,  and  finding  what  temperature 
is  attained  by  the  water ;  (7)  the  colour  of  the  glow,  white-hot 
or  red-hot  or  otherwise,  of  the  heated  object,  observed  with  the 
naked  eye  or,  more  accurately,  by  means  of  a  piece  of  cobalt 
glass  or  of  a  rotatory-polarisation  apparatus  (p.  566);  and 
(8)  by  electric  methods  (p.  628). 

Calorimetry  or  the  quantitative  measurement  of 
Heat.  —  The  Calorie  ((7#)  is  the  amount  of  heat  required  to 
raise  the  temperature  of  1  kilo,  of  water  (or  I/a-  kilos,  of  any  sub- 
stance whose  specific  heat  is  or)  from  0°  C.  to  1°  C.  The  calorie 
or  small  calorie  (ca)  is  the  amount  of  heat  similarly  required 
to  heat  one  gramme  to  the  same  extent.  The  latter  is  the  C.G.S. 
unit ;  the  former  is  much  used  by  French  and  German  writers. 

1.  The  Method  of  Mixtures.  —  This  may  be  illustrated 
by  a  numerical  example.  How  many  calories  of  heat  does  a 
gramme  of  mercury  absorb  when  it  is  heated  from  0°  C.  to 
1°  C.  ?  —  i.e.,  what  is  the  specific  heat  of  mercury? 


xiii.]  CALORIMETRY.  405 

One  gramme  of  mercury  at  100°  C.  and  one  of  water  at 
0°  C.  are  mixed:  the  result  is  a  uniform  temperature  of  3°-194  C. 
The  water  has  gained  3-194  calories  ;  the  mercury  has  lost 
the  same.  Mercury  on  losing  3-194  ca  per  gramme  is  cooled 
through  96°' 8  C. ;  cooling  through  1°  C.  involves  a  loss  of  -033  ca. 
The  specific  heat  of  mercury  is  thus  a  =  -033 ;  and  the  amount 
of  Heat  contained  in  the  mass  of  mercury  mixed  with  the 
water  was  (373  x  -033)  ca,  373  being  its  absolute  temperature 
and  its  mass  being  unity. 

One  gramme  of  water  and  one  of  mercury  are,  together, 
thus  equivalent  in  calorimetric  calculations  to  1-033  grammes 
of  water;  and  their  joint  'water-equivalent'  is  said  to  be  1-033 
grammes,  i.e.,  (the  mass  of  the  water  x  its  sp.  heat)  +  (the  mass 
of  the  mercury  x  its  sp.  heat). 

A  modification  of  this  method  is  that  of  Fig.  141.  A 
globe  filled  with  mercury:  the  free  surface  of  the  mercury  at 


Fig.141. 


a;  the  screw  S,  which  alters  the  position  of  the  surface  a  so  as 
to  bring  it  to  the  zero  of  a  scale  marked  on  the  horizontal  tube 
ab  ;  the  hot  substance,  introduced  into  a  depression  at  o,  heats 
the  mercury,  expands  it,  and  causes  the  capillary  surface  a  to 
assume  a  new  position. 

Otherwise,  instead  of  observing  the  direct  expansion  of  the 
fluid  heated,  its  temperature  may  be  taken  by  a  thermometer. 

Dulong's  water  calorimeter  is  of  this  kind:  a  copper 
chamber  containing  a  living  animal  supplied  with  air  by  afferent 
and  efferent  pipes  :  round  this  a  water-jacket,  the  water  in  which 
assumes  a  certain  observed  temperature.  The  number  of  calo- 
ries taken  up  by  the  water  and  by  the  copper  or,  other  vessels 
(considered  as  equivalent  to  a  times  their  mass  in  water,  <r 


406  HEAT.  [CHAP. 

being  their  specific  heat)  is  found,  and  that  amount  thus  meas- 
ured is  the  amount  of  heat  given  out  by  the  enclosed  animal. 

2.  Latent  heat  methods.  —  The  amount  of  heat  of  a 
hot  body  may  be  measured  by  the  amount  of  ice  melted  by  it  — 
this  being  ascertained  roughly  (Lavoisier  and  Laplace)  by  the 
amount  of  water  which  trickles  from  ice  amid  which  the  hot 
body  is  thrust ;  or,  better  (Sir  J.  Herschel  and  Bunsen),  by 
observing  the  actual  decrease  in  volume  of  a  mixed  mass  of  ice 
and  water  when  some  of  the  ice  is  melted ;  or  by  the  amount 
of  liquid  —  water,  ether,  acetic  aldehyde  —  which  the  heat  of  a 
hot  body  (or  living  animal  —  Rosenthal,  Arch.  f.  Anat.  und 
Physiol.,  1878)  can  evaporate. 

TRANSFERENCE  OF  HEAT. 

When  two  masses  or  parts  of  the  same  mass  are  in  contact, 
the  molecular  agitation  of  each  is  in  part  communicated  to  the 
other :  if  they  be  equally  hot,  each  receives  as  much  heat  as  it 
gives  up :  if  they  be  not  equally  hot,  that  which  has  the  more 
molecular  energy  loses  more  than  it  receives,  while  the  other, 
the  colder,  gains  more  heat  than  it  loses.  The  flow  in  one 
direction  thus  overpowers  that  in  the  other,  and,  on  the  whole, 
heat  is  transferred  from  the  hotter  mass  to  the  colder. 

We  regard  in  general  only  the  difference,  and  not  the  common  part ; 
the  surplus  which  flows  from  the  hotter,  and  not  the  compensated  and  non- 
apparent  flow  from  the  colder  body. 

This  tendency  is  universal.  Heat  always  tends  to  pass  on 
the  whole  from  hotter  to  colder  bodies,  and  if  these  be  in  con- 
tact, the  transference  is  effected  by  conduction  ;  whence  all 
bodies  possess  some  degree  of  conductivity  or  power  of  trans- 
ferring heat  through  their  substance. 

When  two  points  in  a  substance  are  at  temperatures  con- 
stantly differing  by  Sr,  and  are  at  a  distance  c?,  a  flow  of  heat  is 
set  up  between  them.  The  amount  of  heat  which  passes  from 
the  one  point  to  the  other  in  time  t  is  proportional  (1)  to  the 
length  of  time  during  which  the  flow  proceeds  ;  (2)  to  Sr,  the 
difference  of  temperatures  ;  and  (3)  it  is  inversely  proportional 
to  cZ,  the  distance  between  the  points. 

Otherwise,  the  Quantity  of  Heat  which  flows  is  H,  which  <x  t  -  &r/d ;  or 
H  =  ®-t'$T/d.  Here  ®  is  a  coefficient,  the  Coefficient  of  Conductivity,  and 
Sr/d  represents  the  Fall  of  Temperature  per  unit  of  distance,  the  Tempera- 
ture-gradient, G. 


xiii.]  CONDUCTION  OF   HEAT.  407 

The  coefficient  of  conductivity  varies  from  substance  to 
substance,  being  greatest  in  the  metals  ;  some  substances  permit 
a  rapid,  some  only  a  slow  transfer  of  heat;  compare  a  horn 
spoon  and  a  silver  one  inserted  in  a  hot  liquid. 

If  the  one  point  be  maintained  at  the  temperature  r  and 
the  other  at  the  temperature  ry,  intermediate  points  have  tem- 
peratures which  from  point  to  point  sink  uniformly  with 
the  distance  from  the  hotter  point,  if  there  be  no  loss  of  heat 
on  the  way  between  these  points,  as  by  radiation  or  convection. 
There  is  thus  set  up  a  condition  of  Steady  Flow  of  Heat. 

Across  a  plate  of  thickness  d  whose  sides  are  maintained  at 
an  actual  *  and  constant  difference  (r  —  ry),  the  flow  of  heat  per 
unit  of  area  will  in  time  t  be  ®-£-(T  —  r^/d;  across  area  A 
the  flow  will  be  BA.£.(T  - 


This  is  ®A-£  x  the  temperature-gradient  G. 

Let  the  flow  of  heat  be  measured  in  ergs  :  then  ®  is  called  the  Dynam- 
ical Coefficient  of  Thermal  Conductivity.  In  copper,  for  ex- 
ample, ®  =  36,675000;  and  the  Heat-flow,  in  ergs,  =  36,675000  t-Sr/d.  If 
the  heat  flowing  be  measured  in  the  larger  unit,  the  calorie  (=41,593000 
ergs),-  the  proper  coefficient  will  be  a  smaller  one;  it  is  ?9,  the  Calorimet- 
ric  Coefficient  of  Thermal  Conductivity,  and  is  the  one  commonly 
employed;  in  copper  the  Heat-flow  in  calories  is  Hy  =  0-88176  t-8r/d.  If 
the  units  chosen  be  such  that  one  unit  of  heat  can  raise  the  temperature  of 
1  cub.  cm.  of  the  conducting  substance  itself  through  1°  C.,  these  units  are 
each  equal  to  pa-  ca,  where  p  is  the  density  and  o-  the  specific  heat  of  the 
substance:  and  the  Heat-flow,  measured  in  such  units,  is  Hy/  =  p-t-Sr/d; 
where  the  coefficient  p,  called  the  Coefficient  of  Thermometric 
Conductivity  or  of  Thermal  Diffusivity,  is  equal  to  ft/  per  or  to 
®/41,593000po-.  In  copper,  when  p  =  8-6  and  \  =  0-95,  p  =  0-8819.  In 
iron,  ®  =  6,290000  and  ft  =  0-15123  ;  and  since  p  =  7-6  and  a-  =  0-114, 
p  =  0-183.  In  air,  ®  =  2381  and  ft  =  0-0000558  ;  and  since  p  =  0-0013  and 
tr  =  0-1684,  p  =  0-256. 

Example.  —  The  earth  is  found  to  be  about  1°  C.  hotter  for  every  30 
metres  of  vertical  descent  :  the  coefficient  ft  for  rock  is,  on  the  average, 
0-0045  ;  what  is  the  approximate  loss  of  heat  from  the  surface  of  the  earth, 
in  calories  per  sq.  metre  per  annum?  Hy  (in  ca)  =  ft-  A*t-  (T  —  rt)/d  = 
0-0045  x  10,000  sq.  cm.  x  31,556929  seconds  x  1°  C.  •*•  3000  cm.  =  473,000 
ca  per  sq.  metre  per  annum  ;  enough  to  melt  a  layer  of  ice  0-644  cm.  thick. 

The  surface  of  the  earth  is  about  (5-093  x  1018)  sq.  cm.  ;  each  sq.  cm. 
loses  47-3  ca  per  annum;  the  total  yearly  loss  is  (240-9  x  1018)  ca.  The 
specific  heat  of  rock  is  about  0-5  ca  per  cub.  cm.;  the  volume  of  the  earth 
is  about  (1-0866  x  1027)  cub.  cm.;  the  "thermal  capacity  of  the  earth,"  if 
it  were  wholly  similar  to  surface  rock,  would  be  (0-5  x  (1-0866  x  1027)) 
=  (0-5433  x  1027)  ca  per  °C.  The  heat  lost  by  the  earth  at  the  present 
time  would,  on  the  same  assumption,  correspond  to  an  average  fall  in  tern- 

/ 

*  The  flameward  side  of  a  steam-boiler  plate  is  not  at  anything  like  the  tempera- 
ture of  the  flame  beneath  it. 


408  HEAT.  [CHAP. 

perature,  throughout  the  earth,  of  [(240-9  x  1018)  -=-  (0-5433  x  1027)]  °C.  = 
0-00000045°  C.  per  annum. 

The  thermometric  coefficient  p  has  also  the  following  physical  meaning. 
It  serves  as  a  measure  of  the  rate  of  rise  of  temperature  under  a  varied 
distribution  of  heat  in  a  mass;  the  temperature  tends  to  become  uniform 
and  flows  in  the  mass ;  the  rate  of  rise  of  temperature  at  any  point  in  the 
mass  is  proportional  to  the  local  mean  rate  of  change  of  gradient  of  tem- 
perature between  two  points,  unit  distance  apart  in  the  direction  of  the  flow 
of  heat,  and  is  equal  to  p  x  that  mean  rate.  If  two  faces  of  a  slab,  of  area 
A  and  thickness  Sx,  be  supposed  to  have  the  respective  Temperature-Gradi- 
ents G  and  (G  +  SG),  the  outflow  at  the  one  face  would,  in  time  &,  be 
H,  =  #  •  A  •  Bt  •  G  calories ;  and  the  inflow  at  the  other  would  be  equal  to 
ft  •  A  •  &t  •  (  G  +  8G) .  The  surplus  remaining  in  the  slab  would  be  ft  •  A  •  &  •  8G, 
or  ft-A'Sx -St-fiG/ftx  calories.  But  this  slab  contains  A-Sx  cub.  cm.,  of 
density  p ;  its  mass  is  therefore  Ap  •  8x  grammes :  its  specific  heat  is  <r 
ca  per  gramme :  it  will  therefore  require  A .  Bx  •  pa-  calories  to  raise  its 
mean  temperature  through  1°  C. ;  its  mean  rise  in  temperature  will  be 
{(i9.A.&c.&.86?/8ar)-t-(A.8ar.^r)}  degrees  Centigrade  =(tt/p<T)(8t>&G/8x)° 
C.,  in  time  S/.  The  rate  of  increase  of  temperature  per  unit  of  time  will 
therefore  be  (,9//xr)(8#/&i;)0  C.  per  sec.;  and  this  =  p-8G/8x,  =  p-SG 
when  8x  =  1  cm. ;  which  is  the  proposition  stated  above. 

If  a  bar  be  heated  at  one  extremity,  the  amount  of  heat 
which  will  arrive  at  a  sectional  area  a  given  distance  along  the 
bar  will  depend  upon  the  thickness  of  the  bar  and  its  propor- 
tional surface.  A  thin  iron  wire  may  be  melted  at  one  end  but 
not  have  its  temperature  raised  by  1°  C.  at  a  distance  of  6  feet; 
so  much  heat  is  lost  on  the  way,  being  spent  in  warming  the 
surrounding  air  arid  in  keeping  up  radiation  from  the  surface. 
For  the  same  reason,  the  most  volatile  oil  may  be  burned  in  a 
lamp  with  a  sufficiently  long  wick-tube. 

In  such  a  bar,  maintained  at  a  uniform  distribution  of  temperature,  the 
heat  flowing  across  a  given  cross-section  can  be  measured  by  a  process  of 
summation  or  integration.  The  temperatures  at  different  points  beyond 
the  cross-section  are  observed ;  the  rates  of  cooling  of  a  similar  bar  at  differ- 
ent known  temperatures  are  also  observed ;  from  these  data  the  loss  of  heat 
by  radiation  and  convection  can  be  ascertained ;  and  this  is  kept  up  by, 
and  is  equal  to,  the  flow  of  heat  across  the  sectional  area.  The  flow  H  is 
thus  known;  so  is  $r/d,  the  temperature-gradient  at  the  cross-section;  so  is 
A  the  area :  whence  the  value  of  ?9  can  be  calculated. 

In  bars  of  different  thicknesses,  the  distances  from  the  heated  extremity  at 
which  the  same  temperature  can  be  kept  up  by  heating  the  extremity  of  the 
bars  to  the  same  temperature  are  to  one  another  as  the  square  roots  of  the 
thicknesses ;  and  in  bars  of  the  same  thicknesses  but  of  different  lengths 
the  flow  of  heat  into  the  bar  varies  as  the  square  root  of  the  cube  of  the 
length. 

A  hot  point  in  space  conceived  to  be  maintained  permanently  hot  will 
be  the  centre  of  a  flow  of  heat  symmetrical  in  all  directions.  The  points  in 
the  surrounding  space  which  are  at  the  same  temperature  may  be  connected 
and  found  to  lie  on  concentric  spheres,  or  spherical  Isothermal  Surfaces. 


xiii.]  CONDUCTION  OF   HEAT.  409 

The  heat  travels  by  the  shortest  path  from  one  surface  to  another,  by 
Lines  of  Propagation,  or  Lines  of  Flow,  at  right  angles  to  both  ;  and  there 
is  on  the  whole  no  lateral  propagation  over  an  isothermal  surface.  The 
whole  system  of  surfaces  and  lines  closely  resembles  a  system  of  equipo- 
tential  surfaces  and  lines  of  force.  The  difference  of  temperature  per  unit 
of  distance  along  the  lines  of  spherical  propagation  decreases  with  the  dis- 
tance, being  proportional  to  (1 /radius2).  The  greater  the  curvature  of  a 
hot  body,  the  greater  will  be  its  loss  of  heat  by  conduction.  Hence  an 
ellipsoidal  body  maintained  at  a  uniform  temperature  loses  most  heat 
where  the  curvature  is  greatest  —  a  proposition  closely  resembling  one  in 
the  theory  of  electricity. 

We  must  distinguish  a  Flow  of  Heat  from  a  Flow  of 
Temperature.  The  latter  depends,  inversely,  on  the  specific 
heat  pa-  per  unit  of  volume ;  and  if  we  compare  the  passage  of 
heat  through  two  substances  similarly  heated,  we  find  that  even 
though  the  one  substance  have  a  greater  conductivity  than  the 
other,  yet,  if  its  specific  heat  per  unit  of  volume  be  greater  in  a 
still  greater  proportion,  a  given  temperature  may  take  a  longer 
time,  travelling  in  the  better  conductor,  to  reach  a  point  at  a 
given  distance  from  the  source  of  heat,  than  it  does  in  the 
worse  conductor. 

The  rate  of  propagation  of  a  given  temperature  depends  upon  the  ther- 
mometric  conductivity  p  =  ?!//pcr.  Thus  in  copper,  a  given  Temperature 
travels  faster  than  it  does  in  iron :  and  so  does  it  in  still  air,  though  the 
actual  quantity  of  Heat  carried  by  conduction  in  still  air  is  extremely  small. 

When  a  body  is  exposed  to  a  superficial  periodic  variation 
of  temperature,  the  variations  are  propagated  as  waves  of 
temperature  according  to  the  same  law  as  if  they  were  dis- 
placements in  a  vibrating  but  more  or  less  viscous  solid. 

The  waves  diminish  in  amplitude  —  that  is,  in  thermometric  range  — 
as  they  penetrate,  and  that  in  geometrical  progression ;  and  the  depth  at 
which  the  amplitude  is  reduced  in  a  given  ratio  varies  asVT,  and  also  as 
Vp  or  V#/po-.  Yearly  variations  of  temperature  are  thus  felt  at  depths 
beneath  the  earth's  surface  19-11  times  as  great  as  the  daily  variations  are ; 
for  V365  =  19-11. 

Where  a  substance  is  not  ph}^sically  similar  in  all  directions, 
as  in  the  case  of  crystals,  the  conductivity  may  be  unequal  in 
three  directions.  Thus,  a  plate  cut  out  of  any  crystal  belonging 
to  the  binaxial  system,  and  covered  with  a  film  of  wax,  will,  if 
heated  by  a  hot  wire  passed  through  its  centre,  so  conduct  the 
heat  that  the  wax  melts  not  in  a  uniform  circle  —  as  in  glass 
or  a  crystal  of  the  regular  system  it  will  do  —  but  in  an  ellipse. 

Some  solids  are  extremely  bad  conductors  of  heat.  Down 
is  perhaps  the  worst  of  all  conductors;  hare's  fur,  sand,  asbestos, 


410  HEAT.  [CHAP. 

are  examples  of  substances  within  which  warm  objects  may  be 
placed  and  remain  without  losing  their  heat  to  any  material 
extent  for  some  time.  Flannel,  cork,  etc.,  appear  warm  when 
they  are  touched  by  the  bare  skin,  because  they  carry  away 
by  conduction  less  heat  than  the  air  had  been  removing  before 
these  materials  had  been  touched.  Wood  is,  in  the  radial  direc- 
tion, a  bad  conductor :  this  has  a  certain  effect  in  preserving  the 
tree  in  life. 

The  actual  amount  of  the  loss  of  heat  suffered  by  a  cooling 
body  depends  directly  on  the  effective  cooling  surface :  whence 
the  natural  tendency  in  warm  weather  to  lie  at  full  length,  in 
winter  to  roll  the  body  up  into  small  compass. 

The  conductivity  of  the  skin  as  a  whole  is  greatly  diminished  by  a 
layer  of  fatty  tissue.  The  muscles  are  exceedingly  bad  conductors. 

When  a  hot  body  is  surrounded  by  one  or  more  concentric  jackets  with 
layers  of  air  between  them,  the  loss  of  heat  is  remarkably  diminished.  A 
single  layer  of  linen  diminishes  the  loss  of  heat  from  the  human  body  by 
about  two-thirds ;  a  double  layer  effects  a  much  greater  economy  of  heat, 
and  so  forth.  The  practice  in  cold  countries  of  using  double  windows  pro- 
ceeds on  this  principle,  and  hence  also  the  hygienic  advice  to  multiply  the 
number  of  light  garments  in  cold  weather  rather  than  their  weight. 

The  conductivity  of  liquids  is  as  a  rule  greater  than  that  of 
gases,  which  in  the  form  of  true  conduction  of  heat-energy,  as 
distinguished  from  convection,  is  very  small. 

It  is  impossible  to  keep  the  hands  in  water  at  52°  C.,  while  it  is  quite 
possible,  as  observed  by  Banks,  to  remain  for  five  minutes  in  air  near  the 
boiling  point  of  water. 

When  a  hot  body  is  placed  in  air  it  sets  up  a  number  of 
Convection  currents.  Air  becomes  heated  and  rises,  carrying 
away  the  heat  of  the  hot  body :  colder  air  takes  its  place. 

Newton's  law  of  cooling  in  a  current  of  air  is,  that  at  each  instant  the 
amount  of  heat  lost  varies  as  the  difference  of  temperature  between  the 
solid  and  the  air.  This  law  seems  to  be  adhered  to  within  narrow  limits. 

In  an  undisturbed  atmosphere  the  law  of  cooling  by  convection  is,  that 
the  velocity  of  cooling  is  proportional  to  j»arL233,  where  a  is  a  constant  (45 
for  air),  p  the  pressure,  and  r  the  excess  of  temperature  (Dulong  and  Petit). 

In  hydrogen  the  process  of  cooling  is  very  rapid. 

The  carbonic  acid,  etc.,  of  the  atmosphere  are  mixed  thor- 
oughly and  equably,  not  by  diffusion,  which  would  take  several 
hundred  thousand  years  to  accomplish  the  task,  but  by  con- 
vection currents. 

Convection  currents,  as  they  pass  colder  or  warmer  strata 
of  air,  exchange  molecules  with  them  by  diffusion ;  the  temper- 
ature of  the  whole  mass  thus  rapidly  becomes  uniform. 


xiii.]  CONVECTION   OF   HEAT. 

Convection  currents  may  be  demonstrated  by  throwing  some  coloured 
powder  into  cold  water  and  proceeding  to  heat  the  liquid  over  a  lamp ;  by 
looking  at  distant  objects  through  the  heated  gases  which  arise  from  a 
heated  boiler  or  wall :  the  rise  of  smoke  itself  is  an  example  of  solid  par- 
ticles borne  upwards  by  convection  currents  —  particles  which,  when  the 
ascending  air  has  become  cool,  again  fall,  and  may  aid  in  producing  fogs 
by  the  condensation  of  water  around  them. 

Though  two  bodies  be  not  in  contact  with  one  another, 
they  may  yet  exchange  heat  across  the  intervening  space,  and 
the  hotter  body,  giving  out  more  heat  than  it  receives,  is  said 
to  radiate  heat  to  the  colder  body.  This  transfer  of  heat  is 
effected  by  means  of  the  Ether  of  space,  and  we  shall,  in  the 
meantime,  defer  the  consideration  of  the  transfer  of  heat  by 
radiation  until  we  can  take  a  general  view  of  waves  in  the 
Ether. 

Dulong  and  Petit  found  that  between  0°  C.  and  200°  C.,  the  aggregate 
amount  of  radiation  is  proportional  to  (1-00771"  —  1),  where  r  is  the  excess 
of  temperature  above  the  surrounding  enclosure. 

With  diminishing  pressures  of  air  or  gas,  the  rate  of  loss  of  heat  falls, 
at  first  more  rapidly,  then  less  so;  it  then  remains  sensibly  constant 
(Kundt),  but  with  still  further  exhaustions  the  rate  again  falls  (Crookes). 

Transport  of  Heat  from  place  to  place  may  be  effected  by 
storing  up  work-energy  in  springs  which,  on  being  released,  set 
a  mechanism  at  work  which  evolves  heat  by  friction  ;  or  by  stor- 
ing up  heat  as  "  latent  heat,"  or  by  raising  the  temperature  of 
a  substance  whose  specific  heat  is  high.  The  former  method  is 
not  effective,  because  so  large  a  number  of  units  of  work  corre- 
spond to  so  small  an  amount  of  heat ;  the  latter  are  exemplified 
in  heating  by  hot  water  or  by  steam.  A  hot-water  bottle  con- 
tains several  calories  of  heat,  according  to  its  size  and  its  tem- 
perature; these  can  be  liberated  by  conduction  at  any  desired 
situation.  If  filled  with  crystallised  acetate  of  soda,  melted  by 
heat,  the  cooling  is  protracted,  for  the  melted  salt,  as  it  slowly 
solidifies,  gives  out  its  latent  heat.  Steam  at  100°  C.,  when 
condensed,  liberates  at  the  point  of  condensation  546  calories 
of  heat  for  every  gramme  of  water  condensed,  and  can  still,  in 
the  form  of  hot  water,  surrender  more  heat  to  surrounding 
objects. 


CHAPTER  XIV. 

ON  SOUND. 

THE  word  Sound  is  used  in  four  different  senses :  — 

1.  The  physiological  sensation  perceived  by  means  of  the 
ear. 

2.  The  complex  harmonic  motion  of  sounding  bodies  —  the 
Fourier-motion,  the  periodic  or  vibratory  motion  of  elastic  masses 
whose  vibration  is  the  physical  cause  of  sound. 

3.  The    disturbances    of    the    air    which    affect    the    ear. 
"  Sounds,"    says    Newton    (frincip.    ii.    Prop.    L,    Prob.    xii. 
Schol.),  "since  they  arise  in  tremulous  bodies,  are  no  other 
than  waves  (pulsus)  propagated  in  the  air." 

4.  The  energy  of  a  sounding  body.     "  Heat  converted  into 
Sound,"  etc.     It  is  better  in  this  sense  to  say  explicitly,  "  the 
Energy  of  Sound." 

A  sounding  body  is  a  vibrating  body. 

Cause  a  tuning-fork  to  sound  in  the  usual  way  —  by  striking  it  on  the 
knee  or  drawing  a  violin-bow  across  it,  or  by  forcing  a  steel  rod  between  its 
prongs  and  drawing  it  through  the  point  of  the  fork.  Apply  the  point  of 
the  vibrating  tuning-fork  to  the  lips,  to  the  surface  of  water,  to  a  piece  of 
glass.  Bring  a  vibrating  tuning-fork  under  a  light  splinter  of  wood  lying 
upon  two  points  of  support ;  on  contact,  the  light  body  will  be  hurled 
upwards.  Cautiously  bring  a  vibrating  tuning-fork  or  bell  into  contact 
with  a  pith-ball  suspended  by  a  thread. 

Pluck  one  of  the  strings  of  a  violin :  look  at  it  as  it  vibrates  :  touch  it. 
Look  at  a  harmonium  or  concertina  reed  while  it  is  in  action. 

Observe  the  distinct  tremor  caused  by  a  large  organ  pipe  while  sound- 
ing, or  even  by  a  large  drum. 

Relatively  deep,  grave  sounds  are  produced  by  slower 
vibrations;  higher,  shriller  sounds  by  more  rapid  vibra- 
tions. 

Take  a  long  strip  of  iron  —  say  a  strip  4  feet  long ;  fix  it  in  a  vice ;  pull 
it  aside  and  let  it  go ;  it  will  oscillate  transversely  at  a  rate  such  that  the 
oscillations  can  be  counted ;  remove  it,  and  refix  it  so  that  only  2  feet  of  it 
are  now  free  to  move  ;  —  it  will  now  oscillate  four  times  as  frequently :  1  foot 

412 


CHAP,  xiv.]  SOUND-WAVES.  413 

free  —  sixteen  times  as  frequently  as  at  first ;  6  inches  free  —  sixty-four  times 
as  frequently,  and  so  on.  The  oscillations  now  become  so  rapid,  the  number 
of  them  in  a  second  (z.e.,  their  frequency)  becomes  so  great,  that  they  can 
no  longer  be  counted  directly;  now  we  hear  a  sound;  the  shorter  the 
vibrating  part,  the  more  rapid  become  the  vibrations,  the  shriller  the  sound. 

The  transmission  of  sound  from  a  vibrating  body  to 
the  ear  involves,  as  a  rule,  the  formation  of  sound-waves  in 
the  air. 

This  may  be  rendered  impossible,  e.g.,  where  the  sounding 
body — a  bell  suspended  or  placed  upon  wadding  within  the  bell 
of  an  air-pump  from  which  the  air  is  exhausted — has  no  contact 
with  air,  and  therefore  no  means  of  transferring  its  own  vibra- 
tion to  air;  in  such  a  case  the  ear  perceives  no  sound,  even 
though  the  bell  be  struck,  for  there  are  no  air- waves  set  up. 

But  it  may  be  impossible  for  another  reason.  Air  will  not 
oscillate  in  waves  such  as  can  be  propagated  to  a  distance,  unless 
there  be  some  well-marked  compression  or  rarefaction  produced 
at  the  centre  of  disturbance.  Take  as  extreme  instances  of  sound 
produced  by  well-marked  compressions  or  rarefactions  the  effect 
of  the  discharge  of  a  cannon,  which  abruptly  adds  a  mass  of  gas 
to  the  already-present  atmosphere,  and  thereby  produces  great 
and  sudden  compression ;  or  the  rarefaction  produced  by  the 
sudden  collapse  of  a  weak  boiler  when  the  steam  contained  in  it 
has  cooled  down.  Thus  a  vibrating  body,  before  it  can  act  as  a 
sounding  body,  must  produce  alternate  compressions  and  rare- 
factions in  the  air,  and  these  must  be  well  marked.  If,  however, 
the  vibrating  body  be  so  small  that  at  each  oscillation  the  sur- 
rounding air  has  time  to  flow  round  it,  there  is  at  every  oscil- 
lation a  local  rearrangement  —  a  local  flow  and  reflow  —  of  the 
air,  but  the  air  at  a  little  distance  is  almost  wholly  unaffected  by 
this.  The  same  result  follows  if  the  medium  surrounding1  the 

o 

vibrating  body  be  rare  {e.g.,  hydrogen)  or  rarefied  (e.g.,  rarefied 
air  ) ;  then,  on  account  of  the  small  inertia  of  the  medium,  it  is 
easily  induced  to  flow  round  the  vibrating  body ;  in  such  cases 
there  is  but  little  wave-motion  caused  at  any  distance,  and  thus 
there  is  but  little  sound  produced. 

A  string  stretched  between  two  points  of  a  rigid  and  massive  framework 
produces  surprisingly  little  sound  when  caused  to  vibrate :  it  does  not  act 
upon  the  air  otherwise  than  by  setting  up  local  flow  and  reflow.  If  the  same 
string  be  stretched  over  bridges  upon  a  sounding-board,  the  string  gives,  at 
each  oscillation,  an  impulse  to  the  sounding-board  which  causes  it  to  yield 
slightly;  and  thus  the  string  causes  the  sounding-board >to  vibrate.  But 
though  the  amplitude  of  its  vibration  is  small,  the  sounding-board  is  broad, 


414  ON  SOUND.  [CHAP. 

and  the  air  cannot,  by  flowing  round  its  edge,  evade  compression  and  rare- 
faction ;  the  air  is,  accordingly,  alternately  compressed  and  rarefied,  and  thus 
a  system  of  waves  is  effectively  set  up  in  it.  Thus  the  loudness  of  the  sound 
produced  by  a  string  may,  by  the  use  of  a  sounding-board,  be  multiplied 
many  thousandfold.  A  similar  experiment  may  be  performed  with  a  vibrat- 
ing tuning-fork  suspended  in  the  air  by  a  string,  and  the  same  fork  vibrating 
while  its  shank  is  pressed  against  the  panel  of  a  door. 

In  these  cases  the  energy  of  vibration  of  the  string  or  tuning-fork  is  very 
much  more  rapidly  dissipated,  while  the  large-surfaced  sounding-board  is 
enabled  to  produce  an  i  n  t  e  n  s  e  r  or  louder  sound  than  is  produced  when  the 
string  or  the  fork  vibrates  alone ;  and  the  vibration  sooner  comes  to  an  end. 

The  speaking-trumpet  is  in  part  an  application  of  the  same  principle. 
Instead  of  a  comparatively  small  surface,  the  oral  aperture,  being  the  source 
of  sound,  the  much  broader  aperture  of  the  trumpet  is  practically  converted 
into  the  source,  and  the  broad  sound-waves  thence  issuing  are  only  slightly 
weakened  at  their  origin  by  lateral  flow. 

As  a  general  rule  it  is  therefore  advisable,  when  sound  is 
to  be  heard  at  a  distance,  to  make  the  sources  of  sound  of  the 
largest  size  convenient.  Smallness  of  size  may,  however,  be 
compensated  by  quickness  of  vibration. 

Thus  the  chirp  of  certain  insects  is  produced  by  such  extremely  rapid 
movements  —  as  many  as  12,000  to-and-fro  vibrations  per  second  —  that  the 
air  is  alternately  compressed  and  rarefied  on  each  side  of  the  wings  or  in  the 
neighbourhood  of  the  stridulating  organs,  without  having  time  to  flow  round 
them. 

Characteristics  of  Sounds.  —  The  Fourier-motions  which 
may  produce  sounds  differ  amongst  themselves  in  their 

(a)  Frequency  —  the  number  per  second  of  the  slowest 
component-oscillations. 

An  oscillation  is  a  complete  oscillation,  once  to-and-fro.  The  fre- 
quency of  a  seconds  pendulum  is  £  ;  in  one  second  it  performs  half  a  com- 
plete oscillation.  In  French  works  we  find  that  a  "  vibration  simple  "  is  half  a 
complete  oscillation,  a  swing  over  from  one  side  to  the  other ;  and  a  seconds 
pendulum  is  held  to  effect  such  "  vibrations  simples  "  at  the  rate  of  one  per 
second.  The  reason  for  the  apparently  more  artificial  mode  of  defining  an 
oscillation  here  used  will  be  seen  on  considering  the  meaning  of  period  in 
S.H.M.  (p.  82);  a  complete  oscillation  restores  the  oscillating  body  to  its 
starting  point. 

(5)  They  differ  as  to  their  Energy.  Proportional  to  the 
energy  are  the  Intensity  and  the  Square  of  the  Amplitude. 

(c)  They  also  differ  as  to  the  Relative  Amplitudes  of  their 
Components. 

Of  these  three  particulars,  the  first,  the  frequency,  depends 
on  the  vibrating  body  itself,  its  form,  its  material,  etc.,  and  upon 
its  tension,  but  is  very  slightly  affected  by  its  viscosity  ;  the 
second  depends  entirely  on  external  causes ;  the  third  depends 


xiv.]  CHARACTERISTICS  OF   SOUNDS.  415 

partly  on  the  form,  the  tension,  the  rigidity,  etc.,  of  the  vibrat- 
ing body,  partly  on  the  manner  in  which  it  is  set  in  motion. 

By  variations  in  these  particulars  an  infinite  variety  of 
Fourier-motions  may  be  produced  in  vibrating  or  sounding 
bodies ;  and  as  a  natural  consequence  we  might  expect  to  find, 
as  we  do  find,  an  infinite  variety  of  musical  sounds  actually 
occurring  in  nature. 

Musical  sounds  may  differ  from  one  another  in  three  cor- 
responding respects,  viz.  —  Pitch,  Loudness,  and  Quality  or 
Character. 

Pitch.  —  The  pitch  of  a  clear  musical  sound  depends  on  the 
Frequency  of  the  Fundamental  Vibration  of  the  sounding  body. 
Suppose  a  string  to  vibrate  harmonically,  and  its  component 
vibrations  to  occur  261,  522,  783,  1044,  etc.,  times  per  second: 
then  that  string  would  have  a  fundamental  vibration  whose 
frequency  is  261  per  second ;  and  a  sound  of  this  fundamental 

frequency  is  recognised  by  our  musical  sense  as  the  note  3j=pi 

The  loudness  of  a  sound  increases  with  the  amplitude  of 
oscillation  of  the  vibrating  body ;  if  two  strings,  otherwise  simi- 
lar and  similarly  circumstanced,  oscillate  through  ranges  of  |-  and 
|  inch  respectively,  the  latter  has  twice  the  amplitude  and  tends 
to  produce  four  times  as  much  sound  as  the  former :  the  loud- 
ness  or  intensity  of  sound  being,  among  sounds  of  the  same  pitch, 
proportional  to  the  energy  of  vibration,  and  therefore  to  the 
square  of  the  amplitude.  Mark,  however,  that  the  relative 
loudness  of  different  sounds  as  perceived  by  the  ear  is  not  to 
be  measured  by  their  physical  intensity  or  the  square  of  the 
amplitude  of  the  vibrations  at  their  source,  for  the  ear  is  not 
necessarily,  and  is  not  in  fact,  equally  sensitive  to  sound  of  every 
pitch. 

Vis co  sity  of  a  sounding  body,  while  it  scarcely  affects  the 
pitch,  aids  in  causing  the  amplitude  of  the  vibration,  and  there- 
fore the  loudness  of  the  sound  produced,  gradually  to  dwindle 
away. 

As  to  their  Quality  or  Character,  we  find  among  sounds  an 
infinite  variety.  We  can  distinguish  a  sound  produced  by  a 
violin  from  one  of  the  same  pitch  and  loudness  produced  by 
a  clarionet,  a  flute,  or  a  pianoforte ;  we  can  distinguish  the 
sound  of  a  viola  from  that  of  a  violin ;  one  violin  from  another; 
one  player  from  another  on  the  same  violin  ;  one  person's  voice 
from  that  of  another ;  the  voice  of  the  same  person  in  different 


416  ON  SOUND.  [CHAP. 

moods  or  states  of  health.  The  basis  of  all  this  variety  lies  in 
the  endless  differences  that  may  exist  between  Fourier-motions 
which,  though  they  agree  as  to  the  frequency  of  their  funda- 
mental or  slowest  component  and  as  to  the  total  energy  involved 
in  their  movement,  do  not  necessarily  coincide  in  the  relative 
amplitudes  of  their  component  harmonic  motions. 

But  if,  as  this  theory  indicates,  an  extended  series  of  com- 
ponent vibrations  go  to  make  up  the  aggregate  vibration  of  a 
sounding  body,  ought  we  not,  in  the  sound  produced  by  a  sound- 
ing body,  to  hear  a  series  of  tones  corresponding  to  the  series  of 

vibrational  components  ?      If  a  string  produce  the  note  gpEqE, 

corresponding  to  a  fundamental  vibration  whose  frequency  is 
261  per  second,  ought  we  not,  at  the  same  time,  to  hear  other 
sounds  corresponding  to  522,  to  783,  to  1044,  etc.,  vibrations  per 
second?  The  reply  is  that  we  do  actually  hear  such  tones;  but 
we  do  not  attend  to  them,  and  for  practical  purposes  we  are 
therefore  deaf  to  them.  We  are  accustomed  to  interpret  a  sound 
produced  by  a  single  sounding-body  —  the  voice  of  a  person,  for 
example  —  as  a  single  sound;  from  earliest  infancy  we  uncon- 
sciously train  ourselves  to  listen  only  to  the  fundamental  tone  of 
any  single  note:  and  the  presence  of  the  other  tones  of  the  really- 
compound  sound  produced  by  a  single  vibrating-body  has  the 
apparent  result  of  determining  the  Character  of  that  tone  to 
which  alone  we  consciously  listen.  In  many  cases,  when  we 
listen  for  the  higher  component  sounds,  knowing  what  to  listen 
for,  we  can  hear  them,  even  with  the  unaided  ear :  after  practice 
the  ear  acquires  the  power  of  recognising  the  presence  of  these 
harmonics  with  great  readiness — a  power  which  may  easily 
become  oppressive  to  its  possessor.  The  special  training  which 
confers  this  power  differs  only  in  degree  from  that  which  enables 
one  to  discriminate  the  different  notes  which  make  up  a  chord, 
sounded  in  harmony  ;  for  to  the  untrained  ear  even  a  chord,  if  it 
be  well  in  tune,  seems  to  be  a  single  mass  of  sound. 

Noise.  —  If  all  the  keys  of  a  piano  within  the  compass  of 
one  or  two  octaves  be  simultaneously  struck,  the  result  is  a  con- 
fused jangle,  a  Noise.  Here  we  have  the  Superposition  of 
Fourier-motions  resulting  in  an  apparently -irregular  disturb- 
ance of  the  air.  This  may  go  still  farther  ;  the  Fourier-motions, 
which  are  superposed  on  one  another,  may  have  no  relation  of 
frequency  and  little  or  no  individual  persistence.  The  more 
markedly  this  is  the  case,  the  less  musical  will  be  the  sound 


xiv.]  NOISE.  417 

produced,  and  the  more  markedly  will  it  bear  the  character  of 
noise.  The  general  hum  of  a  town  is  made  up  of  sounds  and 
cries,  each  of  which,  taken  singly,  may  perhaps  not  be  unmusi- 
cal; but  because  they  are  not  related  to  one  another  by  any 
simple  numerical  ratio  of  frequency,  they  together  produce  the 
disagreeable  effect  of  a  noise.  Noises,  then,  such  as  the  sound  of 
steam  escaping  from  a  boiler,  wind  rushing  through  trees,  the 
clatter  of  falling  objects,  and  so  forth,  may  be  considered  to  be 
produced  by  the  superposition  of  a  number  of  distinct  musical 
sounds.  Some  of  these  may  predominate  in  intensity  and  in 
persistence  ;  and  thus  a  noise  may  have  a  distinguishable  pitch. 
We  may  recognise  differences  in  pitch  between  the  noises  pro- 
duced by  drawing  the  thumb-nail  at  various  speeds  over  the 
cover  of  a  book  bound  in  cloth,  by  blowing  across  the  mouth  of 
keys  or  tubes  or  flasks  of  various  sizes,  by  letting  boards  of  vari- 
ous sizes  fall  on  a  wooden  floor,  by  blowing  through  glass  tubes 
on  which  bulbs  of  various  sizes  have  been  blown,  and  so  forth. 
Even  where  the  original  disturbance  is  in  the  highest  degree 
irregular,  as  where  bricks  are  pitched  out  of  a  cart,  the  elasticity 
of  the  bricks,  small  though  it  be,  affects  the  pitch  of  the  noise 
produced,  for  the  thuds  produced  by  soft  porous  bricks  are  graver 
than  the  clinks  produced  by  hard  glazed-bricks  of  the  same  size. 

If  we  listen  to  a  continuous  noise  with  the  aid  of  a  resonator 
(p.  430)  tuned  to  some  particular  tone,  we  can  often  recognise 
the  presence  of  that  tone  as  a  component  of  the  noise ;  the 
resonator  will,  if  that  tone  be  present  as  a  component,  sound  it 
forth  —  continuously  if  it  be  continuously  present;  intermit- 
tently if  it  occur  at.  intervals  only. 

Even  a  single  vibrating-body  may,  when  struck,  produce  a 
noise.  A  bell  is  not,  with  ease,  so  cast  as  to  be  perfectly  uni- 
form ;  when  struck  it  tends,  if  not  quite  uniform,  to  divide  into 
unequal  sectors,  each  of  which  pulsates  at  its  own  rate ;  the 
physical  result  is  a  number  of  simultaneous  vibrations  bearing 
no  simple  relation  to  one  another,  and  the  physiological  result 
is  a  mixed  sensation,  a  jangle,  a  kind  of  noise. 

Thus  sounds  originate  in  Fourier-motions ;  a  musical  note 
in  a  single  Fourier-motion  ;  a  noise  in  a  number  of  simulta- 
neous Fourier-motions  whose  fundamental  frequencies  bear  to 
one  another  no  simple  numerical  relation ;  and,  as  we  shall 
afterwards  see,  the  sensation  of  Harmony  in  a  number  of  simul- 
taneous Fourier-motions  whose  fundamental  frequencies  have 
simple  numerical  relations  to  one  another. 

2E 


418  ON  SOUND.  [CHAP. 

The  simplest  possible  sound  would  be  one  produced  by  a 
vibration  in  which  the  Fourier-motion  was  represented  by  one 
component ;  such  a  sound  would  be  a  pure  Tone. 

The  pitch  of  the  sound  or  note  produced  by  a  vibrating 
body  is  the  pitch  of  the  gravest  component,  the  fundamental 
Tone;  and  it  may  be  specified  in  two  ways  :  — 

(1.)  Physically,  by  stating  the  number  of  vibrations  per 
second  which  correspond  to  that  fundamental  tone ; 

(2.)  Musically,  by  referring  the  tone  to  its  place  in  an 
arbitrary  scale  of  pitch  in  conventional  use  among  musicians. 

To  find  the  frequency  of  vibration  corresponding  to  any 
given  note :  —  As  the  note  in  question  let  us  take,  for  the  sake 
of  example,  that  produced  by  an  ordinary  "  A  "  tuning-fork.  A 
card  or  strip  of  metal  is  placed  so  as  to  touch  at  one  end  the 
cogs  of  a  little  cog-wheel,  while  the  other  end  is  firmly  fixed ; 
the  wheel  is  rotated  slowly  —  each  cog  makes  one  click ;  more 
rapidly  —  the  clicks  blend  into  a  hum ;  still  more  rapidly  —  the 
hum  rises  in  pitch,  and  the  faster  the  rotation  the  shriller  becomes 
the  sound;  at  a  certain  rate  of  rotation  the  sound  is  neither 
graver  nor  shriller  than  that  produced  by  the  tuning-fork ;  this 
rate  of  rotation  is  such  that  the  card  is  struck  435  times  per 
second ;  435  impulses  per  second  given  to  the  card,  and  by  the 

card  to  the  air,  produce  the  sound  jE^aiE,  "a'  =  435."  Higher 
sounds  are  due  to  more  rapid,  lower  sounds  to  slower,  vibrations 
than  this.  This  arrangement  is  known  as  Savart's  Wheel. 
Another  contrivance,  devised  to  the  same  end,  is  the  Syren. 
A  rotating  disc  is  pierced  by  holes  arranged  equidistantly  in  a 
circle,  whose  centre  is  in  the  axis  of  rotation  of  the  disc.  A  tube 
brings  a  current  of  air  to  a  spot  near  the  disc,  so  situated  that  in 
some  positions  of  the  disc  the  air  can  blow  clear  through  one  or 
other  of  the  holes,  while  in  others  the  current  of  air  is  almost 
cut  off  by  the  disc  itself.  Rotate  the  disc ;  the  current  of  air  is 
alternately  cut  off  by  the  disc  and  allowed  to  blow  through  it. 
If  there  be  87  holes  in  the  circle  of  holes,  and  if  the  disc  rotate 
five  times  per  second,  there  are  then  produced  435  puffs  of  air 
per  second,  and  the  note  "  a'  "  is  heard :  its  quality  is,  however, 
decidedly  inferior,  for  the  principal  sound  heard  is  the  noise  made 
by  the  current  of  air  when  it  strikes  the  disc.  If  the  current  be 
divided  by  87  pipes,  so  as  to  blow  through  the  87  holes  simul- 
taneously, and  to  be  simultaneously  cut  oft'  from  them  all,  the 
sound  is  very  much  clearer  and  louder  than  when  there  is  only  a 


xiv.]  PITCH.  -    419 

single  stream  of  air  blowing  through  one  hole  at  a  time.  Instead 
of  87  pipes  issuing  from  a  wind-chest,  we  may  employ  a  wind- 
chest  capped  by  a  fixed  disc  containing  87  holes,  arranged  in  a 
circle  like  that  of  the  rotating  disc :  the  rotating  disc  rotates  in 
the  immediate  vicinity  of  the  fixed  one :  simultaneously  the  air 
rushes  through  all  the  apertures  of  the  rotating  disc,  simul- 
taneously it  is  cut  off  from  them  all.  The  number  87  is  in 
practice  never  used;  some  such  number  as  24  or  48  is  chosen. 
Connected  with  the  rotating  disc  is  some  form  of  mechanism  for 
recording  the  number  of  rotations  effected  by  it  in  a  given  time. 
The  rotating  disc  is  caused  to  rotate  at  such  a  speed  as  causes  the 
desired  sound  to  be  produced:  the  number  of  apertures  in  the  disc, 
multiplied  by  the  number  of  rotations  per  second,  gives  the  num- 
ber of  impulses  per  second  imparted  to  the  air,  and  thus  deter- 
mines the  frequency  of  the  tone  in  question.  The  syren  works 
under  water  as  well  as  it  does  in  air. 

The  experiment  already  described  on  page  412  also  gives 
roughly  the  means  of  finding  the  frequency  of  any  given  tone. 
The  thin  strip  of  metal  is,  by  successive  trial,  carefully  with- 
drawn into  the  vice,  until  its  free  part  gives,  when  set  in  vibra- 
tion, a  sound  of  precisely  the  same  pitch  as  the  tone  whose 
frequency  is  to  be  determined.  Say  that  this  length  is  1  inch ; 
and  also  that  if  30  inches  of  the  strip  be  free,  it  executes  29 
complete  oscillations  per  minute.  The  number  of  oscillations 
varies  inversely  as  the  square  of  the  length ;  whence  (1  inch)2 : 
(30  inches)2  : :  29  :  #,  or  x  =  26,100  vibrations  per  minute,  435 
per  second. 

Still  another  method  of  determination  of  the  frequency  of 
vibration  of  sound,  of  a  given  pitch,  is  graphically  to  record  the 
actual  vibrations  of  the  sounding  body.  A  tuning-fork  has  a 
little  feather-barb  attached  by  cement  to  one  of  its  prongs :  the 
extremity  of  the  barb  is  brought  into  contact  with  slightly- 
smoked  paper  spread  over  the  surface  of  a  cylinder.  The  cyl- 
inder is  caused  to  rotate  ;  the  point  of  the  barb  draws  a  straight 
line  on  the  smoked  .paper.  The  fork  is  caused  to  vibrate :  the 
barb  now  describes,  on  the  rotating  cylinder,  a  sinuous  line 
which  records  the  oscillations  of  the  tuning-fork.  An  indepen- 
dent mechanism  can  be  made  to  mark  the  cylinder  once  every 
second,  and  thus  the  absolute  number  of  oscillations  made  by  the 
tuning-fork  during  each  second  can  be  counted  on  the  permanent 
record.  The  same  principle  may  be  applied  to  many  forms  of 
vibrating  body,  such  as  strips  of  metal,  membranes,  etc. 


420 


ON   SOUND. 


[CHAP. 


Musical  Pitch.  —  The  arbitrary  scale  of  pitch  in  common 
use,  and  typified  by  the  white  keys  of  a  pianoforte,  is  the  follow- 
ing:— 

Thirty-two  foot  Octave — Subcontra  Octave. 


g*  

II 

y 



A,, 

B,, 

British       ?„ 

D,,              E,,              F,,              G,, 

German     C,, 

DM              E,,              FM              GN 

A,, 

H,, 

French       ut_, 

re_,            mi_,            fa_,            sol_. 

la_, 

Sl_, 

No.  of  Vi-  )  1fi.qi9c 
brations   f163125 

18-3515625        20'390625            21'75             24'46875 

27-1875 

30-5859375. 

Ratius            16 

:          18        ;        20         :        21-3        :        24 

:        26-6 

:      30. 

Sixteen-foot  Octave  —  Contra  Octave. 

A, 

B, 

British        C, 

^             E,               F,               G, 

German     C, 

D,               E,               F,                G, 

A, 

H, 

French       ut0 

re0              mi0            fa0              so!0 

Ia0 

S»o 

No.  ofVi-)   „„.,.„, 
hrations  f  32  625 

36703125          40-78125             43'5               48'9375 

54-375 

61-171875. 

Ratios              32 

:          36         :           40        :         42-6          :         48 

:       53-3 

:        60. 

Eight-foot  Octave—  Great  Octave. 

British    ~C~ 

D               E               F               G 

A 

B 

German    C 

D               E               F               G 

A 

H 

French      ut, 

re,               mi,            fa,              sol, 

la, 

si, 

No.  of  Vi-  )«,.„- 
brations  I"6525 

73-40625              81-5625                87                 97'875 

108-75 

122-34375. 

Ratios            64 

:         72           :           80        :         85"3        :          96        : 

106-6 

:         120. 

Four-foot  Octave—  Little  Octave. 

~          ^~1 

C 

d                e                f                g 

a 

B 

b 

ut2 

re2              rm*2             fa2              sol0 

la, 

S12 

No.ofVi.)iqo.: 
brations  |1305 

146-8125            163-125              174                  195'75 

217-5 

244-6875. 

Ratios          128 

144          :        160        :         170-6        :         192        : 

213-3 

:       240. 

£ 

Two-foot  Octave—  One-stroked  Octave. 

• 

_          M 

•  /                       .^y. 

7 

d1                e1                f             g1 

a1 

b' 

ut, 

No.ofVi-)9fi; 
brations  j  Mi 

re,              mi,             fa,            so!3 

293-625              326-25                348                 391'5 

la, 

435 

S13 
489-375. 

Ratios         256 

:         288         :          320        :        341'3        :       384         : 

426-6 

:        480. 

One-foot  Octave  —  Two-stroked  Octave 

^  .  

Cj                             G                            ™ 

&- 

-(=2- 

=H 

c" 

d"              e"               f11              g" 

a" 

-^B 

b" 

ut4 

.     re4              mi4            fa4              so!4 

Ia4 

si4 

No.  of  Vi-  )   ,22 
brations  f  522 

587-25               652-5               696                   783 

870 

978-75. 

Ratios          512 

:          576         :         640       :         682-6        :        768         : 

853-3 

:       960 

]  MUSICAL  PITCH.  421 

Six-inch  or  Three-stroked  Octave.  Three-inch  or  Four-stroked  Octave. 


c1"    d1"    e1"     f  "     g"1    a1"    b"1    c""  d""   e""   f""   g""   a""  b""  c"" 
ut5    reg    mig    fa5    so!5    Ia5     si,    ut6    re6    mi6    fa6  so!6    Ia6    si6    ut, 

No.  of  1 


No.  of) 
Vibra- V 
tions  ) 


1044  1174-5  1305  1392  1566  1740  1957'5  2088  2349  2610  2784  3132  3480  3915  4176. 
Ratios  1024:1152  : 1280: 1365  "3: 1536  : 1706'6  : 1920  :  2048  :  2304:  2560:2730  6  : 3072: 341 3 '3: 3840:4096. 

The  starting-point  of  this  notation  is  the  a'  tuning-fork, 
made  to  vibrate  435  times  per  second,  or  the  second  string  of 
the  violin,  made  to  vibrate  in  unison  with  such  a  fork.  Under 
this  system  the  c"  tuning-fork  makes  522  complete  oscillations 
per  second.  This  is  entirely  a  matter  of  convention.  The  num- 
ber 435  was  chosen  by  the  Academic  des  Sciences  of  Paris ;  433 
by  the  Philharmonic  Society  under  Sir  George  Smart  in  1826; 
440  by  the  German  Society  of  Nature-researchers  at  Stuttgart 
in  1834;  452  is  used  in  the  British  Army;  while  a  pitch  a'  =  426-6 
has  been  highly  recommended,  on  the  ground  that  under  such 
a  system  the  tones  Cu,  Ct,  C,  c,  c',  etc.,  are  produced  by  16,  32, 
64,  128,  256,  etc.,  vibrations  per  second — an  arrangement 
which  has  the  advantage  of  giving  very  simple  numbers  to 
deal  with,  but  which  has,  on  the  other  hand,  the  practical  dis- 
advantage of  giving  a  pitch  which  is  too  low  to  please  instru- 
mentalists, and  the  didactic  disadvantage  of  tending  to  conceal 
the  real  arbitrariness  of  the  convention  which  assigns  to  the  af 
or  the  c"  fork  the  particular  number  of  vibrations  chosen  in 
practice.  In  practice  there  is,  indeed,  a  great  lack  of  agree- 
ment; instrument-makers  are  constantly  raising  the  pitch  for 
the  sake  of  increasing  the  brilliancy  of  orchestral  music,  while 
vocalists  are  made  to  suffer.  Modern  concert  .pitch  has  thus 
risen  as  high  as  a'  =  460  vibrations  per  second,  about  1J  semi- 
tone above  what  it  was  in  England  in  the  time  of  Handel 
(V  =  424),  while  the  organ-pitch  in  England  was,  in  the  mid- 
dle of  the  eighteenth  century,  as  low  as  a'  =  388.  If  the  stand- 
ard number  of  vibrations  chosen  for  a'  be  any  other  than  435, 
the  whole  series  of  numbers  given  in  the  table  must  suffer  a 
proportionate  increase  or  reduction.  The  accuracy  of  such  a 
scale  depends  not  upon  precision  of  absolute  numbers  of  vibra- 
tions so  much  as  upon  correctness  of  the  ratios  of  the  several 
numbers  to  one  another. 

The  successive  tones  of  the  scale  of  C  are  related  to  one 


422  ON   SOUND.  [CHAP. 

another,   with   respect   to   their    frequency,   in   the    following 
manner :  — 


f 

-£_,  

r 

-Si— 

n 

C=>  ' 
f 

ts>— 
S 

'..   £3 
1 

t 

d1 

256  : 

288  : 

320 

:  341-3 

:  384 

:  426-6  : 

480  : 

512. 

1   : 

8   : 

4 

:   1 

:  I 

:   \   : 

15 

8 

2. 

Here  C  (V  =  256)  is  a  keynote,  and  upon  it  we  have  raised  a 
diatonic  major  scale,  d  r  n  f  S  1  t  d1. 

Such  a  scale  is  found  by  experience  to  be  satisfying  to  the 
ears  of  the  Western  nations ;  and  whatever  tone  be  chosen  as 
the  keynote,  there  can  always  be  sung  or  played  on  instruments 
of  the  violin  or  of  the  trombone  class  a  scale  of  this  kind,  in 
which  the  intervals  are  felt  to  be  pleasing  and  in  tune,  in  which 
the  intonation  is  felt  to  be  just,  and  in  which  each  tone,  when 
it  is  carefully  listened  to  while  the  keynote  is  borne  in  rnind,  is 
felt  to  have  its  own  peculiar  mental  effect,  this  depending  on  its 
relative  place  in  the  scale,  and  not  on  its  absolute  vibrational 
frequency.  Singers  who  have  sung  much  together,  string  play- 
ers who  have  practised  together  without  pianoforte  accompani- 
ment, naturally  use  the  tones  of  such  a  scale  without  knowing 
or  even  caring  what  the  numerical  ratio  of  the  frequencies  of 
the  various  tones  of  the  scale  may  be. 

Intervals.  —  We  may  now  identify  the  various  intervals 
occurring  within  the  diatonic  scale  — 

Minor  second,  "  semitone "  n  :  f  or  t  :  d1      .  .  .     15     16 

Grave  major-second          .  r  :  m  or  s  :  1       .  .  9    10 

Major  second   .         .         .  d  :  r,     f  :  s,     1  :  t  .  .89 
Grave     (or     Pythagorean) 

minor-third       .         .  r :  f    .         ..        .  .  .     27     32 

Minor  third    • .         .         .  n  :  S  or  1  :  d'      .  .56 

Major  third      .         .         .  d  :  n,     f :  1,     s  :  t  .  .45 

Perfect  fourth  d  :  f,  r  :  S,     m  :  1,     S  :  d',     t  :  rc'      3       4 

Acute  fourth  .         .         .  1:  r1    .         .         .      "  .  .     20     27 

Augmented  fourth  .         .  f :  t    .        ...  .  .     32    45 

Grave  diminished  fifth     .  t,  :  f  .        V       .  .  .     45     64 

Grave  fifth       .         .         .  r:l    .      ...  .  .     27    40 

Perfect  fifth     .  d:  S,  m  :  t,     f  :  d1,     S  :  r1,     1  :  m'     2       3 

Minor  sixth     .         .         .  t,  :  s,     n  :  d1,     1  :  f '  .  .       5       8 


xiv.]  MUSICAL   INTERVALS.  423 


Major  sixth      .      I  .  .  d  :  1,     r  :  t,     S  :  n'  .  •       8 

Acute  major-sixth    .  .  f  :  r1    .         .         . "  «  .16 

Grave  minor-seventh  .  r  :  d',     S  :  f ,     t  :  1'  .  0 

Minor  seventh          .  .  n  :  r',     1  :  S*  •    ,  .  .5 

Seventh.         .         .  .  .  d  :  t,   'fTn*       .  .,  .       8 

Octave     .         .         .  .  d  :  d1,     r  :  r1,  etc.  .  .       1 


5 

27 

16 

9 

15 

2 


Musical  intervals  are  equal  to  one  another  when  the  con- 
stituent tones  in  each  have  the  same  relative  frequency.  Thus 
d  :  s  :  :  1  :  |,  and  n  :  t  :  :  f  :  -^ ;  the  ratio  of  1  to  |  is  equal  to 
that  of  |  to  -^  —  that  is,  it  is  2  :  3 ;  whence  the  musical  inter- 
val between  d  and  s  is  equal  to  that  between  n  and  t. 

Transition.  —  Any  tone  may  be  chosen  as  a  keynote.  Let 
us  choose  g'  =  384  as  our  keynote,  and  then  compare  the  tones 
of  the  scale  of  the  key  of  G  with  those  of  the  scale  of  C. 
Retaining  the  same  ratios,  the  scalre  of  G  is 

d    :     r    :    m   :     f    :    s    :     1     :    t    :    d1. 

1    :    f    :    4    :    f    :    f    :    f    :   -^  :     2. 
384  :  432  :  480  :  512  :  576  :  640  :  720  :  768. 
Comparing  the  two  scales  we  find :  — 

Scale  of  C  (-Key  C"). 

c'  d'  e'  f     g'         a'         bf       c"       d"       e"        f"        g",  etc. 
.     .    .     .     384  :  426-6  :  480  :  512  :  576  :  640  :  682-6  :  768. 

Scale  of  G  ("Key  G"). 
384  :    432    :  480  :  512  :  576  :  640  :    720    :  768. 

The  tones  agree  with  the  exception  of  the  as  and  the/'s. 
The  a'  of  the  scale  -of  C  and  the  a9  of  the  scale  of  G  differ  from 
one  another  in  the  ratio  of  426-6 : 432,  or  80 :  81.  The  two 
tones  are  perfectly  distinct,  and  an  ear  that  has  become  accus- 
tomed to  the  pure  scale  of  C  is  pained,  especially  in  harmony, 
by  the  substitution,  for  the  proper  a1  in  that  scale,  of  the  slightly 
sharper  a'  which  belongs  to  Key  G.  The  difference  between 
the  two  tones  is  called  a  Comma;  and  they  may  be  respec- 
tively written  a'  and  'a'.  The/"  of  Key  C  and  the  correspond- 
ing tone  in  the  scale  of  G  differ  more  widely  from  one  another ; 
their  frequency-ratio  is  682-6  :  720,  and  the  interval  between 
them,  y|-|,  is  sometimes  called  a  semitone. 

In  order  to  play  in  correct  tune  music  written  in  Key  G  as 
well  as  music  written  in  Key  C,  we  would  require^  not  only  the 
tones  of  the  Key  of  C,  but  also  two  additional  tones  in  each 


424  ON  SOUND.  [CHAP. 

octave.  Every  transition  from  one  Key  to  another  "more 
remote  from "  the  Key  of  C  multiplies  the  demand  for  neAV 
tones  ;  and  that  to  an  extent  twice  as  great  as  the  current 
notation,  which  neglects  differences  of  a  comma,  would  seem 
to  indicate. 

In  the  table,  pages  426  and  427,  are  given  the  tones  of  the 
scale  of  C,  together  with  a  number  of  tones  derived  from 
related  keys.  The  relative,  not  the  absolute,  number  of  vibra- 
tions has  been  shown  in  each  case. 

If  a  singer  were  called  upon  to  produce  a  note  of  324  vibrations  per 
second,  the  feat  would  be  impossible.  This  number  is,  however,  1-265625 
X  256  ;  and  hence  if  c'  have  256  vibrations  per  second,  the  note  required  is 
the  re  of  Key  D.  A  tuning-fork  c'  =  256  is  set  in  vibration  ;  call  the  note 
of  the  fork  do  ;  sing  do,  re ;  fix  the  attention  on  re  (df)  ;  call  it  do  without 
changing  its  pitch ;  dwell  on  it  a  moment ;  then  sing  some  such  phrase  as 
do,  mi,  sol,  do,  mi,  re,  do;  and  the  desired  note,  'e',  a  note  differing  by  a 
comma  from  e',  the  mi  of  Key  C,  has  been  produced,  the  sense  of  tonality 
and  key-relationship  having  carried  the  singer  into  the  correct  sound. 

The  column  headed  "  logarithmic  increments  "  contains  fig- 
ures which  measure  the  intervals  between  the  successive  tones ; 
for  it  gives  the  logarithms  of  the  frequency-ratios  between  each 
tone  and  its  predecessor ;  and  the  most  convenient  method  of 
comparing  ratios  is  to  compare  their  logarithms.  When  the  logs. 
are  equal  the  ratios  are  equal;  when  the  ratios  are  equal  the  inter- 
vals are  equal.  Thus  the  intervals  between  C  and  Cft,  Db  and  T), 
T>  and  vDft,  D  and  Dft,  Eb  and  E,  E  and  Eft,  F  and  Fft,  'F  and 
'Fft,  ^Gb  and  rG,  G\>  and  G,  G  and  Gft,  Ab  and  A,  A  and  Aft,  'A 
and  'Aft,  'Bb  and  'B,  Bb  and  B,  B  and  Bft,  are  all  equal,  being 
measured  in  that  column  by  the  logarithm  -0177288,  which  is 
the  log.  of  the  ratio  ||.  Again,  we  find  a  number  of  lesser 
intervals  whose  log.  is  -005395,  and  whose  ratio  is  -f^:  these 
are  C  and  'C,  Cft  and  'Cft,  vDb  and  Db,  ^D  and  D,  rDft  and  Dft, 
T>  and  Eb,  E  and  'E,  F  and  'F,  Fft  and  'Fft,  rGb  and  Gb,  rG  and 
G,  G  and  'G,  vGft  and  Gft,  vAb  and  Ab,  A  and  'A,  Aft  and  'Aft, 
^Bb  and  Bb,  VB  and  B,  B  and  'B,  V  and  c.  Between  these  the 
interval  is  a  Comma. 

The  scale  may  be  seen  to  be  roughly  divisible  into  53  steps 
or  divisions ;  but  these  are  not  equal  to  one  another ;  if  they 
were  equal,  the  logarithms  would  at  each  step -acquire  an  equal 
increment;  for  the  ratio  between  each  tone  and  its  predecessor 
would  be  equal  throughout  the  scale.  Roughly  and  for  dia- 
grammatic purposes  it  is,  however,  convenient  to  represent  the 
interval  between  C  and  D  by  9  steps,  while  that  between  D  and 


xiv.]  MUSICAL   INTERVALS.  425 

E  is  represented  by  8 :  and  the  table  is  so  arranged.  A  thor- 
oughly accurate  table  of  this  kind  would  be  one  engraved  on 
metal,  the  intervals  between  any  two  tones  in  column  3  being 
made  directly  proportional  to  the  log.  increment  between  them. 
The  intervals  marked  Pythagorean  in  the  table  are  thus 
derived :  —  Start  from  c  and  go  upwards  by  successive  fifths, 
<?,  #,  d',  'a',  'e",  'b" ;  going  downwards  we  arrive  at  F,  ^B^,  VE^, 

The  following  exercises  will  perhaps  aid  the  reader  :  — 

(a)  If  the  violin  be  tuned  Spzzsrz,  to  correct  fifths,  starting  with  a' ; 

show  that  these  notes  are  respectively  e",  a',  V,  *g. 

(&)  The  scale  of  "Bfr  major"  is  obtained  by  transition  from  the  key  of 
C  to  that  of  F,  and  from  that  of  F  to  that  of  "  W"  The  scale  of  "  G  minor  " 
has  g  =  la,  and  thus  do  =  &b.  Show  that  the  respective  descending  scales 


'  "        r  "Bfrmajor." 

f,      eV,      V,       c,      'Bt> ) 


The  columns  in  the  table,  pp.  426-427,  headed  "Equally- 
tempered  Scale,"  show  the  nature  of  the  system  of  Equal 
Temperament,  which  is,  as  nearly  as  practicable,  applied  to 
the  pianoforte  and  organ.  The  intervals  are  equal;  the  ratio 
between  a  tone  and  its  predecessor  and  successor  is  in  every 
case  the  same ;  between  each  pair  of  tones  the  logarithmic 


increment  is  equal:  it  is  =  '-  = -0250583.     The  result 

differs  widely  from  pure  intonation  ;  but  we  are  accustomed  to 
it.  On  a  pianoforte  equally  tempered  the  fifths  are  not  appre- 
ciably out  of  tune,  though  they  are  a  little  flat:  but  the  thirds, 
three  of  which  are  forced  to  make  an  octave  instead  of  extend- 
ing only  from  C  to  B$,  are  too  sharp ;  and  though  this  be  not 
offensive  on  the  pianoforte,  to  which  indeed  their  sharpness 
lends  somewhat  of  brilliancy,  yet  in  slow  sustained  harmony 
these  sharp  thirds  are  really  discordant,  as  may  be  well  heard 
on  a  loud  harmonium  tuned  in  the  usual  manner,  and  on  which 
thirds  alone  are  played. 

Loudness.  —  The  physical  Intensity  of  a  sound  depends 
initially  on  the  square  of  the  amplitude  of  the  vibration  of  the 
sounding  body ;  but  the  corresponding  sensation  of  loudness 
depends  not  only  upon  peculiarities  of  sensitiveness  of  the  ear, 
but  also  on  the  amount  of  physical  disturbance  of  its  drum,  and 


426 


ON  SOUND. 


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428  ON   SOUND.  [CHAP. 

if  the  sound  be  conducted  to  the  ear  by  the  air,  it  depends  on 
the  intensity  of  vibration  of  the  air  near  the  ear;  and  this 
varies  riot  only  (1)  as  the  square  of  the  amplitude  of  the  origi- 
nal vibration,  but  also,  in  the  open  air,  (2)  inversely  as  the 
square  of  the  distance  of  the  sounding  object. 

To  compare  the  relative  loudnesses  of  two  sounds  of  nearly  the  same 
pitch,  place  the  sounding  bodies  at  such  distances  that  they  become  just 
inaudible,  and  no  more :  say  that  the  one  becomes  inaudible  at  10,  the  other 
at  50,  yards :  then  the  loudness  of  the  one  at  50  yards'  distance  is  at  the  ear 
equal  to  that  of  the  other  at  10  yards  :  their  initial  intensities  must  be  as 
102 :  502,  or  1  :  25. 

If  the  sound  be  not  propagated  in  free  air,  but  be  confined 
in  a  tube,  the  loudness  of  sound  may  diminish  at  a  much  less 
rate,  for  ultimately  the  waves  become  plane-fronted,  and  move 
down  the  tube  without  any  loss  of  intensity  other  than  what 
is  due  to  such  loss  of  energy  as  is  brought  about  by  friction 
against  the  sides  of  the  tube  or  by  the  viscosity  of  the  air 
itself. 

Hence  sounds  can  be  carried  along  sewers,  speaking- 
tubes,  etc.,  to  great  distances  without  great  diminution  of 
loudness. 

Similarly,  if  sound  be  propagated  by  parallel  or  convergent 
waves  in  the  air,  as  when  it  issues  from  a  wide  aperture,  or  after 
reflexion  from  a  curved  surface,  it  may  lose  little  of  its  inten- 
sity, or  may  even  concentrate  its  intensity  on  some  particular 
place. 

The  loudness  of  a  sound  also  depends,  if  it  be  conveyed  by 
a  gaseous  medium,  on  the  density  of  that  medium  at  the  place 
where  the  vibration  is  imparted  to  it.  The  denser  the  medium 
the  greater  its  inertia,  and  the  more  readily  it  is  compressed 
against  itself:  the  greater  the  compression,  the  greater  the 
amount  of  energy  imparted  to  the  medium,  and  the  louder  the 
sound  produced.  A  body  vibrating  in  vacuo  produces  no  sound  : 
in  rarefied  air  or  hydrogen,  or  any  other  rare  or  rarefied  gas,  it 
produces  a  comparatively  feeble  sound ;  in  carbonic  acid  it  pro- 
duces a  louder  sound  than  in  air.  A  cannon  fired  on  a  moun- 
tain-top produces  little  sound ;  one  fired  beneath  is  heard  dis- 
tinctty  and  loudly  from  a  balloon,  even  at  a  great  height. 

Concentration  of  sound-waves  renders  sounds  louder,  as  in 
ear-trumpets  and  in  those  stethoscopes  the  auditory  extremity 
of  which  fits  into  the  ear. 

Quality  of  Sound.  —  If  a  body  vibrate  so  as  to  produce  a 


xiv.]  QUALITY  OF   SOUND.  429 

sound  of  the  fundamental  pitch  C  =  64,  and  if  all  the  harmonics 
be  present,  the  series  is  the  following :  — 

1.     2.     3.     4.      5.     6.     7.     8.     9.      10.     11.     12.     13.     14.     15,  etc. 
64  128  192  256  320  384  448  512  576    640    704    768    832    896    960 

C      c      g      c'      e'      g'  b\>'-  c"     d"      e"    f"+    g"    a"  +  b\>"-  b",  etc. 
*  *  *        # 

These  are  all  tones  of  the  scale  of  C,  with  the  exception  of 
the  7th  and  its  octave  the  14th,  the  llth,  and  the  13th.  The 
7th  and  the  14th  correspond  to  a  very  flat  B  of  112  vibrations, 
lying  between  Aft  and  'Aft ;  the  llth  to  a  sharp  F  of  88  vibra- 
tions, tying  between  'F  and  Fft ;  the  13th  to  a  flat  A,  lying 
between  Ab  and  A. 

Analysis  of  a  sound  into  its  components  may  be 
effected  by  several  methods,  of  which  we  shall  first  consider 
one  due  to  Prof.  Mayer.  As  our  example  we  take  the  sound 
produced  by  a  vibrating  organ-reed-pipe,  a  sound  which  we 
recognise  as  peculiar  and  characteristic.  We  are  provided  with 
a  set  of  tuning-forks,  one  of  which  vibrates  in  exact  unison 
with  the  fundamental  tone  of  the  organ-pipe,  and  the  rest  of 
which  respectively  vibrate  2,  3,  4,  5,  etc.,  times  as  rapidly.  As 
to  the  organ-pipe,  a  part  of  its  wall  has  been  replaced  by  a 
piece  of  inelastic  thin  morocco  leather,  or  some  similar  sub- 
stance, which  vibrates  exactly  as  the  air  within  the  pipe  does. 
To  one  point  of  this  is  attached  a  bundle  of  silkworm-cocoon- 
threads,  40  inches  or  so  in  length :  each  of  these  is  attached  to 
one  of  the  tuning-forks  and  tightened  somewhat.  The  organ- 
pipe  is  caused  to  sound ;  the  leather  vibrates ;  the  silk  fibres 
are  all  set  in  motion,  and  each  alternately  tugs  and  releases  its 
own  tuning  fork.  If  the  vibration  appropriate  to  any  one  of 
the  tuning-forks  be  present  in  the  original  compound  vibration, 
the  corresponding  fork  is  set  in  motion :  if  it  be  not  present, 
that  fork  remains  silent :  if  the  vibration  be  ample,  the  fork 
sounds  out  loudly :  if  it  be  not,  the  sound  is  feeble.  This 
arrangement  analyses  the  sound  into  its  components,  for  it  can 
be  seen  which  of  the  tuning-forks  are  set  in  vibration  ;  and  if 
the  organ-pipe  cease  sounding,  the  forks  go  on  sounding  for 
some  time,  and  by  their  joint  action  produce  a  compound  sound 
closely  resembling  the  sound  of  the  reed-pipe  which  had  been 
the  means  of  setting  them  in  vibration.  This  action  is  very 
exact :  the  slightest  difference  between  the  natural  rate  of  any 
tuning-fork  and  that  of  the  corresponding  organ-pipe  vibration 


•JTI7BRSITT, 


430  ON  SOUND.  [CHAP. 

causes  the  fork  to  sound  with  comparative  feebleness,  or  not  to 
sound  at  all. 

Resonators  are  extensively  used  as  a  means  of  analysis  of 
sound.  A  resonator  consists  in  its  most  usual  form  of  a  bulb, 
Fig  us  generally  of  glass  or  of  brass,  with  a  large 
aperture,  a,  at  one  side  and  a  small  one,  6, 
at  the  other.  The  air  within  such  a  bulb 
has  a  natural  period  of  vibration  which 
depends  upon  the  cubic  contents  of  the 
resonator  and  upon  the  size  of  the  orifices. 
This  period  can  be  found  by  the  pitch  of  the 
sound  produced,  on  tapping  the  resonator 
with  a  soft  substance,  or  by  blowing  brief  blasts  of  air  across 
its  mouth.  If  the  air  convey  a  system  of  waves  which  agree 
in  period,  either  absolutely  or  approximately,  with  the  natural 
free  vibration  of  the  air  in  the  resonator,  the  air  in  the  resonator 
will  absorb  the  energy  of  those  waves,  will  be  set  in  motion, 
and  will  act  as  a  sounding  body.  If  we  be  provided  with  a  set 
of  such  resonators,  the  air  in  one  of  which  freely  vibrates  in 
unison  with  an  a'  tuning-fork,  and  in  the  others  respectively 
2,  3,  4,  5,  6,  7,  etc.,  times  as  rapidly,  —  then,  on  listening  to  an 
a'  organ -reed-pipe,  one  ear  being  closed  and  the  other  adapted 
to  each  resonator  in  succession  (this  being  done  by  fitting  the 
nipple  b  of  the  resonator  (Fig.  142)  into  the  ear),  we  shall,  if 
the  proper  sound  of  any  of  the  resonators  be  contained  in  the 
complex  sound  to  which  we  listen,  hear  that  resonator  loudly 
sing  out  its  proper  tone ;  while,  if  it  be  not  present,  we  shall 
simply  hear  the  ordinary  sound  of  the  pipe  through  the  reso- 
nator, without  any  reinforcement.  And  further,  if  we  fill  our 
ears  with  the  sound  of  the  tone  thus  sung  out  by  the  resonator, 
and  remember  its  pitch,  we  shall,  when  the  note  is  again  sounded 
out  by  the  organ-pipe,  have  no  difficulty,  even  without  a  reso- 
nator, in  hearing  the  harmonic  tone  :  and  by  dint  of  practice  we 
may  hear  at  will,  or  even  independently  of  will,  many  if  not 
all  of  those  component  harmonic  tones  which,  by  accompanying 
that  fundamental  tone  to  which  alone  in  ordinary  circumstances 
we  are  accustomed  to  listen,  help  to  make  up  the  note  of  the 
organ-pipe. 

A  very  convenient  form  of  resonator  may  be  made  of  a  common  tall 
lamp  chimney  or  a  similar  piece  of  tubing.  If  it  be  held  vertical,  as  its 
lower  end  is  immersed  in  water  to  various  depths  its  natural  pitch  varies : 
and  a  tube  thus  gradually  lowered  into  water  is  capable  of  resounding  in 
succession  to  the  different  harmonics  of  a  fundamental  note,  so  that  the  ear, 


XIV.] 


RESONATORS. 


431 


placed  near  the  tube,  can  recognise  their  several  presences.  In  Wintrich's 
resonator  an  aperture  at  the  side  may  be  closed  by  the  finger.  By  aid  of 
the  same  resonator  an  observer  is  thus  enabled  to  listen  alternately  to  the 
grave-pitched  muscle-sound  of  the  heart  and  to  the  sharper  valve-stretching 
sound.  See  Gscheidlen,  Physiol.  Methodik. 

Resonators  may  be  otherwise  employed.  If  the  small  aper- 
ture b  be  stopped  with  wax,  and  if  the  resonator  be  brought 
near  a  sounding  body,  it  will  absorb  the  energy  of  any  vibration 
corresponding  to  its  own  proper  tone,  and  may  then  be  removed 
and  listened  to :  thus  each  one  of  a  set  of  resonators  may  be 
made  to  select  one  tone  out  of  the  group  of  tones  present 
in  an  ordinary  musical  sound,  and  to  bring  it  to  the  ear  to  be 
listened  to. 

A  resonator  and  a  sounding  body  to  which  it  is  in  response  seem  to  be 
mutually  repelled,  in  consequence  of  the  stresses  set  up  in  the  intervening 
air  (Dvorak). 

Resonators  may  further  be  used  to  transfer  the  energy,  which 
they  thus  take  up,  to  relatively-massive  bodies  such  as  tuning- 
forks.  A  resonator  may  be  made  in  the  form  of  a  thin  wooden 
box  with  open  ends :  a  tuning-fork  precisely  in  tune  with  it  is 
fitted  on  its  upper  surface  ;  a  sound  causes  the  air  in  the  box 
to  vibrate,  the  air  acts  upon  the  box,  and  the  box  upon  the 
tuning-fork ;  if  all  be  in  exact  unison,  the  energy  accumulates 
in  the  tuning-fork,  which  comes  to  vibrate  energetically  and  to 
produce  a  loud  sound. 

Again,  the  oscillation  of  the  air  in  a  resonator  may  be 
rendered  visible  by  the  following  device:  —  A  cavity  in  a 
block  of  wood  '(Fig.  143)  is  divided  into  two  parts  by  a  mem- 
brane, such  as  thin  gold- 
beater's  skin.  The  one 
moiety  of  the  cavity  is 
connected  with  the  cavity 
of  a  resonator :  the  other 
is  connected  with  a  sup- 
ply of  coal-gas  which 

enters  at  B  and  passes  out  Gas 

at  C  on  its  way  to  be  burned  at  the  jet  D.  This  contrivance  is 
called  Koenig's  manometric  capsule.  When  a  sound  is  pro- 
duced outside  F,  containing  as  one  of  its  component  tones  the 
proper  tone  of  the  resonator,  the  air  in  the  resonator  oscillates 
in  sympathy  with  that  component,  the  diaphragm  vibrates  with 
it,  and  the  flame  at  D  is  rendered  alternately  higher  and  lower 


432  ON   SOUND.  [CHAP. 

by  the  action  of  the  vibrating  diaphragm  on  the  stream  of  gas. 
The  flame  obviously  alters  its  character :  and  the  change  under- 
gone by  it  can  be  studied  by  looking  at  it  while  the  head  is 
turned  rapidly  from  side  to  side,  the  eyes  being  kept  fixed  rela- 
tively to  the  head ;  or  by  looking  at  the  flame  through  an  opera- 
glass,  which  is  rapidly  moved  across  the  field  of  view ;  or,  best  of 
all,  by  looking  not  directly  at  the  flame  but  at  its  image  in  a 
rapidly-rotating  mirror :  in  all  which  cases  the  flame  or  its  image 
appears  to  spread  out,  not  into  a  uniform  band  of  light,  but 
into  a  band  with  serrated  edges,  or  even  into  a  chain  of  bead- 
like  separate  images. 

A  sufficiently-extensive  set  of  resonators  would  thus  enable 
us  to  effect  the  analysis  of  sounds  of  any  degree  of  complexity  : 
but  resonators  do  not  furnish  us  with  as  delicate  a  means  of 
investigation  as  the  means  first  described,  unless  indeed  they  be 
each  allied  with  a  tuning-fork ;  they  respond  in  general  with 
excessive  readiness  to  any  tone  in  proximity  to  their  own  natural 
tone. 

Synthesis  of  Sound.  —  Von  Helmholtz  showed  that  any 
quality  of  sound  may  be  built  up  by  the  superposed  effect,  upon 
the  ear,  of  simultaneously  sounding  tuning-forks  of  the  proper 
number,  pitch,  and  relative  loudness. 

Complex  Sound-Waves.  — The  pitch,  the  loudness,  and  the 
quality  of  a  sound  may  be  studied  together  by  causing  sound- 
waves to  impinge  directly  upon  some  sensitive  body  without  any 
intermediate  process  of  selection  or  filtration.  Thus,  if  instead 
of  a  resonator,  as  in  Fig.  143,  a  cone  be  adapted  to  a  mano- 
metric  capsule,  and  if  sound  be  produced  at  the  mouth  of  the 
cone,  sound-waves  will  impinge  directly  upon  the  membrane  in 
A.  The  membrane  will  go  through  a  complex  motion  some- 
what resembling  the  original  compound-vibration  of  the  sounding 
body,  and  the  flame  will  demonstrate  this  by  its  variations  of 
height.  The  image  of  this  oscillating  flame  will  appear  in  a 
mirror,  if  the  mirror  be  made  to  revolve,  as  a  band  of  light, 
serrated  by  large  teeth,  whose  outline  is  broken  by  subsidiary 
serrations ;  the  number  and  size  of  the  greater  serrations  indi- 
cate the  frequency  and  amplitude  of  the  fundamental  vibration  : 
those  of  the  subsidiary  serrations  vary  with  the  number  and 
variety  of  the  subsidiary  vibrations.  This  experiment  may  be 
roughly  carried  out,  if  there  be  no  revolving  mirror  at  hand, 
by  whirling  the  gas-flame  itself  (a  rat's-tail  jet  at  the  end  of  a 
flexible  tube)  before  the  eye. 


xiv.],  COMPLEX   SOUND-WAVES.  433 

It  is  interesting  to  carry,  out  this  experiment  by  singing  into  the  open 
end  of  the  cone ;  even  among  notes  of  the  same  pitch  sung  to  the  same 
vowel,  the  association  of  different  forms  of  the  flame-image  with  different 
qualities  of  tone  and  different  subjective  sensations  is  very  striking ;  and 
it  is  possible  for  a  singer  to  attain  to  the  production  of  very  pure  tone  — 
such  pure  tone  having,  however,  a  somewhat  hollow  quality  —  by  finding  out 
for  himself  how  to  control  the  larynx  so  as  to  keep  the  serrations  visibly 
open  and  simple. 

If  such  a  membrane  have  a  small  mirror  attached  to  it,  the 
mirror  will  share  in  the  vibrations  of  the  membrane,  and,  if  it  be 
jointed  on  a  hinge,  will  reflect  a  beam  of  light  in  such  a  fashion 
as  to  produce  a  curve  upon  photographic  paper  uniformly  rolled 
past  the  vibrating  membrane  ;  this  curve  will  indicate  the  fre- 
quency, the  amplitude,  the  complexity,  of  the  vibrations  of  the 
membrane. 

Sound-waves,  however  complex,  may  again  be  caused  per- 
manently to  record  the  succession  and  variation  of  their  own 
impulses.  Ldon  Scott's  Phon autograph  is  a  conical  vessel, 
closed  at  its  narrower  end  by  a  membrane ;  to  the  membrane  is 
attached  a  writing-point ;  the  extremity  of  the  writing-point  is 
brought  into  contact  with  a  smoked  revolving-cylinder.  So 
long  as  there  is  no  sound,  the  writing-point  describes  a  uniform 
line  on  the  rotating  cylinder:  when  sound-waves  enter  the  cone 
the  membrane  is  set  in  vibration,  and  the  writing-point  now 
describes  an  undulating  line,  which  varies  in  its  form  according 
to  the  frequency,  the  amplitude,  and  the  complexity  of  the  origi- 
nal vibration. 

It  must  be  observed  that  membranes  thus  made  use  of  do  not  exactly 
reproduce  the  original  motion  at  any  point  of  their  surface :  those  compo- 
nents are  exaggerated  which  approximately  or  exactly  coincide  in  frequency 
with  some  normal  mode  of  free  vibration  of  the  membrane  itself. 

Edison's  Phonograph  is  a  phonautograph  whose  writing- 
point  is  somewhat  blunt;  and  it  records  the  vibrations  of  its 
membrane  by  being  driven  through  variable  distances  into  a 
sheet  of  soft  tin  fixed  on  a  rotating  cylinder,  or  into  the  sub- 
stance of  a  rotating. cylinder  or  disc  of  wax:  it  leaves  a  perma- 
nently-deforming mark,  a  groove  of  varying  depth.  If  the 
membrane,  after  having  made  such  a  mark,  be  raised  from  the 
rotating  cylinder,  and  the  cylinder  turned  back  to  its  initial  posi- 
tion ;  if  the  membrane  be  now  readjusted  in  its  former  position, 
or,  better,  a  little  nearer  the  cylinder ;  and  if  the  cylinder  be 
again  rotated  in  the  former  direction,  with  the  same  velocity  as 
at  first,  —  the  depressions  in  the  groove  previously  produced, 


434  ON   SOUND.  [CHAP. 

being  of  variable  depth,  cause  the  blunt  writing-point,  under 
which  they  pass,  to  move  alternately  towards  and  away  from  the 
cylinder ;  this  compels  the  membrane  to  execute  vibrations,  and 
in  so  doing  to  set  up  vibrations  and  sound-waves  in  the  air, 
which,  being  received  by  the  ear,  produce  a  sound  similar  to  the 
original.  Not  exactly,  however:  the  process  is  not  perfectly 
reversible.  Some  consonants  are  not  well  reproduced,  especially 
the  explosives  (b,  p,  t,  d,  k,  g)  and  the  sibilants  (s,  z,  th);  and 
further,  it  is  generally  found  that  there  has  been  some  exaggera- 
tion of  some  of  the  higher  components  in  the  course  of  trans- 
mission through  the  membrane,  the  effect  of  which  is  to  render 
the  sound  reproduced  one  whose  quality  is  somewhat  metallic, 
nasal,  or  even  squeaky  and  Punchinello-like. 

LAWS  OF  VIBRATION  OF  SOUNDING  BODIES. 

These  laws  form  properly  a  part  of  Kinetics ;  but  the  means 
of  research  into  the  phenomena  of  Vibration  which  lies  most 
readily  at  our  disposal  is  the  observation  of  the  pitch  of  the 
sound  produced  by  vibrating  bodies ;  for  which  reason  some 
part  of  the  consideration  of  these  laws  has  been  deferred  to  this 
place. 

In  general,  any  vibration  of  a  vibrating  or  sounding  body  is 
a  periodic  motion,  a  Fourier-motion ;  though  in  particular  cases 
we  may  find  that  the  vibration  is  not  a  single  Fourier-motion 
either  simple  or  complex,  but  may  be  resolved  into  a  number  of 
such  motions,  simultaneous  and  superposed. 

Transverse  Vibrations  of  Strings. — If  a  string  be  stretched 
and  drawn  aside  from  its  mean  position,  it  tends  to  return  to 
that  position.  In  Fig.  144  let  the  string  AB,  subjected  to  a 

Fig.  144. 


tension  of  T  dynes  (=  the  Weight  of  T/g  grammes),  be  drawn 
into  the  position  ACB,  the  particle  C  being  supposed  for  sim- 
plicity's sake  to  have  been  initially  at  O,  the  centre  of  AB  :  the 
tension  tending  to  bring  back  the  particle  C  to  O  is  the  com- 
ponent of  the  total  tension  resolved  in  the  direction  CO :  this 
varies  directly  as  CO  —  that  is,  the  restitution-force  varies 
directly  as  the  displacement  —  the  criterion  of  Harmonic 


xiv.]  TRANSVERSE   VIBRATIONS  OF  STRINGS.  435 

Motion  :  and  it  can  be  shown  as  a  consequence  of  the  fact  that 
the  string  is  fixed  at  A  and  B,  that  the  string  will  oscillate  in 
some  such  manner  that  its  aggregate  motion  can  be  analysed 
into  a  number  of  simple  oscillations  whose  periods  are  commen- 
surable :  in  other  words,  that  the  motion  of  the  string  is  a  Fou- 
rier-motion. According  to  the  mode  of  disturbance  —  striking, 
plucking,  bowing  —  of  the  string,  or  the  duration  of  these  opera- 
tions, there  may  be  an  infinite  variety  in  the  relative  amplitudes 
of  these  component  simple  oscillations.  Some  of  these  com- 
ponents may  even  be  altogether  absent  :  where,  for  example,  a 
string  is  plucked  at  its  centre,  it  is  not  possible  that  any  of  those 
components  which  have  a  point  of  rest  at  the  centre  of  the 
string  should  be  present,  and  the  vibration  of  a  string  so  plucked 
is  one  in  which  all  the  even  components  are  absent.  In  gen- 
eral, a  vibrating  string  does  not  present  any  component-oscil- 
lation any  one  of  whose  nodes  would  be  at  the  point  of 
disturbance. 

Frequency  of  Oscillations.  —  The  velocity  of  propagation  of  a 
transverse  wave  along  a  uniform  stretched  string,  perfectly  flexible,  is 
v  —  ~\/t/p  (p.  268)  ;  here  t  =  T/?rr2,  where  r  is  the  cross-sectional  radius  of 


the  string.  Hence  v  =  VT/TT/O  H-  r.  But  the  length  of  each  wave  is  fixed  by 
the  condition  of  the  string  ;  the  string  is  bound  at  each  end,  and  if  we  con- 
fine our  attention  to  the  slowest  component,  the  fundamental  tone,  we  see 
that  A,  the  wave-length,  is  equal  to  2AB  or  2/,  twice  the  length  of  the  string. 
Then,  since  n,  the  number  of  complete  oscillations  per  second,  is  equal  to 
i//  A,  we  have  n  =  v/\  =  v/'2l  =  VT/irp  -r-  2rl  ;  or,  if  m  =T/g  be  the  number 
of  grammes  whose  Weight  stretches  the  string,  n  =  Vmg/Trp  -H  2rl. 

Problem.  —  A  wire  of  steel  (p  =  7-8),  1  m.  long  and  1-2  'mm.  thick,  is 
stretched  by  the  Weight  of  40  kilogrammes  and  set  in  transverse  vibra- 
tion :  what  will  be  the  frequency  of  its  fundamental  vibration?  What 
its  pitch?  —  Ans.  n  =  Vmg/wp  -f-  2rl  =  {V(40000  x  98_l)_-i-  (3-1416  x  7-8) 
-f-  (2  x  0-06  x  100)}  =  10545  vibrations  per  sec.  ;  H==EE  when  c'  =  253-1. 

So  for  a  perfectly-flexible  string:  the  effect  of  rigidity  of  wire  or 
string  is  to  diminish  the  number  of  vibrations,  and  to  cause  the  motion  to 
assume  the  character  of  a  number  of  superposed  harmonic  motions  of 
incommensurable  period. 

The  vibrations  of  a  violin  string  differ  much  from  those  of  a 
pianoforte  string.  In  the  violin  the  oscillating  string  sometimes 
travels  in  the  same  direction  as  the  bow,  sometimes  away  from 
it.  When  the  bow  and  the  string  travel  in  the  same  direction, 
the  bow  drags  the  string  with  it,  distorts  it,  pulls  it  out  to  an 
extent  greater  than  that  which  it  would  have  travelled  if 
allowed  freely  to  vibrate.  When  the  string  returns  the  bow 
fails  to  retain  it,  loses  it,  and  as  it  is  returning  bites  and  catches 


436  ON  SOUND.  [CHAP. 

it  again  by  means  of  some  rough  resinous  particle  at  some  other 
part  of  the  bow. 

The  friction  and,  consequently,  the  adhesion  between  the  string  and  the 
bow  are  relatively  somewhat  greater  when  both  move  in  the  same  direction, 
for  at  low  relative-speeds  friction  tends  to  increase. 

The  string  is  thus  distorted  and  assumes  successively  a 
number  of  forms,  of  which  no  one  is  curved :  and  the  form  of 
the  vibrating  string  at  any  instant  presents  an  angle  between 
two  straight  lines,  a  form  differing  considerably  from  the  curve 
of  sines.  But  it  is  periodic,  and  it  is  true  Fourier-motion. 

The  mathematical  problem  is  —  What  superposition  of  commensurate 
S.H.M.'s  (compare  Fig.  48)  will  produce  a  vibration-curve  such  that,  for  a 
certain  distance,  the  flexures  so  balance  one  another  as  to  produce  a  straight 
line,  and  then  so  aid  one  another  as  to  produce  an  abrupt  angle,  again  fol- 
lowed by  a  straight  line  ?  This  can  be  solved,  and  the  result  is  that  tke 
vibration  of  a  bowed  string  must  be  composed  of  a  fundamental  vibration, 
of  weak  components  2d  to  6th,  and  of  ample  higher  components.  This 
agrees  with  the  result  of  resonator-analysis  of  the  sound  of  a  violin. 

In  the  sound  of  a  violin  the  upper  harmonics  are  loud  and 
piercing ;  the  nearer  harmonics  are  feeble,  and  the  fundamental 
tone  stands  apparently  alone,  but  rendered  penetrating  in  qual- 
ity by  the  high  mass  of  harmonics.  Purity  of  violin  tone 
depends  upon  perfect  periodicity  of  the  peculiar  motion  of  the 
string ;  this  is  difficult  to  attain,  for  a  good  elastic  violin, 
uniform  strings,  a  uniform  bow,  uniformly  resined  and  evenly 
handled,  are  necessary:  and  any  stumbling  of  the  bow  over  the 
string,  or  any  irregular  movement  of  the  string  under  the  bow, 
is  revealed  by  scratchiness  of  tone. 

The  sharper  the  angle  made  at  the  point  of  disturbance,  the 
richer  the  tone  in  high  harmonics.  A  string  plucked  with  a 
quill,  as  in  the  old  harpsichord,  has  thus  a  metallic  tinkling 
quality,  and  its  fundamental  tone  is  relatively  very  feeble. 

A  string  struck  suddenly  at  one  point  has  a  form  differing 
greatly  from  that  of  the  curve  of  sines.  Part  of  the  string 
remains  unaffected,  while  the  part  struck  is  distorted.  This  dis- 
tortion travels  along  the  string,  and  results  in  a  periodic  motion 
abounding  in  high  components ;  the  tone  produced  is  tinkling. 
If  the  same  cord  or  wire  be  struck  gently  by  a  soft  elastic 
hammer,  the  blow  being  deliberate,  and,  as  it  were,  gradually 
insisting  upon  the  displacement  of  the  string  at  the  point  struck, 
the  disturbance  is  more  evenly  spread  over  the  whole  string,  the 
fundamental  component-vibration  is  more  prominent,  the  higher 
components  are  relatively  more  feeble,  and  the  tone  is  purer. 


xiv.]  TRANSVERSE   VIBRATIONS  OF  STRINGS.  437 

In  a  pianoforte  string  struck  by  an  elastic  soft-hammer  the 
harmonics  up  to  the  sixth  are  present;  the  seventh  is  obliterated, 
or  nearly  so,  by  the  hammer  being  made  to  strike  the  string  at 
a  spot  one-seventh  of  its  length  from  the  end  of  the  string  — 
that  is,  at  a  spot  which  would  have  been  a  node  of  the  seventh 
component  if  that  component  had  been  allowed  to  exist  in  the 
compound  vibration:  and  the  components  beyond  the  seventh 
are  feebly  represented. 

The  Monochord  is  a  box  of  thin  light  wood,  containing 
air  which  communicates  with  the  exterior  air  by  lateral  aper- 
tures. Upon  this  box  rest  two  bridges  ("banjo-bridges"),  one 
near  each  end.  Over  the  bridges  is  stretched  a  wire;  of  this, 
one  end  is  firmly  fixed  to  one  end  of  the  box,  while  the  other 
is  either  passed  over  a  pulley  and  made  to  support  a  weight, 
or  else  is  connected  with  a  tuning-peg,  which  may  be  turned  by 
a  tuning-key.  The  tension  on  the  stretched  wire  may  thus 
be  varied. 

Experiments  with  the  Monochord. — For  experiments 
it  is  better  to  use  a  form  of  monochord  in  which  there  are  two 
wires,  of  which  one  is  tightened  by  a  peg,  the  other  by  the 
weight  of  a  suspended  mass ;  in  the  latter  of  the  two  wires  the 
total  tension  on  the  wire  can  be  directly  measured,  in  the  former 
it  must  be  inferred. 

1.  Suppose  a  wire  1*2  mm.  thick,  whose  free  vibrating  part 
is  1*2  metres  long,  to  be  stretched  by  the  weight  of  48  kilogr., 
and  the  pitch  of  the  sound  produced  to  be  G  =  96  vibrations. 
What  weight  ought  to  be  added  in  order  to  raise  the  pitch  to  d? 

The  pitch  is  raised  G :  d  —  i.e.  a  fifth :  the  vibrations  are 
rendered  more  numerous  in  the  ratio  2:3;  the  tension  must  be 
increased  in  the  ratio  22 :  32,  or  4 :  9  ;  the  stretching-weight  must 
be  increased  from  that  of  48  to  that  of  108  kilogr. ;  the  mass 
which  would  have  to  be  added  is  60  kilogrammes. 

2.  Two  wires  of  equal  thickness  are  stretched  —  one  by  the 
tuning-peg  and  the  other  by  the  weight  of  a  heavy  mass  —  so  as 
to  vibrate  in  unison.     The  weighted  wire  is  removed  and  replaced 
by  one  of  the  same  length,  but  of  a  different  thickness,  stretched 
by  the  same  weight.      A  thinner  wire   gives  a  higher  note,  a 
thicker  one  a  lower. 

If  in  Ex.  1  a  wire  1  mm.  thick  be  employed,  what  will  be 
the  pitch  of  the  sound  produced?  The  frequency  varies  inversely 
as  the  radius :  it  therefore  exceeds  that  of  a  wire  1*2  mm.  thick 
in  the  ratio  of  6  :  5 ;  the  note  produced  will  be  Bb. 


438  ON  SOUND.  [CHAP. 

3.  A  brass  wire  and  a  steel  wire  of  equal  gauge  are  equally 
stretched  :  they  are  free  to  vibrate  in  equal  lengths.     Brass  has 
a  density  p  =  8-38,  steel  =  7*8.     The  brass  wire  gives  a  sound 
lower  in  pitch  than  that  given  by  the  steel  wire  :  the  respective 
frequencies  are  in  the  ratio  of  V7-8  to  V8-38  or  1  :  1-03651. 

A  catgut  string,  whose  density  is  small,  gives  a  higher  note 
than  a  steel  wire. 

4.  In  order  to  vary  the  free   vibrating-length   of   wire,  a 
movable  bridge  is  arranged  under  the  string.     If   this  be  so 
placed  that  60  cm.  of  the  wire  are  free  to  vibrate  instead  of  120 
as  before,  the  sound  produced  will  be  the  Octave;  if  40  (=i|^), 
the  Twelfth;  if  30  C^1^),  the  Fifteenth;  if  24  (  =  1fa),  the 
Seventeenth  —  and   so   forth  —  above    the    fundamental    note 
emitted  by  the  freely-vibrating  string  of  120  cm.  length. 

The  number  of  vibrations  of  a  string  varies  inversely  as  its 
length.  Hence  if  we  wish,  with  a  string  which  sounds  C,  to  pro- 
duce the  note  D,  whose  frequency  is  f  x  that  of  C,  we  must 
allow  the  string  to  vibrate  not  as  a  whole,  but  only  in  -|  of  its 
length.  To  produce  the  scale  on  one  string,  the  parts  of  the 
string  which  are  allowed  to  vibrate  are  as  follows  :  — 

d       r        m        f        s        1         t         d1 


i     t     i    t    t    A     * 

The  application  of  this  principle  is  familiar  in  violin  playing. 

5.    Nodes  and  Loops  can  also  be  shown  on  the  monochord. 

If  the  wire,  120  cm.  in  its  vibrating  length,  be  lightly  touched 

Fig.145. 


at  20  cm.  from  the  end,  and  if  the  twenty-centimetre-part  of  the 
wire  be  set  in  vibration  by  a  bow,  the  whole  wire  is  found  to  be 
in  vibration  from  end  to  end ;  but  not  as  a  whole.  It  divides 
itself  into  segments  or  vibrating  loops,  separated  by  nodes  or' 
points  of  rest.  Each  segment  is  20  cm.  long:  and  the  sound 
given  out  is  that  which  might  be  emitted  by  half-a-dozen 
separate  wires  each  20  cm.  long  —  that  is,  it  bears  to  the  note 
emitted  by  the  whole  string  the  same  proportion  as  g1  does  to  C 


XIV.] 


MONOCHORD. 


439 


—  two  octaves  and  a  fifth.  Similarly  for  other  fractional  divi- 
sions of  the  wire  or  string.  Lightly  stopping  the  string  has  the 
effect  of  destroying  or  of  checking  the  formation  of  all  those 
modes  of  vibration  which  have  not  a  node  at  the  point  touched ; 
hence  the  6th,  12th,  18th,  etc.,  components  of  the  vibration  of 
the  whole  string  are  unchecked,  while  the  other  components  are 
rendered  impossible. 

The  nodes  and  loops  of  a  string  vibrating  in  this  way  are 
rendered  manifest  by  paper  riders  placed,  some  at  the  nodes, 
some  on  the  loops ;  when  the  string  enters  into  vibration  those 
riders  which  had  been  placed  on  the  nodes  will  retain  their 
place,  while  those  on  the  loops  will  be  jerked  off. 

Nodes  and  loops  on  vibrating  strings  may  be  illustrated  on 
a  large  scale  as  follows:  —  Take  an  indiarubber  tube  10  feet 
long,  filled  with  sand,  or  a  long  spiral  of  iron  or  brass  wire,  and 
fix  one  end  of  this  to  a  wall ;  hold  the  other  end  in  the  hand. 
On  moving  the  hand  gently,  the  natural  period  of  oscillation  of 
the  cord  can  be  easily  found.  Give  with  the  hand  a  series  of 
transverse  impulses,  so  timed  as  to  aid  the  natural  oscillations  : 
the  tube  or  spiral  will  enter  as  a  whole  into  wide  oscillations. 
Now  give  such  impulses  twice  as  often :  the  cord  will  divide  into 
two  segments,  pivoting  in  a  striking  manner  round  the  central 
point.  Do  so  three  times  as  often  as  at  first:  the  cord  will 
divide  into  three  segments  or  loops,  pivoting  on  two  nodes.  By 
increasing  the  frequency  of  the  movements  of  the  hand,  the  cord 
can  be  made  to  oscillate  in  4,  5,  6,  7,  or  even  8  or  9  segments, 
according  to  the  dexterity  of  the  experimenter.  The  experi- 
ment is  a  striking  one ;  and  it  may  be  varied  by  causing  the 
hand  to  move  in  a  circle  or  an  ellipse  instead  of  a  straight  line. 

Melde's  Experiments. — A  tuning-fork  is  provided  with 
a  little  hook  on  one  of  its  prongs  ;  to  this  hook  is  attached  a  fine 
white  silk  thread.  This  thread 
is  passed  over  a  wheel  and  at- 
tached to  a  suspended  mass 
partly  immersed  in  water ;  the 
quantity  of  this  mass  can  be 
coarsely  adjusted  by  the  addi- 
tion or  removal  of  sand ;  its 
effective  weight  can  be  finely 
adjusted  by  varying  the  quan- 
tity of  water  in  W  (Fig.  146). 
The  fork  is  set  in  vibration;  waves  appear  to  travel  up  and 


Fig.HS. 


440  ON  SOUND.  [CHAP. 

down  the  thread ;  if  the  string  be  illuminated  by  a  beam  of  light 
in  a  dark  room,  the  effect  is  singularly  beautiful.  As  the  ten- 
sion is  increased,  the  segments  vary :  at  length  the  thread  vibrates 
as  a  whole,  and  seems  to  form  an  opalescent  spindle.  Its  fre- 
quency of  vibration  is  half  that  of  the  fork  ;  the  thread,  when 
at  its  limit,  is  pulled  back  by  the  retreating  fork  into  its  mean 
position,  but  is  relaxed  and  allowed  to  swing  over  upon  the 
return  of  the  fork ;  whence  two  oscillations  of  the  fork  corre- 
spond to  one  of  the  thread.  If  the  tension  be  reduced  to  ^,  the 
vibrating  part  of  the  string  must  be  shortened  to  J  in  order  to 
keep  time  with  the  fork,  or  else,  if  the  string  be  not  shortened, 
the  string  will  divide  into  two  equal  segments  or  loops  separated 
by  a  node  :  if  the  tension  be  reduced  to  -J,  the  string  divides  into 
three  loops  with  two  nodes  ;  and  so  forth. 

If  the  tuning-fork  be  turned  round  through  90°,  so  as  not 
now  to  tighten  and  relax  the  thread,  but  to  give  it  a  series 
of  transverse  impulses,  a  similar  series  of  phenomena  will  be 
observed;  but  the  fundamental  vibration  is  now  simply  synchro- 
nous with  that  of  the  tuning-fork. 

When  the  thread  is  suspended  between  two  tuning-forks 
whose  frequencies  bear  an  aliquot  ratio,  the  tuning-forks  being 
placed  at  such  distances  from  one  another  as  to  tighten  the 
thread  to  the  required  amount,  the  motion  of  the  thread  becomes 
periodic,  and  presents  a  complex  of  beautiful  loops  and  nodes 
which  are  obtained  with  comparative  ease. 

Transverse  vibrations  of  cords  may  be  studied  with  respect 
to  the  motion  of  each  particle  by  casting  a  beam  of  light  along 
a  vibrating  cord,  and  looking  at  a  particular  bright  or  bright- 
ened spot  on  the  cord.  The  bright  spot  appears,  when  the  cord 
is  looked  at  end-on,  to  give  in  quick  succession  a  large  variety 
of  such  forms  as  we  have  already  seen  to  be  produced  by  the 
composition  of  S.H.M.'s.  A  bright  spot  on  the  cord  may  also 
be  looked  at  through  a  microscope  whose  object-glass  is  borne 
upon  a  vibrating  tuning-fork ;  the  apparent  motion  of  the  spot 
produced  by  the  motion  of  the  object-glass  (this  being  parallel 
to  the  length  of  the  cord)  is  compounded  in  the  eye  with  its 
real  motion ;  the  apparent  up-and-down  motion  of  the  spot, 
as  looked  at  transversely,  is  spread  out  into  an  open  curve, 
and  thus  becomes  more  intelligible,  for  the  eye  can  more 
readily  comprehend  open  curves  than  simple  up-and-down 
movements. 

Longitudinal  vibrations  of  a  string  may  be  excited  by 


xiv.]  LONGITUDINAL   VIBRATIONS.  441 

drawing  one  point  of   a  violin  bow  along  the  string:  a  very 
shrill  tone  is  produced. 

The  velocity  of  propagation  is  v  =  v/g7p ;  the  wave-length  is  twice  the 
length  of  the  string,  or  A.  =  21 ;  the  number  of  fundamental  vibrations  per 
second,  n  —  v/\  =  Vg/p  -=-  2/. 

Problem.  —  A  steel  wire  (elasticity  g  =  2, 520,000000  #,  and  density  7-8), 
of  one  metre  in  length,  is  clamped  at  the  two  ends  and  set  in  longitudinal 
vibration.  What  will  be  the  pitch  of  the  sound  produced? — Ans.  The 
frequency  is  n  =  ^%fp  +  21  =  (V2520,000000  x  981  H-  7-8  -=-  200}  =.  2815 
vibrations  per  second  =  /""  + . 

As  a  rule  the  longitudinal  vibrations  of  a  string  or  wire  are 
much  more  frequent  than  the  transverse  ones,  and  thus  produce 
a  much  shriller  sound,  and  further,  they  are  not  so  much  affected 
by  tension  applied  to  the  string,  for  a  variation  of  tension  which 
would  materially  modify  the  frequency  of  transverse  vibration 
would  have  little  effect  upon  either  the  elasticity  g,  or  the 
density  p,  upon  which  the  longitudinal  vibrations  depend. 

Before  the  transverse  vibrations  could  be  as  frequent  as  the  longitudinal 
ones,  it  would  be  necessary  that  the  tension  should  be  t  =  g,  the  amount 
ideally  necessary  to  double  the  length  of  the  wire. 

A  violin  e"-string  gives,  when  'rubbed  longitudinally  by  one  point  of 
the  bow,  a  sound  in  the  neighbourhood  of  ('/$)"" ;  while,  when  let  down  so 
as  to  sound  only  e',  it  gives  out  a  longitudinal  vibration-sound  not  so  low  as 
(/$)"" ;  the  longitudinal  vibration  hardly  falls  a  comma,  while  the  trans- 
verse falls  an  octave.  All  the  catgut  strings  of  a  violin  may  be  observed  to 
give  out  nearly  the  same  longitudinal  note,  for  this  does  not  depend  on  their 
thickness. 

From  this  we  see  how  important  it  is  to  use  the  bow  in  such 
a  way  as  to  bring  out  transverse  vibrations  only,  and  by  no 
means  to  wield  it  so  that  any  component  of  its  motion  over  the 
string  can  excite  longitudinal  vibrations,  resulting,  as  these  do, 
in  shrill  discordant  tones. 

By  means  of  the  monochord  we  may  learn  that  a  string, 
while  vibrating  longitudinally,  divides  into  loops  separated  by 
nodes,  just  as  it  does  while  executing  transverse  vibrations. 

Longitudinal  vibrations  of  rods  resemble  those  of  strings 
or  wires.  A  glass  rod  grasped  by  its  centre  and  rubbed  longitu- 
dinally by  a  resined  cloth  will  enter  into  longitudinal  vibration 
and  will  produce  a  shrill  sound.  A  glass  tube  treated  in  the 
same  manner  may  be  made  to  vibrate  so  vehemently  that  it 
shivers  into  segments. 

Transverse  vibrations  of  rods  obey  the  rule  that  if  0  be  the 
thickness  of  the  rod,  I  its  length,  g  its  Young's  Modulus  of 
elasticity,  and  p  its  density,  then  n  <x  Vg/p  •  6/P,  or  n  =  const,  x 


442  ON   SOUND.  [CHAP. 

Vg//o  •  6/12.  The  constant  varies  according  to  the  form  of  the 
cross-section  and  the  mode  of  clamping  or  support ;  and  for  the 
different  harmonics  it  presents  values*  which  bear  no  simple 
numerical  relation  to  one  another. 

As  examples  of  rods  free  at  both  ends  and  vibrating  trans- 
versely, we  may  take  the  common  glass  or  metal  harmonicon  — 
plates  of  glass  or  metal  supported  by  threads  at  the  nodal  lines 
and  struck  by  hammers.  As  examples  of  rods  clamped  at  one 
end  and  vibrating  transversely,  we  may  take  reeds  such  as  those 
of  the  harmonium  or  concertina.  Their  pitch  is  raised  by  filing 
off  their  substance  towards  their  free  ends ;  it  is  lowered  by 
thinning  them  towards  their  base.  Tuning-forks  afford  another 
example  of  vibrating  rods ;  they  are  tuned  in  the  same  way. 

In  rods  of  the  same  thickness  the  frequency  of  vibration  varies  inversely 
as  the  square  of  the  length,  as  the  formula  n  —  c  "N/g/p.  O/l'2  indicates;  but 
if  the  thickness  6  and  the  length  I  vary  together,  so  that  different  rods  have 
the  same  shape,  the  frequency  depends  on  the  relative  length  only.  Thus 
a  tuning-fork  4  inches  long  and  one  2  inches  long,  of  the  same  shape, 
produce  notes  which  differ  by  an  octave :  the  same  rule  applies  to  reeds. 

A  rod  does  not  vibrate  as  a  whole,  in  halves,  thirds,  etc.,  but  the  com- 
ponent vibrations  have  incommensurable  frequencies,  and  each  component 
has  its  own  rate  of  travelling  through  the  solid,  so  that  the  periodic  nature 
of  the  vibration  is  disturbed.  Such  a  vibration  of  a  rod  is  an  extreme  case 
of  the  vibration  of  a  rigid  or  thick  wire. 

A  rod  of  circular  section  can  vibrate  transversely  with  indifference  in 
all  directions.  One  whose  section  is  oblong  can  oscillate  more  widely  in  a 
plane  at  right  angles  to  the  broad  face  than  it  can  in  a  plane  parallel  to  that 
face  —  e.g.,  a  vibrating  reed,  in  which  the  latter  oscillation  is  absolutely 
insignificant.  If  a  rod  of  circular  section  be  filed  at  one  side,  all  component 
oscillations  which  tend  to  bend  the  rod  upon  the  filed  face  are  retarded ; 
those  which  are  at  right  angles  to  these  are  unaffected.  Thus  a  rod  of  steel, 
grasped  by  a  vice  at  a  particular  spot,  may  be  so  tuned  that  when  it  is  set 
in  vibration  by  a  violin  bow  its  point  may  execute  vibrations  in  directions 
at  right  angles  to  each  other,  and  bearing  to  each  other  any  predetermined 
ratio  of  frequency. 

Take  a  knitting-needle ;  fix  it  in  a  vice ;  mark  on  the  needle  the  height 
at  which  it  stands  in  the  vice ;  touch  the  free  tip  of  the  needle  with  a  little 
gum:  scatter  a  little  starch  or  powdered  antimony  over  the  needle  tip; 
some  will  adhere.  Suppose  the  ratio  desired  is  4:5;  refer  to  Fig.  38.  File 
the  rod,  always  towards  one  aspect,  until  the  movement  of  the  tip  of  the  rod, 
as  revealed  by  the  brilliant  particles  of  starch  or  antimony,  comes  to  present 
the  curve  sought.  If  the  filing  have  been  carried  too  far,  a  little  metal  may 
be  removed  from  the  needle  at  an  aspect  at  right  angles  to  that  of  the  pre- 
vious operation.  If  after  a  needle  has  been  tuned  in  this  way,  so  that  its 
component  vibrations  have  been  rendered  commensurate,  it  be  grasped  by 
the  vice  at  a  point  a  little  above  or  below  the  original  point,  the  intervals 

*  See  Lord  Rayleigh's  Theory  of  Sound,  Vol.  I.,  Chap.  viii. 


xiv.]  TRANSVERSE   VIBRATIONS.  448 

cease  to  be  commensurate,  and  the  curves  seen  pass  through  a  series  of 
changes  exemplified  by  those  of  Fig.  41. 

A  tuning-fork  or  vibrating  reed,  made  to  write  its  own  vibrations  on  a 
rotating  cylinder,  describes  a  sinuous  curve  which  is  almost  identical  with 
the  curve  of  sines.  This  shows  that  the  motion  is  peiidular.  Again,  if 
two  harmonium  reeds  have  their  tips  silvered  so  as  to  reflect  light,  and  if 
they  be  arranged  at  right  angles  to  one  another ;  and  if  a  lamp  and  lens  be 
so  placed  that  a  beam  of  light  falls  first  upon  one  reed-tip,  then  upon  the 
other,  and  finally  upon  a  screen ;  then  if  one  reed  be  set  in  vibration,  the 
spot  of  light  opens  out  into  a  line,  while  if  both  vibrate  the  line  opens  out 
into  some  figure  of  the  order  of  those  shown  in  Figs.  35-40.  This  figure 
retraces  itself  and  retains  its  form  if  the  reeds  be  accurately  tuned  to  an 
aliquot  ratio  of  frequencies,  while  on  the  other  hand,  if  the  reeds  be  not  so 
in  tune,  the  figure  undergoes  rapid  changes  —  changes  painful  to  the  eye, 
as  the  accompanying  beats  are  to  the  ear. 

Torsional  Vibration  of  Rods.  —  When  a  rod  is  clamped  by 
one  end  in  a  vice,  and  a  violin  bow  drawn  round  it,  it  may  be 
caused  to  execute  vibrations  in  which  it  successively  twists  and 
untwists  itself  round  its  own  axis  ;  and  it  is  found  to  do  so  with 
a  frequency  Vn/g  x  that  of  the  longitudinal  vibrations  in  the 
same  rod. 

Vibration  of  Discs  or  Plates.  —  A  disc  of  metal  or  of 
glass  may  be  caused  to  vibrate  by  means  of  a  violin  bow  drawn 
across  its  edges.  The  point  of  support  of  the  disc  is  necessarily 
a  nodal  point ;  any  number  of  points  may  be  supported  or  fixed, 
and  all  these  must  also  be  nodal  points.  The  disc  or  plate  may, 
under  such  arbitrary  conditions,  adjust  itself  so  as  to  vibrate, 
according  to  circumstances,  with  great  variety  of  nodal  lines 
and  vibrating  segments. 

A  disc  of  brass  or  glass  may  be  fixed  at  its  centre  to  a  heavy  stand.  If 
the  circumference  be  touched  at  any  point  while  the  whole  is  set  in  vibration 
by  a  violin  bow,  the  point  touched  will  be  a  nodal  point ;  the  spot  where  the 
violin  bow  is  applied  tends  to  become  the  centre  of  a  loop ;  according  to  the 
relative  situations  of  the  points  held  fixed  and  of  the  point  of  application 
of  the  disturbing  cause,  will  vary  the  manner  and  the  pitch  of  the  resultant 
vibration. 

Different  discs  of  the  same  shape  and  vibrating  in  similar 
ways  have  relative  vibrational  frequencies  varying  as  O/l2-  Vg/p  ; 
the  same  law  as  holds  good  in  the  transverse  vibration  of  bars. 

The  form  of  the  nodal  lines  may  be  studied  by  strewing  sand  and  lyco- 
podium  powder  upon  a  vibrating  disc  or  plate :  the  sand  collects  on  the  nodal 
lines;  the  lycopodium,  by  reason  of  the  disturbance  of  the  air,  is  blown 
towards  the  centre  of  each  vibrating  segment. 

Contiguous  sectors  are  in  opposite  phases  of  vibration.  If 
the  ear  be  placed  immediately  opposite  the  centre  of  figure  of  a 


444  ON   SOUND.  [CHAP. 

circular  vibrating  disc,  there  will  be  but  little  sound  heard :  the 
receding  and  the  approaching  sectors  neutralise  each  other's 
effects  upon  the  air  and  upon  the  ear.  If  the  hand  be  held 
above  a  vibrating  disc  so  as  partly  to  cut  off  the  effect  of  one  of 
the  sectors,  the  sound  heard  opposite  the  centre  of  the  disc  is 
enhanced ;  if  two  contiguous  sectors  be  thus  shaded  from  hear- 
ing, the  sound  is,  as  at  first,  very  feeble  ;  if  two  sectors  not 
contiguous  but  vibrating  in  the  same  sense  be  thus  covered,  the 
sound  produced  is  much  louder.  If  a  Koenig's  manometric 
capsule  be  provided  with  a  tube  which  bifurcates,  and  if  the 
branch  tubes  each  terminate  in  a  cone,  one  cone  may  be  placed 
over  a  vibrating  sector,  while  the  other  may  be  moved  about 
over  the  disc.  As  it  passes  round  the  circumference  of  the 
disc,  it  will  be  found  that  the  gas-flame  of  the  capsule  is  alter- 
nately much  agitated  and  steady  —  agitated  when  both  cones 
are  'over  sectors  vibrating  in  the  same  sense  ;  steady,  or  nearly 
so,  when  they  are  over  sectors  vibrating  in  opposite  phases. 
There  is,  also,  always  some  oscillatory  tangential  twist  of  the  disc. 

Vibration  of  Membranes.  —  If  a  membrane  be  subject  to  a 
tension  t  equal  over  its  whole  circumference,  such  as  that  of  a 
drum,  it  vibrates  as  a  whole  ;  its  higher  component  vibrations 
are  not  commensurable  with  the  slower  ones,  and  the  higher 
tones  faintly  to  be  heard  in  the  sound  of  a  drum  discord  with 
the  fundamental  tone,  [woe  Vt/p^-r.] 

If  the  membrane  be  not  equally  stretched  in  all  directions, 
it  will,  when  set  in  vibration,  so  arrange  itself  as  to  vibrate 
feebly  along  the  line  of  least  tension,  and  strongly  in  the 
direction  of  greatest  tension.  Thus  a  square  piece  of  thin 
indiarubber,  clamped  by  the  two  opposite  edges  and  stretched, 
will  vibrate  at  the  same  rate  as  an  indiarubber  cord  of  the  same 
free  length  and  exposed  to  the  same  tension  t  per  unit  of  sec- 
tional area ;  and  it  may  be  idealised  as  a  collection  of  india- 
rubber  cords,  arranged  side  by  side,  attached  to  each  other,  and 
vibrating  in  unison. 

Vibration  of  Bells.  —  A  bell-shaped  body,  set  in  motion  by 
being  struck,  or  by  being  rubbed  with  the  resined  or  wetted 
finger  carried  round  the  circumference,  enters  into  vibration 
simultaneously  radial  and  tangential.  The  bell  divides  into  an 
even  number  of  sectors ;  of  these  one  half  dilate,  while  the 
other  half  (individually  alternating  with  the  former)  contract 
radially.  At  the  same  time  sectors  of  the  bell,  moving  tan- 
gentially,  twist  to-and-fro  round  the  axis  of  the  bell ;  alternate 


xiv.]  VIBRATION   OF   BELLS.  445 

sectors  are  opposed  to  one  another  in  the  direction  of  their 
twist ;  hence  at  some  of  the  nodes  which  separate  the  sectors 
there  is  compression,  at  others  dilatation  of  the  substance  of 
the  bell.  The  loops  of  the  radial  motion  are  the  nodes  of  the 
tangential  motion ;  thus  where  there  is  least  expansion  or  con- 
traction there  is  the  greatest  amount  of  twist.  In  the  circum- 
ference of  a  vibrating  bell  there  are  generally  four  loops 
corresponding  to  each  motion. 

The  effect  of  loading  a  vibrating  body  is  to  lower  its 
pitch ;  if  the  load  be  distributed  uniformly,  all  the  components 
are  lowered ;  if  it  be  suspended  from  points  of  the  vibrating 
body,  those  component  vibrations,  if  any  there  be,  which  have 
their  nodes  at  those  points  remain  unaffected. 

The  preceding  propositions  have  related  to  the  vibrations 
into  which  a  body  may  enter  when  it  is  disturbed  and  then  left 
to  itself.  The  vibration  in  such  cases  is  called  free  vibration, 
the  period  of  which  depends  on  the  nature  and  the  form  of  the 
vibrating  body  itself.  If  a  body  capable  of  vibration  be  acted 
upon  by  a  series  of  impulses  ab  externo,  the  result  depends  upon 
the  period  of  recurrence  of  these  impulses.  These  may  be  so 
timed  as  to  aid  the  natural  free  vibrations  of  the  body,  adding 
energy,  and  therefore  increasing  the  amplitude  at  every  oscil- 
lation ;  or  they  may  be  so  timed  as  sometimes  to  aid,  sometimes 
to  thwart  the  natural  oscillations,  and  thus  to  produce,  on  the 
whole,  no  effect  so  far  as  concerns  the  amplitude  of  these.  In 
the  former  case  the  interval  between  two  successive  impulses  ab 
externo  is  equal  to  the  period  of  the  natural  vibration ;  while, 
when  this  interval  differs  materially  from  the  period  of  the  free 
vibration,  the  amplitude  of  vibration  is  not  increased,  but  the 
energy  communicated  in  the  successive  impulses  is  dissipated  in 
heat. 

A  heavy  bell  has  a  natural  period  of  pendular  swing,  and  if 
a  person  gently  pull  the  bell-rope,  a  very  slight  swing  may  be 
obtained,  perhaps  barely  perceptible.  If  the  pull  be  repeated 
while  the  rope  is  tending  to  slacken  in  the  ringer's  hand,  the 
original  small  swing  is  increased,  perhaps  doubled;  a  succession 
of  well-timed  pulls  causes  ultimately  a  wide  oscillation  of  the 
bell ;  and  when  the  bell  has  been  set  fairly  ringing  the  ampli- 
tude of  the  oscillation  is  kept  up,  and  all  loss  of  energy  replaced, 
by  a  series  of  well-timed  impulses.  If,  however,  the  impulses  be 
so  timed  that  they  sometimes  act  in  favour  of  the  oscillations  of 


Fig.147. 


446  ON  SOUND.  [CHAP. 

the  bell,  and  are  sometimes  delivered  against  a  tightening  rope, 
the  bell  will  either  not  ring  at  all  or  will  do  so  very  irregularly. 

If  a  body  capable  of  freely  vibrating  with  n  oscillations  per 
second  be  placed  in  material  communication  with  a  body  actually 
vibrating  n  times  per  second,  the  former  will  take  up  energy  and 

enter  into  vibration.  If  a  cylinder 
AB  be  fixed  at  the  extremity  of  the 
rod  CD ;  if  another  cylinder  EF,  of 
like  material  and  dimensions, be  fixed 
at  the  other  end  of  CD ;  and  if  AB 
_,,™,  "^D"^  foe  set  in  longitudinal  vibration,  — 

then  EF  will  be  set  in  vibration  of  equal  period.  Here  it  must 
be  observed —  (1)  that  expansion  of  AB  occurs  at  the  same  time 
with  compression  of  EF,  and  vice  versd  ;  and  (2)  that  the  centre 
of  mass  of  the  whole  system  remains  unchanged  in  its  position. 

Two  organ-pipes  of  exactly  the  same  pitch,  mounted  on  the 
same  wind-chest,  may  fall  into  a  similar  opposition  of  phase,  and 
retain  it  for  long  periods  of  time :  so  long  as  they  do  this,  they 
can  together  emit  but  little  sound,  and  the  sound  produced  is 
rendered  louder  by  silencing  one  of  the  pipes. 

Two  similar  strings  of  equal  length,  equally  stretched  over 
the  same  solid  framework  parallel  to  one  another,  will  so  adjust 
their  vibrations  as  simultaneously  to  approach  or  to  diverge 
from  one  another. 

A  tuning-fork  m&y  be  regarded  as  made  up  of  two  equal 
rods,  connected  with  a  common  basis  ;  the  prongs  simultaneously 
approach  and  diverge  from  one  another  when  the  fork  is  in 
vibration. 

Two  clocks  which  keep  good  time  together  will,  when  placed 
on  the  same  table,  beat  synchronously. 

A  tuning-fork  can  be  set  in  vibration  by  another  tuning-fork 
of  exactly  the  same  pitch  vibrating  within  the  same  room. 
The  material  communication  between  the  two  forks  is  effected  by 
the  air;  the  sounding-fork  causes  waves  in  the  air;  these  cause 
well-timed  impulses  against  the  second  fork ;  the  effect  of  these 
is  cumulative,  and  the  second  fork  takes  up  the  vibration  of  the 
first. 

In  the  same  way  a  mass  of  air  in  a  vessel. or  flask,  since  it 
has  a  natural  period  of  vibration,  may,  if  air-waves  dash  against 
the  open  mouth  of  the  vessel  at  such  intervals  as  correspond  to 
its  natural  vibration,  be  caused  by  the  accumulated  effect  of 
these  waves  to  enter  into  violent  oscillation.  This  principle  we 


xiv.]  EESONANCE.  447 

have  seen  made  use  of  in  resonators ;  and  from  the  analogy  of 
these  instruments  all  phenomena  of  this  kind  may  be  called 
phenomena  of  Resonance  ;  the  law  of  which  is,  that  any  undula- 
tion or  vibration  is  taken  up  —  its  energy  is  absorbed  —  by  any 
body  capable  of  freely  vibrating  synchronously  with  it,  free  to  do 
so,  and  exposed  to  its  periodic  impulses.  If  the  undulation  or 
vibration  which  concusses  the  vibratile  body  be  complex-har- 
monic, those  components  only  which  correspond  to  the  natural 
period  of  the  vibratile  body  are  taken  up  by  that  body.  On  this 
principle  resonators  are  used  to  detect  the  higher  components  of 
a  complex  sound. 

Forced  Vibrations.  —  Under  certain  circumstances  a  vibra- 
tile body  may  be  compelled  to  surrender  its  own  preference  for 
a  particular  mode  of  vibration,  and  to  vibrate  with  more  or  less 
accuracy  in  an  arbitrary  manner  imposed  upon  it  by  external 
force. 

Huyghens  discovered  that  two  clocks  which  did  not  keep 
time  separately,  kept  time  together  when  placed  on  the  same 
table ;  the  more  rapid  clock  forced  up  the  speed  of  the  slower 
one,  while  it  was  itself  delayed.  Two  prongs  of  a  tuning-fork, 
slightly  unequal  in  size,  will  force  one  another  to  agree  in  their 
periods  of  vibration.  If  the  one  prong  be  powerfully  pulled  to- 
and-fro  by  an  external  mechanism  so  predominant  that  its  period 
cannot  be  altered  by  any  resistance  offered  by  the  fork,  the  fork 
as  a  whole  wrill  be  forced  to  vibrate  at  a  rate  determined  by  the 
exterior  mechanism  or  outside  force ;  and  it  will  maintain  this 
rate  as  long  as  it  is  compelled  to  do  so,  but  no  longer  ;  this  being 
the  case  of  a  tuning-fork  controlled  by  an  electromagnetic  inter- 
rupter, which  is  found  to  return  to  its  normal  rate  of  vibration 
as  soon  as  the  electric  current  ceases.  The  nearer  the  rate  of 
the  forced  vibration  is  to  the  rate  of  the  free  vibration  of  the 
fork,  the  wider  is  the  oscillation. 

A  resonating  chamber  of  air  will  also,  on  the  same  principle, 
resound  to  some  extent  under  the  influence  of  a  tuning-fork  of 
pitch  slightly  differing  from  its  own  natural  pitch.  Here  we 
observe  that  a  steel  fork  can  force  the  less  massive  air  slightly 
to  alter  the  period  of  its  vibration ;  towards  the  end  of  each 
swing  of  the  particles  of  the  air,  they  are  in  very  slow  motion, 
and  can  with  comparative  ease  be  constrained  to  shorten  or  to 
prolong  the  period  of  their  oscillation.  Air-waves,  on  the  other 
hand,  cannot  force  a  massive  tuning-fork  in  this  ,way,  and  in 
order  that  a  tuning-fork  may  take  up  a  vibration  conveyed  to  it 


448 


ON  SOUND. 


[CHAP. 


through  air,  the  vibration  of  the  primary  sounding-body  must  be 
in  exact  unison  with  the  free  vibration  of  the  tuning-fork  acted 
upon. 

There  is  difficulty  in  constraining  a  denser  substance  to  take  up  a  vibra- 
tion communicated  to  it  by  a  rarer  one.  The  mechanical  action  of  the  ear 
involves,  however,  a  problem  of  this  kind ;  air-waves  have  to  be  communi- 
cated to  the  denser  sense-organs  ;  and  this  is  effected  by  diminishing  their 
amplitude  and  consequently  increasing  the  force  of  their  impulse  —  a  prin- 
ciple now  familiar  to  us. 

Perhaps  the  best  examples  of  bodies  which  can  be  forced  into 
vibrations  of  any  period  are  found  among  membranes ;  to  some 
degree  of  amplitude  any  period  of  vibration  can  be  forced  upon  a 
membrane,  though  all  stretched  membranes  —  as,  e.g.,  the  orches- 
tral drum  —  have  natural  periods  of  free  vibration  which  can  be 
modified  by  varying  the  tension.  If  a  vibratile  body  which  has 
a  natural  period  of  vibration  be  concussed  by  an  external  Fourier- 
motion,  the  resultant  forced-motion  of  the  vibratile  body  will  still 
be  Fourier-motion ;  but  those  components  of  the  original  motion 
which  nearly  coincide  with  the  natural  oscillations  of  the  vibra- 
tile body  will  be  represented  in  the  resultant  forced  complex- 
vibration  by  components  proportionately  exaggerated  —  a  princi- 
ple we  have  already  seen  to  affect  the  working  of  the  Phonograph. 

If  a  membrane  be  of  unequal  width  and  be  stretched  in  one 
direction,  a  forced  vibration  imposed  upon  it  will  affect  only 
certain  parts  of  its  area.  In  Fig.  148,  ABC  is  a  membrane, 

Fig.148. 


triangular  in  form  and  exposed  to  a  tension'  parallel  to  CB. 
Consider  a  single  very  narrow  strip,  «/,  of  this  membrane ; 
imagine  it  to  be  isolated  from  the  rest  of  the  membrane.  Such 
a  strip  would  have  a  certain  length,  thickness,  tension,  density ; 
it  would  therefore,  if  it  entered  alone  into  transversal  vibrations, 


xiv.]  FORCED  VIBRATIONS.  449 

produce  a  note  of  a  certain  definite  pitch.  Let  that  note  be 
sounded  in  the  neighbourhood  of  the  membrane  ;  the  membrane 
will  vibrate  strongly  at  <?/,  the  disturbance  rapidly  shading  off 
into  rest  on  either  side  of  ef ;  and,  further,  there  will  be  some 
disturbance  in  those  parts  of  the  membrane  whose  length  is  2<?/, 
3ef,  and  so  forth.  Each  external  sound  is  responded  to  by  a 
different  part  of  the  membrane,  which  plays  the  part  for  the 
time  being  of  an  imperfectly-isolated  string.  At  right  angles 
to  the  direction  of  tension  there  is  but  little  vibration. 

If  the  external  sound  be  complex,  several  such  strips  are  set 
in  motion. 

Musical  Instruments.  —  For  the  ends  of  musical  art  it  is 
necessary  that  the  vibrating  body  used  as  the  source  of  sound 
should  be  capable,  at  the  will  of  the  performer,  of  producing 
several  sounds.  In  the  old  Russian  horn-bands  each  player  had 
only  one  sound  at  his  disposal,  and  by  dint  of  practice  and  drill 
learned  to  produce  his  solitary  note  at  the  right  instant ;  but  this 
kind  of  orchestral  music  is  quite  exceptional. 

We  have  already  seen  that  all  the  notes  of  the  scale  may  be 
produced  on  the  monochord  by  varying  the  length  of  the  free 
vibrating  part  of  the  string. 

Some  stringed  instruments — the  Lyre,  the  Harp,  the  Violin, 
etc.,  when  played  pizzicato,  the  Banjo,  the  Guitar  —  are  played 
by  plucking  the  strings.  In  some  cases  —  the  Lyre,  the  Harp 

—  the  number  of  sounds  which  can  be  produced  is  limited  by 
the  number  of  strings  present ;  in  others  —  the  Banjo,  the  Guitar 

—  each  string  is  made  to  produce  a  number  of  sounds  which 
depend  upon  the  number  of  frets  by  which  the  finger  of  the  exec- 
utant is  guided  in  shortening   the    string;    in   instruments  of 
the  Violin  class  there  is  no  mechanical  aid  to  the  performer's 
fingers,  and  he  is  left  to  his  own  judgment  as  to  the  precise 
amount  by  which  any  given  string  should  be  shortened  in  order 
that  it  may  emit  the  particular  sound  of  which  he  has  already 
formed  a  mental  idea.     All  these  instruments  are  provided  with 
sounding-boards  which    increase  the  surface  by  which  vibra- 
tions are  communicated  to  the  air ;  and  when  their  strings  are 
plucked,  the  sound  produced  is  of  short  duration  and  rich  in 
high  harmonics,  poor  in  lower  ones. 

The  strings  of  the  Harpsichord  were  plucked  by  quills 
which  were  actuated  by  hammers.  The  sound  was  poor  in 
quality,  being  feeble  in  the  fundamental  tone  £nd  dispropor- 
tionately strong  in  the  higher  harmonics;  and  it  was  feeble 


450  ON  SOUND.  [CHAP. 

in  intensity  as  compared  with  the  pizzicato  notes  of  a  violin, 
because  the  sounding-board  was  wanting  in  flexibility  and  had 
little  effect  on  the  air. 

The  Pianoforte  contains  very  strong  wire  tightly  stressed ; 
the  total  stress  on  a  Broad  wood  grand  pianoforte  exceeds  the 
weight  of  35,000  Ibs.,  that  on  a  Steinway  is  72,000  Ibs. ;  whence 
the  modern  pianoforte  is,  as  regards  its  framework,  necessarily  a 
very  much  more  massive  instrument  than  its  predecessors.  The 
longer  the  string  corresponding  to  a  given  note,  and  the  greater 
the  tension  upon  it,  the  more  precisely  will  the  harmonic  tones 
be  in  tune  with  the  fundamental,  and  the  fuller  and  richer  will 
be  the  sound  produced.  The  sound  of  a  pianoforte  string  struck 
in  the  usual  way  is  rich  in  harmonics  up  to  the  sixth ;  the  seventh 
is  purposely  prevented  by  the  choice  of  the  spot  at  which  the 
hammer  strikes ;  the  eighth  and  those  beyond  are  feebly  repre- 
sented. The  sounds  of  the  higher  strings  approximate  in  char- 
acter to  pure  tones.  The  compass  of  the  modern  pianoforte  is 
from  A,,(=  27-1875)  in  the  thirty-two-feet  octave  to  «""(  =  3480), 
or  even  to  c'""  (  —  4176),  the  sound  of  an  organ-pipe  an  inch-and- 
a-half  long. 

In  a  vibrating  pianoforte-string  those  components  disappear  whose 
periods  are  f ,  f,  -f ,  etc.,  of  the  period  of  contact  between  the  hammer  and 
the  string.  Hence  the  varying  quality  of  tone  obtained  from  the  same 
string  by  hammers  of  different  degrees  of  repair  or  varying  in  the  hard- 
ness or  elastic  softness  of  the  leather,  or  by  striking  the  pianoforte  keys 
in  different  ways.  The  slower  the  stroke,  the  longer  the  contact,  the  greater 
the  disappearance  of  higher  harmonics. 

The  Violin,  whose  strings  are  tuned  tox#,  W,  a',  e",  has  a 
compass  ranging  from  \7  to  about  e"" ;  the  Viola  is  tuned  to  V,  ^, 
V#f,  a' ;  the  Violoncello  is  tuned  to  XC,  VG,  Y?,  a  ;  the  Double-bass 
is  tuned  to  [E//]?  A/r  T)p  XG, ;  these  instruments  give  the  strings 
of  the  orchestra  an  aggregate  compass  of  from  Ey<  or  An  to  about 
e"",  or  about  seven  octaves.  In  the  Violin  three  strings  are  of 
catgut;  their  pitch  depends  upon  their  thickness;  the  fourth 
string  is  weighted  by  a  spiral  of  silver  wire  —  an  arrangement 
which  to  a  great  degree  obviates  rigidity.  The  belly  of  the 
violin  acts  as  a  sounding-board.  The  air  in  the  cavity  aids 
in  the  resonance  and  improves  the  tone,  for  it  is  easily  forced 
to  take  up  the  vibrations  of  the  solid  parts  of  the  instrument, 
especially  those  component  vibrations  which  are  already  the 
most  prominent;  and  this  action  of  the  internal  air  is  impor- 
tant, as  may  be  found  on  covering  the  J  holes  with  tissue  paper, 
for  then  the  tone  of  the  instrument  is  materially  deteriorated. 


xiv.]  MUSICAL   INSTRUMENTS.  451 

The  quality  of  the  tone  is  found  also  to  depend  greatly  upon 
the  empirical  form  of  the  bridge. 

Transverse  vibrations  of  reeds  are  utilised  in  the  Musical 
Box:  reeds  of  various  lengths  are  struck,  vibrate  in  the  free 
air,  and  produce  sound  of  little  intensity.  In  the  Harmonium 
and  Concertina  the  passage  of  air  from  a  bellows  into  a  reso- 
nating air-chamber  is  obstructed  by  reeds,  which  almost  com- 
pletely close  the  air  passage.  The  pressure  accumulates  and 
bends  the  reed ;  the  air  escapes  and  the  pressure  is  relieved ; 
the  reed  returns  and  swings  past  its  mean  position,  again  to 
be  driven  outwards.  The  vibrations  of  such  reeds  contain,  as 
harmonics,  tones  bearing  to  the  fundamental  ratios  incommen- 
surable, but  approximating  to  ^,  -|,  -L6-,  etc.  Such  reeds  vibrating 
in  the  open  air  produce  very  little  sound :  the  air  of  the  reso- 
nating cavity  acts  as  an  intermediary,  and  communicates  the 
vibration  by  a  broader  surface  to  the  external  air. 

In  Mr.  Baillie  Hamilton's  String-organ  very  various  quali- 
ties of  rich  and  full  tone  are  produced  by  connecting  in  various 
ways  a  vibrating  reed  with  a  string  stretched  over  a  sounding- 
board. 

In  reed  organ-pipes  a  reed  vibrates  at  the  bottom  of  a  tube, 
the  cavity  of  which  it  alternately  opens  to  and  closes  against  the 
access  of  air  from  the  wind-chest:  the  air  in  the  cavity  of  the 
organ-pipe  resounds  to  the  vibration  of  the  reed,  and  does  this 
most  loudly  when  its  own  natural  pitch  coincides  with  the  pitch 
of  the  reed.  The  Clarionet,  the  Oboe,  the  Bassoon,  are  instru- 
ments of  this  order;  in  them  as  in  the  organ-pipe  the  reed  must 
be  cut  to  a  proper  size :  but  each  can  produce  several  tones,  for 
the  vibration  of  a  reed  made  of  cane  is  more  irregular  than  that 
of  a  metal  reed,  and  as  the  size  of  the  resonating  air-cavity  can 
be  altered  by  manipulating  the  finger-holes  and  keys,  to  each 
altered  size  of  the  resonating  cavity  there  corresponds  a  differ- 
ent tone  or  component  of  the  motion  of  the  reed,  which  the 
tube  is  capable  of  selecting,  and  to  which  it  resounds. 

In  the  ordinary  Organ-pipe  a  sheet  of  air  is  blown  across 
the  embouchure  of  the  pipe,  and  vibrates  like  an  invisible  reed 
just  outside  the  pipe :  the  air  inside  is  set  in  vibration.  The 
pitch  of  the  sound  produced  by  it  is  not  so  high  as  we  might  be 
led  to  infer  from  the  dimensions  of  the  column  of  air  in  the 
pipe,  its  elasticity,  and  its  density  alone :  the  reason  is  that  the 
column  of  air  in  the  pipe  is  not  an  isolated  mass  of  air.  While 
vibrating  it  has,  at  its  extremity,  to  thrust  aside  the  surrounding 


452  ON   SOUND.  [CHAP. 

air  at  each  expansion ;  the  inertia  of  the  surrounding  air  ham- 
pers the  vibration,  and,  as  it  were,  loads  and  retards  the  vibrating 
column. 

Nodes  in  an  organ-pipe  may  be  demonstrated  by  Koenig's 
manometric  capsules  attached  to  the  sides  of  the  pipe.  Near 
the  centre  of  an  open  pipe  there  is  a  node,  a  maximum  of  varia- 
tion of  density,  and  hence  the  air  at  the  node  is  alternately 
squeezed  together  and  relaxed,  the  membrane  of  the  manomet- 
ric capsule  placed  at  the  node  will  vibrate,  and  the  flame  will 
either  be  extinguished  or  will  present  variations  of  height. 

If  an  organ-pipe  be  strongly  blown,  the  sound  will  rise 
slightly :  if  the  force  of  the  current  exceed  a  certain  amount, 
the  sound  suddenly  breaks  into  the  upper  octave  —  a  phenome- 
non familiar  to  flute-players  —  and  near  the  centre  of  the  pipe 
is  now  a  loop,  while  there  are  now  two  nodes.  Hence,  in  a  pipe 
so  overblown,  the  manometric  capsule  fixed  at  a  point  near  the 
centre  almost  ceases  to  indicate  variations  of  density,  and  the 
flame  remains  almost  steady  :  never  absolutely  so  in  practice, 
for  there  is  not  even  any  one  point,  at  the  centre  of  any  loop,  at 
which  all  variations  of  pressure  entirely  vanish. 

If  a  pipe  be  filled  with  hydrogen  the  sound  produced  is 
nearly  two  octaves  above  that  produced  by  air  at  the  same 
temperature.  The  velocity  of  the  sound-waves  is  ~\//c/e  •  k//o 
=  ^/k/c  •  p/p  (p.  368) ;  in  air  at  atmospheric  pressure  n,  this  is 
V(l-4058n//t?air);  in  hydrogen,  it  is  V(l-4139n/phyd),  for  in 
that  gas  &/<?  =  1-4139,  the  specific  heats  being  3409  and  2-411; 

in  the  case  of  hydrogen  p  =  T^T7x  tne  density  of  air;  whence 

the  frequency  of  vibration  in  hydrogen  is  to  that  in  air  as 
1 :  VT4-44  x  (1-4139  -r-  14058)  =  1 :  3-8107  or  : :  C  :  'b  +. 

If  the  barometric  pressure  vary,  the  pitch  is  unaffected : 
for  Vk/P  =  "VjD/p ;  but  the  density  p  increases  or  diminishes 
pari  passu  with  the  pressure  p  :  thus  p/p  is  constant,  the  velocity 
is  constant,  and  the  pitch  is  constant. 

If  the  temperature  r  vary,  the  velocity  v,  =  -V/c/c  -  k/p 
=  ^/k/c  -  p/p  =  -\/k/c  •  i&r,  and  therefore  also  the  pitch,  varies  as 
the  square  root  of  the  Absolute  temperature.  A  pipe  which 
gives  a  sound  a'  =  435  at  the  temperature  of  10°  C.  will  vibrate 
446-79  times  per  second  at  the  temperature  of  25°  C.,  for  as 
V283  :  V298  : :  435  :  446-79 ;  the  pitch  is  thus  sharpened  in  the 
ratio  1 : 1-027105,  or  more  than  two  commas ;  while  the  reeds 
are  affected  to  a  very  much  less  degree,  and  thus  an  organ  tuned 


xiv.]  ORGAN-PIPES.  453 

in  a  cool  room  becomes  discordantly  out  of   tune  with  itself 
when  the  room  becomes  warm. 

Problem.  —  Of  what  length  would  an  organ-pipe  be  which  sounds  C, 
at  0°  C.  and  760  mm.  bar.  pressure  ?  C,  =  32-625  vibrations.  — A  ns.  Since  the 
frequency  of  vibration  is  n  =  v/\  =  ^k/c  •  k/p  H-  2  /  =  ^/k/c-  n/p  •+  2 1,  the 
length  I  =  v'k/c  •  u/p  -=-  2  n  =  Vl-41  x  1,01.3663-0-00129.32  --  65-25  =  508-7  cm. 
A  pipe  of  this  length  really  gives  a  sound  somewhat  lower  than  C, ;  the 
wider  the  pipe  the  lower  the  pitch.  A  Crpipe  is  called  a  16-foot  pipe, 
though  really  somewhat  longer. 

The  common  Flute,  the  Fife,  the  Piccolo,  the  Flageolet,  are 
modifications  of  the  open  organ-pipe :  the  size  of  the  vibrating 
column  of  air  is  altered  in  these  instruments  by  opening  or 
closing  lateral  apertures.  In  the  Flageolet  the  mouthpiece  is 
so  constructed  that  the  stream  of  air  cannot  but  pass  in  the 
direction  empirically  found  to  produce  the  best  sound:  in  the 
Flute,  Fife,  and  Piccolo  the  direction  of  the  exciting  stream  of 
air,  and  to  some  extent  the  corresponding  quality  of  tone,  and 
even  the  pitch,  are  under  the  control  of  the  player. 

Brass  instruments  vary  in  shape  from  the  Bugle  (conical 
from  mouthpiece  to  bell)  to  the  Trumpet  (slowly- widening  tube 
and  suddenly-widening  bell),  with  the  intermediate  forms,  the 
Cornet  and  the  French  Horn.  The  lips  of-  the  player  are  made 
to  vibrate  :  the  cavity  of  the  instrument  resounds.  The  scale 
of  the  instrument  is  confined  to  harmonics,  which  are  obtained 
by  varying  the  method  and  force  of  blowing  and  the  tension  of 
the  lips ;  but  since  the  size  of  the  cavity  of  the  instrument  may 
be  modified,  any  tone  of  the  scale  within  the  compass  of  the 
instrument  may  be  brought  in  as  a  harmonic  of  some  funda- 
mental note.  The  size  of  the  cavity  may  be  modified  by  slides, 
as  in  Trombones  —  instruments  which  are  capable  of  giving  pure 
intonation ;  by  pistons  which  lengthen  the  tube  by  predetermined 
amounts,  as  in  the  Cornet  and  Cornopean,  the  Euphonium,  and 
Bombardon ;  by  levers  which  shorten  the  tube,  as  in  the  Ophi- 
cleide.  In  some  cases,  however,  as  in  the  French  Horn  and 
Trumpet,  the  instrument  is  confined  to  one  note  and  its  har- 
monics ;  but  its  cavity  may  be  altered  in  size  by  the  addition  of 
additional  pieces  of  tubing  or  "  crooks,"  whose  dimensions  are 
so  adjusted  as  to  cause  the  fundamental  tone  of  the  instrument 
to  change  by  an  interval  predetermined  for  each  crook ;  and  the 
note  produced  may  be  to  some  extent  modified  both  in  pitch 
and  in  quality  by  the  action  of  the  lips,  and  of  the*  right  hand 
placed  in  the  bell  or  the  open  mouth  of  the  instrument. 


454  ON   SOUND.  [CHAP. 

The  pitch  of  these  instruments  is  that  of  an  open  pipe. 
Their  tone  is  rich  in  quality  and  full  of  high  harmonics ;  and 
though  we  are  inclined  to  associate  their  peculiar  quality  with 
metal,  it  depends  only  on  the  form  of  the  internal  cavity  and 
of  the  bell  mouth,  not  on  the  material  of  the  walls  of  the  instru- 
ment, provided  that  these  be  rigid.  A  cornet  sound  will  be 
produced  by  a  cornet-shaped  tube  of  guttapercha,  if  the  tube  be 
so  thick  as  to  be  rigid. 

Stopped  Organ-Pipes.  —  A  longitudinal  column  of  air 
steadied  at  one  end  should  vibrate  at  one  half  the  rate  of  a  col- 
umn free  to  expand  at  both  ends;  but  a  stopped  organ-pipe  does 
not  give,  as  this  rule  would  indicate,  the  octave  of  the  open  pipe 
of  the  same  length,  but  its  seventh,  or  even  its  minor  seventh. 
The  air  within  it  is  even  more  loaded  and  hampered  by  the  sur- 
rounding air  than  that  in  an  open  pipe.  When  overblown  it 
breaks  into  harmonics.  (See  Fig.  73.) 

Other  Sources  of  Sound  —  Singing  and  Sensitive  Flames. 
—  A  tube  terminating  in  a  blowpipe-nozzle  is  connected  with 
the  gas  supply :  the  gas  is  ignited  at  the  nozzle,  and  a  small 
flame  is  thus  produced.  The  nozzle  is  slowly  passed  up  a  wide 
glass  tube,  such  as  an  Argand  lamp-cylinder.  At  some  partic- 
ular position  in  the  tube,  the  flame  alters  its  character  and  begins 
to  sound  forth  a  note  somewhat  higher  than  the  natural  note 
(in  cool  air)  of  the  tube  itself ;  or,  should  it  not  spontaneously 
burst  into  song,  it  may  be  induced  to  do  so  by  singing  to  it  a 
note  whose  pitch  is  nearly  that  which  it  will  emit  when  in 
action.  When  the  action  has  commenced,  the  flame  is  found, 
upon  observation  with  a  rotating  mirror,  to  be  alternately  ex- 
tinguished and  rekindled ;  its  image  appears  to  be  broken  up 
into  separate  beads.  A  flame  of  hydrogen  is  prompter  in  its 
action  than  a  flame  of  coal  gas  ;  for  it  excites  more  disturbance 
in  the  column  of  cool  air  surrounding  it.  When  the  action  has 
commenced,  the  vibrations  of  the  flame  and  of  the  column  of 
air  react  on  one  another. 

The  flame  must  be  of  a  certain  height  adapted  to  each  size 
of  tube.  If  the  apparatus  be  arranged  so  that  the  flame  is  put 
about  one-quarter  up  the  tube,  and  if  the  height  of  the  flame  be 
carefully  altered  by  slowly  turning  the  controlling  gas-tap,  it 
will  be  found  that  at  a  certain  definite  height  of  flame  the  action 
will  spontaneously  commence,  doing  so  with  small  initial  inten- 
sity, while,  when  the  gas-flame  is  lowered  to  one-half  of  this 
effective  height,  the  sound  breaks  into  the  octave  above. 


xiv.]  SINGING  FLAMES.  455 

When  a  jet  of  gas  is  allowed  to  flow  vertically  upwards 
under  a  pressure  of  about  f  inch  of  water,  through  a  minute 
aperture,  above  which  a  sheet  of  fine  wire-gauze  is  arranged 
horizontally  at  a  distance  of  about  2  inches,  the  gas  may  be  lit 
on  the  upper  side  of  the  wire-gauze,  through  which  the  flame 
will  not  descend.  The  distance  of  the  jet  from  the  gauze  may 
be  so  adjusted  that  the  flame  is  yellow  at  the  tip,  and  at  the  tip 
only.  The  flame  is  now  a  sensitive  flame,  and  responds  by 
sinking  down  to  the  gauze  when  sound-waves  strike  it.  If  it 
be  surrounded  by  a  wide  tube,  it  will  sing  spontaneously.  If, 
while  the  flame  is  so  surrounded,  the  gas  be  turned  down  until 
the  singing  just  ceases,  the  flame  becomes  extraordinarily  sensi- 
tive to  all  very  high  sounds,  such  as  hissing,  the  rattling  of 
keys,  etc.,  and  it  sings  loudly  as  long  as  these  stimuli  are  main- 
tained. A  simple  flame  issuing  from  an  exceedingly  narrow 
steatite  burner,  under  a  very  great  pressure  of  gas  ("  10  inches 
of  water  "),  is  sensitive  to  sounds  too  acute  to  be  perceptible  to 
the  human  ear ;  it  alters  its  form  under  their  influence. 

Trevelyan's  Rocker.  —  A  mass  of  lead  so  shaped  as  to  rest 
upon  two  long  but  very  narrow  linear  feet.  Placed  upon  a  hot 
body,  the  points  of  contact  of  the  Tocker  with  the  hot  body  are 
suddenly  expanded  by  heat ;  the  rocker  is  jerked  upwards ; 
before  it  has  fallen  back,  the  heated  points  have  cooled  and 
returned  to  their  normal  dimensions.  The  process  is  repeated. 
The  whole  oscillates  rapidly  and  makes  a  humming  noise. 

Radiophony.  —  When  a  beam  of  light  or  radiant  heat  falls 
upon  a  body  capable  of  absorbing  heat,  that  body  becomes 
warmed,  and  expands.  A  flash  of  light  produces  an  instanta- 
neous expansion,  which  immediately  dies  away.  An  intermit- 
tent beam  produces  a  succession  of  expansions  and  contractions ; 
in  other  words,  the  surface  of  the  body  vibrates.  The  ampli- 
tude of  its  movement  may,  with  beams  of  light  of  moderate 
intensity,  exceed  the  ten-millionth  part  of  a  centimetre.  Lord 
Rayleigh  has  shown  that  this  amplitude  is  sufficient  for  the 
production  of  sound ;  and  the  power  of  converting  the  energy 
of  an  intermittent  beam  of  light  or  radiant  heat  into  that  of 
sound  has  been  shown  by  Prof.  Graham  Bell  to  belong  to  all 
matter,  with  a  few  doubtful  exceptions. 

If  an  intermittent  beam  be  focussed  upon  a  mass  of  lamp- 
black, at  each  flash  of  light  it  becomes  warm,  and  the  air  within 
it  is  dilated ;  if  it  be  contained  in  a  test  tube  the  open  end  of 
which  is  connected  with  the  ear  by  an  indiarubber  tube,  as  the 


456  ON   SOUND.  [CHAP. 

successive  flashes  produce  successive  dilatations  and  pulses  in 
the  air,  these  pulses  are  perceived  by  the  ear  as  sound ;  if  the 
lampblack  be  contained  within  a  resonator,  the  frequency  of 
whose  natural  vibration  is  equal  to  that  of  the  frequency  of  suc- 
cession of  the  flashes,  the  resonator  emits  a  loud  sound,  audible 
at  a  distance. 

PKOPAGATION  OF  SOUND. 

Propagation  of  Sound  occurs  in  all  elastic  media,  and  is 
effected  by  waves  of  alternate  Compression  and  Rarefaction. 

Propagation  of  Sound  in  Solids.  —  Sound  being  a 
vibration  is  propagated  along  the  ground ;  we  may  put  our 
ears  to  the  ground  to  listen  for  distant  railway  trains,  distant 
vehicles,  distant  firing  or  marching.  It  is  conducted  along 
wood,  as  in  the  ordinary  stethoscope ;  or  in  the  experiment  of 
closing  the  teeth  upon  a  long  piece  of  wood,  to  the  other  end 
of  which  an  assistant  holds  a  vibrating  tuning-fork ;  or  of  caus- 
ing a  long  strip  of  wood  to  rest  by  one  end  against  the  panel 
of  a  door,  while  the  other  end  is  in  contact  with  a  vibrating 
tuning-fork ;  the  vibration  is,  in  this  case,  communicated  to  the 
panel,  which  acts  as  a  sounding-board,  and  itself  sounds  out 
loudly.  Sound  is  conducted  by  wires ;  taps  on  a  telegraph 
wire  are  audible  at  a  great  distance  to  an  ear  applied  to  the 
wire,  provided  that  there  be  no  intervening  tunnel  or  bridge 
to  form  a  resonating  cavity  and  to  absorb  the  energy  of  the 
vibration.  In  the  Wire  Telephone  the  central  points  of  two 
stretched  membranes  or  boards  of  thin  wood  are  connected  by 
a  long  wire,  which,  if  sufficiently  heavy  and  tense,  may  be  sus- 
pended from  posts,  or  even  stretched  upon  carpet  or  bent  round 
corners,  and  may  thus  serve  the  purposes  of  domestic  telegraphy. 
When  sound-waves  impinge  upon  one  membrane  of  the  wire 
telephone,  as  when  it  is  directly  spoken  at,  the  complex  motion 
of  the  air  is  transferred  to  the  membrane ;  by  the  membrane  it 
is  transferred  to  the  wire ;  by  the  wire  to  the  second  mem- 
brane ;  by  that  membrane  to  the  air,  and  by  this  to  the  ear  of 
the  distant  listener.  If  the  membranes  be  of  silk,  and  connected 
with  the  wire  by  synchronised  spiral  springs,  the  arrangement 
is  very  effective,  and  sound  is  carried  to  very  great  distances, 
even  under  unpromising  circumstances  (Mechanical  Pulsion 
Telephone).  A  slender  apparatus,  consisting  of  two  parch- 
ment membranes  stretched  on  rings,  with  an  intervening  silk 
thread,  is  sold  as  a  toy  under  the  name  of  the  Lover's  Telegraph. 


xiv.]  PROPAGATION  OF  SOUND.  457 

Propagation  in  Liquids.  —  Divers  while  under  water  hear 
the  sound  of  waves  beating  against  the  shore.  A  tumbler  of 
water  standing  on  a  resonance  box  will,  if  the  handle  of  a  vibrat- 
ing tuning-fork  be  dipped  in  the  water,  convey  the  vibration 
to  the  resonance  box,  the  air  in  which  will  resound.  An  inverted 
bell  filled  with  water  and  set  in  vibration  will  cause  the  water 
to  assume  beautiful  wave-forms  —  an  experiment  which  may  be 
performed  with  an  inverted  propagating-glass  set  in  vibration 
by  a  wetted  finger  drawn  round  its  edge,  or  with  a  capacious 
wine-glass  set  in  vibration  by  a  violin  bow.  In  the  latter  case 
the  interest  of  the  experiment  is  increased  by  substituting  for 
water  some  strongly-alcoholic  liquid ;  the  agitation  breaks  the 
surface  of  the  liquid  into  drops  which,  by  evaporation,  lose  some 
alcohol  and  dance  011  the  surface  of  the  vibrating  liquid. 

An  organ-pipe  may  be  blown  by  water  under  water;  the 
water  vibrates  in  the  place  of  air. 

Propagation  in  Air  and  other  Gases.  —  In  general, 
sound  travels  in  air  in  concentric  spherical  waves.  If  it  be 
restricted  to  tubes,  the  waves  may  become  plane-fronted. 

Though  these  waves  are  invisible  their  existence  is  beyond 
doubt,  for  when  they  strike  any  solid  object  they  produce 
mechanical  effects,  and  the  phenomena  of  sound,  so  far  as  these 
depend  upon  propagation  through  the  air,  obey  the  laws  of 
wave-motion. 

The  breaking  of  glass  windows  by  the  discharge  of  artillery, 
the  destruction  of  the  drum  of  the  ear  which  has  been  known 
to  be  caused  by  the  explosion  of  dynamite,  the  destruction  of 
property  by  the  explosion  of  a  gunpowder  magazine,  are  only 
exaggerated  instances  of  that  conveyance  of  energy  by  the  air 
which  is  associated  with  the  production  of  sound.  In  Edison's 
Phonomotor  a  membrane  is  stretched  over  a  frame ;  at  its  pos- 
terior aspect  it  is  connected  with  a  broad  hook  which  rests  on 
the  broad  margin  of  a  heavy  wheel :  the  margin  of  this  wheel  is 
provided  with  small  roughnesses  so  shaped  that  it  is  easy  for  the 
broad  hook  to  slip  over  them  in  one  direction,  but  in  one  direc- 
tion only ;  on  its  return  it  is  caught  in  the  small  teeth  and  tends 
to  pull  the  wheel  round.  The  membrane  is  spoken  at ;  it  trem- 
bles ;  the  hook  catches  some  of  the  teeth ;  on  its  return  it  gives 
an  impulse  to  the  wheel ;  continuous  sound  causes  the  wheel  to 
rotate,  and  this  with  considerable  speed  and  power;  and  if  a 
little  crank  be  fixed  to  the  rotating  wheel,  the  energy  of  the 
human  voice  may  be  made  to  perform  obvious  mechanical  work. 


458  ON  SOUND.  [CHAP. 

Sound-waves  in  air  are  amenable  to  the  laws  of  ordinary 
tridimensional  Wave-motion  already  discussed. 

In  Figs.  43-48  we  find  curves  whose  forms  depend  upon  the 
assumption  that  when  two  vibrations  concur,  the  amplitude  of 
the  resultant  is  obtained  by  addition  of  the  amplitudes  of  the 
components.  To  a  first  approximation  this  is  true,  but  it  leads 
to  a  curious  result.  If  the  amplitudes  and  periods  of  two  vibra- 
tions be  equal,  the  resultant  vibration  (Fig.  44),  having  twice 
the  amplitude,  will  have  four  times  the  energy  of  either ;  the 
motions  cannot  be  so  superposed  without  a  draft  of  energ}^  from 
elsewhere.  If  two  equal  waves  arrive  in  the  same  phase  at  the 
entrance  of  the  same  channel,  there  is  found  to  be  reflexion  of  a 
negative  wave  from  the  mouth  of  that  channel. 

Superposition  of  vibrations  is  familiar  in  Acoustics  as  a  cause 
of  Beats.  Two  tuning-forks  or  reeds  are  brought  into  exact 
unison ;  they  emit  jointly  a  smooth  sound.  Suppose  their  pitch 
to  be  c"  =  522.  Load  one  with  wax  until  it  vibrates,  say,  521 
times  per  second.  Once  in  the  course  of  a  second  they  will 
aid  one  another,  and  their  action  on  the  air  at  any  given  spot 
coincides ;  once  in  the  course  of  every  second  they  will  thwart 
one  another ;  their  joint  effect  will  be  an  alternate  fading-away 
and  swelling-out  of  the  sound ;  but  the  pitch  of  that  sound  will 
be  52 1|-  vibrations  per  second.  As  the  loading  of  the  one  fork 
increases,  the  beats  increase  in  number  until  they  become  too 
rapid  to  be  counted ;  but  before  this  occurs  the  two  notes  have 
ceased  to  be  blended  into  one  note  of  average  pitch,  and  the 
effect  is  that  of  n  painful  discord. 

As  differences  of  phase  accumulate,  the  wave-form  goes  through  periodic 
changes :  and  Lord  Kelvin  is  of  opinion  that  the  ear  can  distinguish 
differences  in  the  wave-form  due  to  differences  of  phase,  for  between  any 
two  beats  the  sound  heard  has  a  certain  rotatory  character. 

The  easiest  as  well  as  the  most  accurate  way  of  tuning  a 
fork  to  a  given  note  is  to  have  at  hand  a  standard  fork  which 
makes  four  vibrations  less  per  second  than  correspond  to  that 
note ;  then  adjust  the  fork  to  be  tuned  until  it  makes  exactly 
four  beats  per  second  with  that  artificial  standard. 

Diffraction.  —  Sound-waves  have  in  general  the  properties 
of  waves  whose  wave-length  is  not  small  in  comparison  with 
the  apertures  through  wrhich  they  pass,  the  surfaces  by  which 
they  are  reflected,  or  the  obstacles  round  which  they  flow.  A 
sound-wave  coming  through  a  chink  would  suffer  great  lateral 
diffraction,  as  shown  in  Fig.  56 ;  and  the  effects  of  obstacles 


xiv.]  DIFFRACTION.  459 

intervening  in  the  path  of  a  wave  of  sound  may  not  be  such  as 
even  to  produce  a  sound-shadow.  If,  however,  the  apertures,  or 
surfaces,  or  obstacles  in  question  be  very  large  in  comparison 
with  the  wave-length,  there  may  be  true  sound-shadows  and  lim- 
ited beams  of  acoustic  disturbance,  reminding  us  of  the  shadows 
and  beams  of  light  met  with  in  Optics.  Sensitive  flames  may 
be  used  to  detect  such  sound-shadows ;  and  the  optical  effects 
of  Diffraction  may  be  imitated  acoustically.  The  air-waves  pro- 
duced by  the  note  C  (  =  64)  have  a  length  of  (33,200  -=-  64  =  ) 
518-75  cm.,  or  between  16  and  17  feet ;  those  produced  by  the 
note  c"'"  have  a  length  of  about  an  inch  and  a  half.  When  a 
mixture  of  long  waves  and  ripples  strikes  an  obstacle,  the  long 
waves  may  pass  round  it,  while  the  obstacle  may  intercept  the 
ripples,  for  relatively  to  these  it  may  be  wide  enough  to  cast 
a  sound-shadow.  Thus  the  sound  of  a  brass  band  suddenly 
changes  in  quality  when  the  band  comes  round  a  corner  into 
sight.  It  is  plain  that  for  acoustic  purposes  all  the  auditors  at  a 
concert,  though  they  are  not  absolutely  prevented  from  hearing 
by  not  seeing  the  performers,  ought  to  be  in  full  view  of  the 
orchestra. 

It  sometimes  happens  that  a  person  at  the  same  level  as  a 
source  of  sound,  and  in  full  optical  view  of  it,  hears  nothing  :  the 
sound-waves,  passing  through  air-strata  of  different  density, 
curve  upwards.  Being  very  broad  they  present  no  diffraction, 
and  pass  over  the  head  of  the  observer,  who  may  again  come 
within  their  range  by  elevating  his  position.  This  occurs  when 
the  upper  strata  are  cooler;  when  they  are  warmer,  the  sound 
descends. 

Reflexion  of  Sound  follows  the  ordinary  laws  of  the  reflexion 
of  waves.  Sound  can  be  reflected  by  a  mirror ;  a  high-pitched 
bell  ringing  round  the  corner  of  a  house  can  be  rendered  audible 
by  a  sufficiently-large  mirror  placed  at  a  proper  angle.  If  a 
stretched  sheet  of  tracing-paper  be  placed  at  the  same  angle  it 
will  reflect  a  proportion  of  the  sound  and  transmit  some  of  it.  A 
dry  handkerchief  will. transmit  a  considerable  amount  of  sound; 
so  will  even  a  half-inch  layer  of  felt ;  but  if  wetted  these  become 
better  reflectors,  while  they  become  almost  impervious  to  sound. 
The  reflexive  power  of  flame  is  nearly  the  same  as  that  of  tracing- 
paper  ;  and  hot  air  above  a  gas-flame  can  reflect  sound  almost  as 
well  as  the  gas-flame  itself.  In  clear  weather  the  air  is  rarely 
uniform ;  there  are  ascending  and  descending  currents  of  hotter 
and  colder  air ;  at  each  surface  of  each  of  these,  sound  is  partly 


460  ON   SOUND.  [CHAP. 

reflected,  partly  transmitted,  and  so,  ere  long,  it  is  wholly  dissi- 
pated. Sound  is  often  heard  better  in  foggy  or  even  in  rainy  or 
snowy  weather  than  in  clear,  for  then  the  air  is  more  uniform. 
Sound  coining  through  a  fog  (vesicles  or  minute  drops),  or  a 
shower  of  rain,  or  a  shower  of  snow,  must  be  to  a  certain  extent 
lost  by  repeated  reflexion ;  but  this  effect  is  often  balanced  by 
the  increase  of  intensity  arising  from  the  concentration  of  the 
waves  in  the  narrowed  channels  between  the  drops  or  flakes. 

In  many  buildings  there  are  whispering  galleries  or  places 
where  a  faint  whisper  uttered  at  a  particular  spot  is  heard  at  a 
distant  part  of  the  edifice.  This  phenomenon  may  arise  in  two 
ways :  —  (1)  Reflexion  from  the  vaulted  roof,  which  acts  like  a 
concave  mirror  and  causes  the  waves  received  by  it  to  converge 
after  reflexion  upon  a  particular  focus  —  a  phenomenon  very 
common  in  ellipsoidal  roofs,  a  whisper  uttered  at  one  focus  of 
the  ellipsoid  being  reflected  to  the  other  focus,  and  distinctly 
heard  there ;  a  similar  phenomenon  also  occurs  in  elliptical 
rooms,  where  the  sound  is  reflected  by  the  walls;  or  (2)  by 
the  sound  undergoing  successive  reflexions,  and  thus  travelling 
round  the  walls. 

Reflexion  of  sound  is  familiarly  illustrated  by  the  Echo. 
Sound  striking  a  broad  cliff  or  wall  is  reflected,  the  reflected 
waves  sometimes  travelling  with  singular  absence  of  diffraction 
and  precision  of  direction.  If  a  person  can  utter  ten  syllables 
per  second,  and  if  he  speak  loudly  at  that  rate  in  presence  of  a 
cliff  or  high  wall  directly  opposite  him  and  at  a  distance  of  1660 
cm.  (55  feet),  just  as  he  is  commencing  the  second  syllable  the 
reflected  sound  of  the  first  syllable  begins  to  arrive  at  his  ear, 
and  at  the  instant  when  he  ceases  to  speak,  the  sound  of  the  last 
syllable  spoken  begins  to  be  heard ;  the  sound  travels  to  the  cliff 
and  back  during  the  tenth  part  of  a  second.  If  the  cliff  were 
3320  cm.  distant,  the  speaker  would  hear  two  syllables  repeated. 
When  sound  is  re-echoed  from  cliff  to  cliff,  or  to-and-fro  between 
two  smooth  walls  directly  opposite  to  one  another,  the  result 
may  be  a  multiple  echo,  which  repeats  a  sound  several  times. 
The  roll  of  thunder  is  partly  due  to  multiple  reflexion  from 
cloud  to  cloud,  partly  to  the  varying  distance  of  the  points  of 
disturbance,  and  the  successive  breaking  on  the  ear  of  sound- 
waves produced  along  a  long  line. 

When  the  distance  of  the  reflecting  surface  from  the  source 
of  sound  is  too  small  to  produce  a  distinct  and  separate  echo, 
the  echo  may  be  heard  merely  as  a  reinforcement  of  the  sound 


XIV-J  REFLEXION.  461 

produced;  whence  the  practice  of  placing  sounding-boards 
behind  and  above  pulpits  and  orchestras. 

The  action  of  the  Ear-trumpet  depends,  in  the  first  place,  upon 
multiple  reflexion;  sound-waves  on  their  arrival  are  reflected  by 
the  bell  into  the  tube ;  then  they  travel,  plane-fronted  or  nearly 
so,  down  the  tube  to  the  ear,  narrowing  in  breadth  and  increas- 
ing in  intensity  as  they  do  so. 

Refraction  of  Sound.  —  If  a  lens  be  constructed  of  two 
large  sheets  of  collodion  cemented  together  at  their  edges  and 
inflated  with  carbonic  acid,  sound-waves  diverging  from  a  watch 
placed  at  one  side  of  this  lens  may,  after  passing  through  it, 
converge  upon  a  focus  on  the  other  side  of  it;  this  shows  that 
refraction  occurs  when  the  sound-waves  enter  and  quit  the 
denser  gaseous  medium,  the  carbonic  acid. 

Interference  of  Sound.  —  If  A  and  B  in  Fig.  75  represent 
the  position  of  two  tuning-forks  kept  accurately  in  unison,  being 
driven  by  the  same  interrupted  electric-current,  the  ear,  placed 
successively  at  a',  6',  <?',  etc.,  perceives  alternate  sound  and  silence. 
The  same  occurs  when  A  and  B  in  that  figure  represent  aper- 
tures in  the  side  of  a  padded  box  within  which  an  organ-pipe  or 
bell  is  caused  to  produce  sound. 

Velocity  of  Sound.  —  There  is  but  little  information  as  to 
the  properties  of  sound-waves  travelling  in  a  substance  whose 
elasticity  is  not  the  same  in  all  directions. 

In  Scotch  fir  the  longitudinal  propagation  is  more  rapid  than  that 
across  the  fibres  of  the  wood  in  the  ratio  of  5:4;  hence  sound-waves  in 
that  wood  must  be  spheroidal.  This  is  ascertained  by  observing  the  form 
of  the  nodal  lines  in  a  vibrating  plate  of  that  wood. 

In  practice  we  meet  with  spherical  waves  such  as  those  in 
the  air,  and  plane-fronted  waves  such  as  those  which  run  along 
wires. 

The  velocity  v  =  V(ft  +  |tt)/p  in  tridimensional  solids,  Vft/p  in  non- 
viscous  liquids,  Vk/c  •  fc/p  in  gases  ;  Vg/p  along  a  rod  or  wire  when  the 
wave  is  plane-fronted,  and  Vt/p  along  a  wire  when  the  displacement  of  the 
wire  is  transversal. 

The  velocity  of  sound  in  Solids  may  be  found  by  determin- 
ing the  pitch  of  longitudinal  vibrations  set  up  in  long  thin  rods. 
The  length  of  the  rod  is  half  a  wave-length,  \/2;  the  pitch 
gives  n,  the  number  of  vibrations  per  second ;  the  equation 
v  =  n\  gives  v,  the  velocity  of  sound,  in  cm.  per  second. 

The  velocity  of  sound  in  Liquids  may  be  determined  by 
direct  experiment,  as  by  sounding  a  bell  under  water,  and 


462  ON  SOUND.  [CHAP. 

by  listening  at  a  distant  station  for  the  arrival  of  the  sound,  the 
precise  instant  of  the  production  of  which  is  signalled ;  or  by 
comparing  the  pitch  of  organ-pipes  blown  in  different  liquids. 

The  formula  v  =  Vft/p,  for  a  non-viscous  liquid,  would  give  for  water 
the  velocity  of  143,900  cm.  per  second,  for  fe  =  (compressibility)-1  =  2-07  x  10W 
and  p  —  1 ;  the  observed  value  is  148,900  cm. 

The  velocity  of  sound  in  Gases  —  as,  for  example,  air  —  has 
been  directly  determined  by  firing  cannon  at  a  known  distance, 
and  by  observing  the  interval  of  time  which  elapses  between 
seeing  the  flash  and  hearing  the  report.  The  objections  to  this 
method  are,  that  such  violent  concussions  as  those  of  cannon  pro- 
duce aerial  vibrations  which  can  be  shown  to  travel  faster  than 
disturbances  of  less  intensity,  and  that  the  velocity  is  not  equal 
in  all  directions  round  the  cannon  or  at  all  distances  from  it. 

The  length  of  an  organ-pipe  producing  a  note  of  a  given 
pitch  has  been  taken  as  a  means  of  measurement.  The  column 
of  air  in  the  pipe,  if  it  vibrated  alone,  would  be  half  a  wave- 
length in  length ;  but  the  air  is  not  isolated  ;  the  sound  actually 
produced  is  graver  than  that  corresponding  to  an  isolated 
column  of  air,  and  the  velocity  so  measured  is  not  even  approxi- 
mately correct. 

On  similar  principles  the  length  of  a  pipe  closed  at  one  end 
and  subjected  at  the  other  to  the  aerial  impulses  derived  from  a 
vibrating  tuning-fork —  this  length  being  so  adjusted  by  a  mov- 
able piston  that  the  air  in  the  tube  resounds  its  loudest  —  is 
taken  as  X/4,  and  the  velocity  of  sound  is  again  obtained  — 
only  approximately,  however  (Fig.  149). 


Fig.149. 


The  velocity  of  sound  in  air  may  also  be  determined  by 
methods  based  on  interference.  At  A  (Fig.  150)  sound  enters; 
the  waves  pass  along  the  two  channels  B  and  C  to  E.  B  can  be 
lengthened  ;  it  is  lengthened  until  no  sound  is  heard  at  E  ;  B  is 
now  half  a  wave-length  longer  than  C.  At  E  may  be  placed  the 
ear,  a  resonator,  a  manometric  capsule,  or  any  other  indicator  of 
sound-waves. 


xiv.]  VELOCITY   OF  PROPAGATION.  463 

No  energy  passes  down  DE  when  no  sound  is  heard  at  E  :  the  waves  are 
reflected  at  D  and  pass  back  to  A,  the  B-waves  returning  along  C,  the 
C-waves  along  B ;  they  arrive  at  A  in  the  same  phase.  Even  if  they  arrive 
at  D  in  the  same  phase  and  pass  on  to  E,  there  must  be  reflexion  of  a  nega- 
tive wave  from  D  along  B  and  C. 

A  glass  tube  vibrating  longitudinally  will  emit  a  sound ; 
the  length  of  the  glass  is  half  a  wave-length.  If  the  tube  con- 
tain air,  the  air  will  be  set  in  vibration ;  but  it  must  divide 
itself  into  segments,  each  half  as  long  as  an  air-wave  correspond- 
ing to  the  same  tone.  Finely-powdered  silica  placed  in  the  tube 
will  be  distributed  by  the  vibrating  air  in  such  a  way  as  to 
accumulate  at  the  nodes  of  the  aerial  vibrations.  The  number 
of  air-segments  thus  indicated  shows  the  comparative  speed  of 
waves  in  air  and  in  glass.  This  is  Kundt's  method. 

The  theoretical  value  for  the  velocity  in  air  is  found  from 
the  equation  v  =  ^Jk/c  -  fe/p,  where  k/c  can  be  found  by  thermo- 
dynamic  experiments,  fc(  =  p)  and  p  by  direct  observation. 

The  best  average  value  for  the  velocity  of  sound  in  air  seems 
to  be  about  33,200  cm.  per  second;  the  extremes  being  330-6 
(330-7  Regnault)  and  333-7  metres. 

The  velocity  of  sound  in  air  is  unaffected  by  variations  of 
pressure,  as  we  have  already  seen  ;  it  varies  as  the  square  root  of 
the  absolute  temperature;  it  is  affected  by  humidity,  for  damp 
air  is  lighter  than  dry  air  under  the  same  pressure.  It  is  also 
affected  by  wind,  which  not  only  retards  the  passage  of  sound 
to  windward,  but  may  also  distort  the  waves  and  cause  them  to 
pass  upwards.  In  general,  the  greater  the  intensity  of  sound 
the  greater  its  velocity :  sound  therefore  continuously  slackens 
in  speed  of  transmission  as  it  becomes  fainter. 

It  is  still  a  moot  point  whether  acute  sounds  are  more  or 
less  rapidly  conveyed  through  air  than  grave  sounds :  they  are 
certainly  not  conveyed  so  far,  for  they  are  more  affected  by  the 
viscosity  of  the  air,  and  thus  the  higher  components  of  an  inhar- 
monious sound  are  abolished  by  distance,  and  the  whole  effect 
is  softened. 

The  velocity  of  sound  being  known,  it  becomes  possible  to  measure 
certain  distances  by  its  aid.  A  lightning  flash  is  seen,  practically,  instan- 
taneously ;  if  the  sound  take  5  seconds  to  travel,  it  must  have  travelled  1660 
metres,  or  about  a  mile.  Again,  a  stone  is  dropped  down  a  well ;  the  sound 
of  the  splash  is  heard  in  6  seconds :  at  what  depth  is  the  surface  of  the 
water  ?  The  stone  falls  for  t  seconds  through  a  space  equal  to  \gt* ;  the 
sound  comes  up  for  y  seconds  through  a  space  equal  to  33,20%  cm.  From 
the  two  equations  t  +  y  =  6,  £  x  981*2  =  33,2(%,  we  find  t  =  5-543,  and  the 
depth  equal  to  15,172  cm. 


464  ON   SOUND.  [CHAP. 

Propagation  of  Sound  in  gases  according  to  the  Kinetic  Theory.* 

—  Suppose  AB  to  be  a  cylinder,  CD  an  oscillating  piston.     The  molecules 

which  make  up  the  gas  in  AB  are  repre- 

'  sented  by  dots ;   for  the  nonce  we  shall 

*.  *  *  I        suppose  these  molecules  to  be  arranged 
•  •  •  •        in  straight  files,  and  we  shall  consider 

I!.*.... only   one   such   file.      The   members   of 

!!rill!!!l!  such  a  file  remain  in  the  same  straight 

'  line,  striking  and  rebounding  from  one 

another.  We  keep  in  mind  the  proposi- 
tion that  two  perfectly-elastic  bodies,  when  they  enter  into  collision,  rebound 
from  one  another  with  exchanged  velocities ;  and  that  other,  that  a  per- 
fectly-elastic body  striking  a  rigid  obstacle  returns  with  a  velocity  equal  to 
the  relative  velocity  with  which  it  struck  the  obstacle.  If  the  piston  be 
at  rest,  it  does  not  change  the  velocities  of  those  particles  which  strike  it, 
except  in  their  direction.  If  it  be  moving  towards  them  as  they  strike  it, 
they  rebound  from  it  with  a  velocity  greater  than  that  with  which  they 
approached  it ;  this  increased  velocity  they  communicate  by  exchange  to 
others  with  which  they  come  into  collision  :  they  then  return  to  the  approach- 
ing piston,  and  again  rebound  with  increased  velocity.  These  molecules 
thus  act  as  carriers  of  energy ;  they  borrow  from  the  advancing  piston  a 
certain  amount  of  energy,  which  they  pass  on  to  the  molecules  beyond ; 
from  molecule  to  molecule  this  energy  is  transmitted,  and  a  transitory 
crowding  together  of  the  molecules,  commencing  at  the  piston,  is  propa- 
gated through  the  whole  gaseous  mass.  Conversely,  when  the  piston  is  in 
retreat,  those  molecules  which  overtake  it  rebound  with  a  diminished  veloc- 
ity; they  exchange  this  diminished  velocity  with  those  molecules  which 
they  tardily  encounter,  and  which  they  do  not  turn  back  until  these  have 
travelled  farther  than  they  normally  could  have  done ;  a  diminution  of 
velocity  and  an  accompanying  rarefaction  are  propagated  throughout  the 
gas.  If  the  particles  all  lay  in  straight  lines,  the  speed  of  propagation  of 
sound  would  be  the  average  speed  at  which  the  particles  move ;  just  as  the 
rate  of  propagation  of  a  message  by  couriers  would  be  the  average  rate  at 
which  they  ride.  Nothing  would  be  gained  in  point  of  speed  by  multiply- 
ing the  number  of  such  couriers  if  their  horses  were  not  susceptible  to 
fatigue ;  and  so  it  is  a  matter  of  indifference  what  the  number  of  molecules 
is  by  the  intervention  of  which  the  exchange  and  transmission  of  energy 
are  effected,  provided  always  that  the  collisions  of  the  molecules  occupy 
time  inappreciable  in  comparison  with  the  intervals  spent  by  them  in  trav- 
ersing their  free  paths,  and  further,  that  the  size  of  the  molecules  be  very 
small  in  comparison  with  the  average  length  of  the  free  path.  Thus  the 
velocity  of  sound  does  not  depend,  within  the  same  gas,  upon  the  density 
or  on  the  pressure. 

If  the  speed  be  changed,  the  case  is  different;  an  increased  average 
speed  causes  an  increased  velocity  of  propagation.  This  may  occur  within 
the  same  gas  when  the  temperature  is  altered ;  the  absolute  temperature 
measures  the  kinetic  energy  of  the  molecules;  to  the 'square  root  of  this 
the  mean  velocity  is  proportional ;  the  rate  of  sound-propagation  would,  if 
the  particles  all  lay  in  straight  lines,  be  equal  to  this  mean  velocity ;  whence 


*  See  Tolver  Preston,  Phil.  Mag.  iii.  (1877). 


XIV-]  KINETIC   THEORY   OF  GASES.  465 

the  rate  of  sound-propagation  would  vary  as  the  square  root  of  the  absolute 
temperature. 

On  comparing  two  gases  we  find  that  the  mean  velocity  of  the  molecules 
varies  inversely  as  the  square  root  of  the  molecular  weight,  and  therefore  as 
the  square  root  of  the  density ;  whence  the  velocities  of  sound  in  two  differ- 
ent gases  would,  on  the  above  hypothesis,  be  inversely  as  the  square  roots  of 
the  respective  densities. 

We  cannot,  however,  affirm  that  the  particles  of  a  gas  lie  in  straight 
lines  or  files  ;  they  move  on  the  whole  with  perfect  symmetry  with  reference 
to  every  point.  Professor  Clerk  Maxwell  showed  that,  taking  this  into 
account,  we  ought  not  to  expect  the  rate  of  propagation  of  sound  to  be 
equal  to  the  average  velocity  of  the  particles,  but  proportional  to  it; 
and  that,  on  the  assumption  that  the  particles  were  small  as  compared  with 
their  mean  distances,  and  that  each  one  was  smooth  and  round,  so  as  not  to 
be  set  in  rotation  by  impacts,  then  the  rate  of  propagation  of  sound  should 
bear  to  the  mean  velocity  of  the  particles  the  ratio  of  \/5:3,  or  -745:1. 
Kundt  and  Warpurg  found  exactly  this  ratio  in  the  case  of  the  vapour  of 
mercury.  In  hydrogen  the  mean  velocity  is  184,260  cm.  per  sec. ;  the  veloc- 
ity of  sound  in  hydrogen  is,  according  to  the  mean  of  several  observations, 
126,917-6  cm.  per  sec. ;  the  ratio  is  -6888:1,  less  than  that  given  above.  The 
inference  is  that  the  molecules  are  to  some  extent  set  in  rotatory  as  well  as 
in  translatory  movement. 

Doppler's  Principle.  —  From  a  sounding-body,  approaching 
or  approached,  sound-waves  reach  the  ear  in  greater  number  than 
when  the  source  of  sound  and  the  listener  are  relatively  at  rest; 
and  conversely,  if  the  sounding-body  recede  or  be  receded  from, 
fewer  sound-waves  will  reach  the  ear.  To  a  person  standing  at 
a  railway  station  while  an  express  rushes  whistling  through,  the 
pitch  of  the  whistle  seems  suddenly  to  fall  as  the  engine  passes 
him.  Even  the  puffs  of  an  approaching  goods-engine  seem  to  the 
ear  appreciably  more  numerous  than  those  of  a  receding  one. 

Problem.  —  Show  that  with  a  velocity  of  2000  cm.  per  sec.  (say  45 
miles  an  hour)  the  fall  of  pitch  exceeds  a  major  second,  and  is  the  same 
whatever  may  be  the  pitch  of  the  note ;  the  medium  being  air. 


THE  HUMAN  EAR. 

Aerial  waves  are  communicated  to  the  air  in  the  external 
auditory  meatus.  This  is  short  in  comparison  with  the  length 
of  the  average  sound-wave.  Its  own  proper  sound  is  about 
#"'",  and  sounds  in  the  neighbourhood  of  this  tone  are  painfully 
reinforced  by  the  resonance  of  the  meatus. 

The  movements  of  the  air  in  the  meatus  do  not  materially 
differ  from  those  of  a  single  point  in  the  wave-front :  the  physi- 
cal problem  to  be  solved  in  the  organ  of  hearing  is*  one  of  the 
same  kind  as  would  be  presented  if  the  eye  were  called  upon,  by 


466  ON   SOUND.  [CHAP. 

the  inspection  of  a  single  point  on  the  surface  of  a  multifariously- 
rippled  sheet  of  water,  to  discriminate  all  the  component  undula- 
tions of  the  extended  surface. 

The  movements  of  the  air  in  the  meatus  are  communicated 
to  the  drum  of  the  ear,  the  membrana  tympani,  which  is  affected 
by  the  direct  impact  of  the  moving  air-particles. 

The  drum  of  the  ear  may  receive  some  vibrations  by  direct  transmission 
from  the  bones  of  the  skull. 

We  remark  here  —  (1.)  The  natural  note  of  so  small  a 
membrane  is  very  high ;  but  weighted  as  it  is  by  the  chain  of 
bones  of  the  internal  ear,  it  can  take  up  vibrations  of  a  much 
less  frequency  than  this  note. 

(2.)  The  vibration  of  the  membrane  is  a  forced  one,  and,  as 
regards  the  relative  amplitudes  of  very  high  components,  does 
not  precisely  coincide  in  character  with  that  of  the  air. 

(3.)  At  the  same  time  the  form  of  the  membrane  is  such 
that  it  vibrates  more  at  its  edges  than  at  its  centre,  and  the 
tendency  of  the  membrane  to  set  up  vibrations  of  its  own,  or 
to  alter  those  forced  upon  it,  is  mitigated. 

(4.)  The  membrane  is  normally  under  tension :  it  is  pulled 
inwards  by  the  handle  of  the  malleus;  considerable  pressures 
upon  it  cause  very  small  inward  movements,  especially  since 
its  radial  fibres  have  very  slight  extensibility. 

(5.)  It  is  easier  for  a  rarefaction  of  the  air  in  the  meatus 
to  cause  an  outward  movement,  which  slackens  the  membrane, 
than  for  a  condensation  to  drive  the  membrane  inwards  and 
thus  to  tighten  it  —  a  fact  of  importance  in  reference  to  combi- 
national tones,  of  which  hereafter. 

(6.)  When  the  membrane  does  move  inwards,  it  pushes 
inwards  the  handle  of  the  malleus,  which  is  firmly  attached  to 
it:  but  only  through  a  very  small  distance.  This  small  ampli- 
tude of  movement,  about  one-fortieth  of  that  of  the  air  in  the 
meatus,  implies  that  the  handle  of  the  malleus  is  wielded  with 
considerable  force  —  one  step  in  the  increase  of  the  force  of  the 
aerial  vibrations  on  their  way  to  the  internal  ear. 

(7.)  The  movements  of  the  drum  of  the  ear  are  astound- 
ingly  small.  The  greatest  displacement  seems  to  be  about 
0-1  mm.  or  l-250th  of  an  inch.  A  sound  produced  by  an  /ft 
( =  181)  pipe  under  an  air  pressure  of  40  mm.  of  water  can 
be  distinctly  heard  at  a  distance  of  115  metres.  Topler  and 
Boltzmann  calculated  that  at  such  a  distance  the  movements 


xiv.]  THE   HUMAN  EAR.  467 

of  the  air  must  be  reduced  to  -000,04  mm. ;  but  those  of  the 
more  massive  drum,  with  its  appendages,  cannot  be  more 
than  -000,001  mm.  or  the  twenty-five-millionth  part  of  an  inch 

—  an  oscillation  so  minute  as  to  be  beyond  direct  microscopic 
observation. 

The  drum  of  the  ear  sets  in  motion  the  handle  of  the 
Malleus.  The  malleus  is  a  small  bone,  somewhat  resembling 
a  hammer,  with  a  head  and  a  handle.  It  is  so  suspended  by 
ligaments,  head  upwards,  that  when  its  handle  is  thrust  inwards, 
its  head  is  made  to  rotate  to  a  limited  extent.  The  head  of  the 
malleus  is  connected  by  a  smooth  joint  of  peculiar  form  with  a 
second  bone,  the  Incus.  The  action  of  the  joint  is  such  that 
when  the  handle  of  the  malleus  is  forced  inwards,  the  head,  as 
it  rotates,  locks  in  the  incus  and  forces  it  round ;  while,  if  the 
handle  of  the  malleus  be  driven  violently  outwards,  the  head, 
rotating  in  a  reversed  direction,  does  not  pull  the  incus  with  it, 
but  glides  over  it,  rotating  through  as  much  as  5°  before  the 
two  bones  again  begin  to  move  as  one  piece.  If  air  be  driven 
through  the  Eustachian  tube  from  the  mouth-cavity,  as  it  always 
is  during  swallowing,  it  presses  against  the  membrane  from 
within  ;  so,  if  it  were  not  for  this  peculiar  joint,  there  would  be  a 
decided  risk  of  the  chain  of  bones  —  malleus,  incus,  and  stapes 

—  being  torn  away  from  their  connection  with  the  internal  ear. 
While  the  two  bones  are  thus  unlocked,  as  during  swallowing, 
there  is  an  impairment  in  their  power  of  transmitting  vibrations, 
and  there  arises  a  partial  deafness,  especially  for  loud  sounds. 

The  incus  has  a  process  or  long  projection  which,  when  the 
handle  of  the  malleus  moves  inwards,  moves  inwards  also. 
The  point  of  this  is  attached  to  a  little  stirrup-shaped  bone,  the 
Stapes.  Motion  is  thus  communicated  through  malleus,  incus, 
stapes ;  but  the  stapes  moves  only  two-thirds  as  much  as  the  end 
of  the  handle  of  the  malleus  —  another  step  in  the  increase  of 
force  and  diminution  of  amplitude  of  the  vibration  conveyed 
to  the  ear  by  the  air. 

Communication  of  sound  through  the  bones  of  the  head  seems  to  be 
effected,  for  the  most  part,  through  the  malleus,  incus,  and  stapes.  Com- 
munication of  sound  to  the  bones  of  the  head  may  be  facilitated  by  means 
of  the  audiphone.  This  is  a  plate  of  thin  vulcanite,  which  is  bent  and 
kept  by  strings  under  a  certain  degree  of  tension,  and  the  edge  of  which  is 
placed  in  contact  with  the  teeth.  Even  a  piece  of  stiff  brown  paper,  loosely 
doubled  and  grasped  by  its  opposite  edges  between  the  teeth,  will  act  as  an 
audiphone  and  take  up  waves  of  sound  from  the  air,  and  wilf  convey  them 
to  the  bones  of  the  head.  A  certain  proportion  of  the  sound  travels  directly 


468  ON   SOUND.  [CHAP. 

to  the  nervous  apparatus  embedded  in  the  skull,  which  is  directly  shaken ; 
sound  is  thus  rendered  to  some  extent  audible  to  those  whose  auditory 
ossicles  fail  in  their  function. 

The  footplate  of  the  stirrup  is  blended  with  a  membrane 
occupying  a  small  aperture  —  the  fenestra  ovalis  —  in  the  hard 
mass  of  the  temporal  bone.  The  footplate  of  the  stirrup  has 
itself  a  form  closely  resembling  that  of  a  footprint.  If  now  the 
reader  will  place  his  foot  on  a  soft  carpet  and  forcibly  drive 
into  the  carpet  the  outer  edge  of  the  foot,  he  will  see  that  the 
inner  edge  of  his  foot  is  tilted  upwards ;  this  describes  the 
motion  of  the  stapes  when  driven  inwards  against  the  membrane 
of  the  fenestra  ovalis. 

Beyond  this  membrane  lies  the  fluid  of  the  internal  ear, 
contained  in  membranous  bags,  which  float  in  channels  hollowed 
out  in  the  temporal  bone.  This  system  of  membranous  bags 
containing  fluid  consists  of  the  vestibule,  the  cochlea  (in  front), 
and  the  semicircular  canals  (behind).  Here  we  have  to  do 
with  the  two  former.  The  fluid  lying  in  the  vestibule  is  imme- 
diately behind  the  membrane  of  the  fenestra  ovalis ;  the  vibra- 
tions of  the  stapes  are  communicated  to  it.  This  fluid  is  in 
direct  communication  with  the  fluid  lying  in  a  part  of  the 
cochlea  called  the  scala  vestibuli. 

The  structure  of  the  cochlea  seems  at  first  sight  somewhat 
complex.  It  is  a  snail-shell-like  object;  if  unwound  and  laid 
flat  it  might  be  diagrammatically  represented  by  Fig.  152.  S  is 
the  stapes,  blending  with  the  membrane  of  the  fenestra  ovalis  ; 
V  is  the  vestibule ;  S.V.  is  the  scala  vestibuli,  a  cavity  extend- 
ing to  the  tip  of  the  cochlea;  there  it  is  continuous  with  S.T., 
the  scala  tympani ;  this  ends  at  f.r.  the  fenestra  rotunda,  an 
aperture  closed  by  a  strong  membrane.  The  fluid  filling  S.V. 
(the  perilymph)  is  continuous  with  that  in  S.T.,  and  hence 
strong  pressure  on  the  stapes  will  cause  f.r.,  the  membrane  of 

Fig.152. 


S.V. 


the  fenestra  rotunda,  to  bulge  outwards.  The  scala  tympani  and 
the  fenestra  rotunda  are  perhaps  a  safety  arrangement.  The 
vibrations  which  are  important  to  us  are  those  of  the  fluid  in 
the  scala  vestibuli. 


xiv.]  THE   HUMAN   EAR.  469 

Between  the  scala  vestibuli  and  the  scala  tympani  lies  a  par- 
tition, incomplete  at  the  apex  of  the  cochlea.  This  partition  is 
partly  of  bone,  partly  of  membrane  :  the  purely  membranous 
part  is  called  the  membrana  basilaris. 

Transverse  section  of  the  cochlea  shows  us  further  that  we 
have  not  only  to  do  with  the  two  scalse  and  the  intervening  par- 
tition, but  also  with  a  third  cavity.  Such  a  section  is  shown 
diagrammatically  in  Fig.  153.  The  Flg  153 

scala  vestibuli  rests  only  upon  the 
bony  part  of  the  partition  ;  the  third 
cavity,  the  scala  cochlece,  S.C.,  lies 
mainly  above  the  membranous  part, 
the  basilar  membrane,  m.b.  The 
scala  cochlea}  contains  fluid  (endo- 
lymph),  and  is  practically  a  closed 
cavity.  When  the  liquid  in  the 
scala  vestibuli  vibrates,  the  endo- 
lymph  in  the  scala  cochlece  is  at  each  impulse  forced  into  similar 
movement  before  the  liquid  in  S.V.  has  had  time  to  pass  into 
S.T. ;  and  it  in  its  turn  acts  upon  the  basilar  membrane. 

The  basilar  membrane  is  triangular  in  form,  being  widest  at 
the  tip  of  the  cochlea.  It  takes  up  vibrations  of  definite  pitch, 
in  response  to  which  it  vibrates  not  as  a  whole  but  locally  (see 
Fig.  148)  ;  just  as  when  we  sing  to  a  piano  with  the  dampers 
down,  only  those  strings  respond  which  are  in  unison  with  the 
sound  produced  by  the  voice.  It  responds  to  vibrations  of  con- 
siderable slowness  compared  with  its  natural  vibrations,  for  not 
only  does  it  lie  between  two  liquids,  but  it  is  also  somewhat 
heavily  loaded;  upon  it  are  mounted  certain  rigid  structures 
arranged  in  two  rows,  the  rods  of  Corti;  and  upon  these  are 
arranged  a  number  of  nerve-cells,  the  cells  of  Corti,  each  of  which 
is  connected  with  a  single  nerve-fibre.  Each  of  these  nerve-fibres 
can  only  be  stimulated  to  sensation  by  the  vibratile  movement 
of  that  cell  of  Corti  from  which  it  runs ;  and  it  is  deaf  to  every 
sound  but  that  one  to  which  the  vibration  of  the  particular  under- 
lying part  of  the  basilar  membrane  corresponds.  As  there  are 
from  16,000  to  20,000  such  cells  of  Corti,  and  therefore  the  same 
number  of  nerve-fibres  which,  although  they  merge  into  a  com- 
mon strand  —  the  auditory  nerve  —  and  enter  the  brain  side  by 
side,  are  isolated  from  one  another,  we  may  say  that  we  have  riot 
one,  but  from  16,000  to  20,000  distinct  Senses  of  Hearing,  each 
with  its  own  special  Organ  of  Hearing ;  the  Ear,  as  we  have  de- 


470  ON  SOUND.  [CHAP. 

scribed  it,  being  merely  a  mechanical  means  for  the  transmission 
of  vibration  from  the  external  world  to  these  sense-organs,  and 
its  due  distribution  among  them. 

There  is  yet  some  difficulty  in  seeing  how  the  rods  of  Corti,  3000  pairs 
in  number,  affect  differently  the  half-dozen  cells  of  Corti  borne  by  each  pair ; 
hence  some  deny  that  more  than  3000  different  sounds  can  be  perceived 
otherwise  than  by  a  process  of  comparison ;  even  this  would  enable  us  to 
distinguish  tones  differing  by  less  than  the  thirtieth  part  of  a  semitone.  In 
some  animals,  also,  the  basilar  membrane  appears  too  unwieldy  a  structure 
to  act  well  in  the  way  described  (Rutherford). 

When  an  external  sound  does  not  coincide  with  any  of  the 
above  16,000  or  20,000,  the  cells  most  nearly  corresponding  to 
it  will  be  disturbed ;  one  of  these  cells  will  vibrate  more  than 
its  neighbours.  The  phenomenon  is  now  one  of  unconscious 
comparison  of  their  relative  disturbances.  Within  the  middle 
range  of  hearing,  sounds  differing  by  one  vibration  in  three  sec- 
onds can  be  distinguished  by  some  persons ;  when  the  notes 
chosen  are  very  high  in  pitch,  grave  errors  in  the  discrimination 
of  pitch  may,  on  the  other  hand,  be  readily  committed. 

When  a  compound  sound  is  produced,  the  basilar  membrane 
is  set  in  motion  in  a  number  of  limited  regions  at  the  same  time, 
and  the  effect  is  mingled  in  the  brain.  The  nature  of  the  com- 
pound sensation,  by  which  we  thus  recognise  the  different  quali- 
ties of  sound,  is  a  question  which  passes  the  bounds  of  Physics. 

The  ear  has  a  certain  power  of  persisting  in  vibrations  once  set  up  in  it ; 
but  only  in  small  degree,  and  to  an  extent  the  greater  the  lower  the  pitch  of 
the  note.  If  the  sound  c'  be  broken  up  by  alternate  flashes  of  sound  and 
silences  of  equal  duration,  when  these  each  number  130  per  second  the  sound 
seems  continuous.  Each  pitch  has  its  own  duration  of  persistence ;  and 
Mayer  has  pointed  out  (see  Phil.  Mag.,  1876,  vol.  ii.)  that  in  a  mixture  of 
sounds,  alternately  admitted  to  the  ear  and  shut  off  from  it  by  apertures  in 
a  rotating  disc,  some  may  be  rendered  evidently  intermittent  by  the  action 
of  the  disc,  while  others  may  appear  continuous,  and  that  thus  we  have  a 
new  means  of  analysing  compound  sounds. 

As  to  the  limits  of  hearing,  sounds  may  be  perceived  by  the 
human  ear  which  range  from  16  (Preyer),  or  34  (von  Helm- 
holtz),  to  about  32,000  (Despretz),  or  even  40,000  ( Appunn  and 
Preyer)  ;  there  being  between  different  individuals  curious  dif- 
ferences in  the  power  of  perception,  especially  of  high  sounds. 

A  small  number  of  abrupt  clicks  cannot,  with  ease,  -blend  into  a  con- 
tinuous sound ;  if  they  seem  to  do  so,  it  is  generally  some  high  harmonic 
that  is  really  heard.  A  very  small  number  of  true  pendular  vibrations 
seems  to  produce  sound  when  sufficient  surface  is  acted  upon,  or  the  ear 
otherwise  firmly  enough  set  in  vibration ;  whence  the  very  deep  hum  of  a 
river-steamer  slackening  speed,  or  that  sound  of  contracting  muscle  — 19 


xiv.]  THE   HUMAN  EAK.  471 

vibrations  per  second  —  which  is  heard  when  the  forefingers  are  pressed  into 
the  ears  and  the  elbows  pressed  against  the  table. 

The  sensible  loudness  of  sounds  does  not  coincide  very 
closely  with  their  physical  intensity.  This  arises  partly  from 
modifications  in  the  form  of  the  vibration  induced  by  so  compli- 
cated a  transmission  through  the  auditory  apparatus,  partly  from 
causes  purely  physiological. 

It  is  curious  that,  as  Mayer  has  shown,  high  notes  are  heard  with  diffi- 
culty in  the  presence  of  lower  ones.  Hence  sixteen  violins  in  an  orchestra 
produce  by  no  means  so  great  an  effect  as  sixteen  violins  alone.  A  lower 
note  tends  to  drown  a  higher  one,  especially  if  the  higher  note  be  thoroughly 
in  tune,  or  form  a  correct  acoustic  interval  with  the  lower.  In  a  single 
musical  sound  the  fundamental  tone  drowns  the  harmonics,  even  though 
the  latter  may,  as  in  a  tinkling  piano,  very  greatly  exceed  it  in  physical 
intensity. 

The  sensitiveness  of  different  ears  to  sound  may  be  com- 
pared by  measuring  the  relative  distances  at  which  a  given 
sound  becomes  inaudible ;  as  the  squares  of  these  distances,  so 
are  the  sensibilities  of  the  listening  ears.  If  one  ear  can  hear  a 
certain  sound  at  3  feet,  the  other  only  at  3  inches,  then  the 
duller  ear  is  (362  -s-  32)  =  144  times  less  sensitive  than  its  fellow. 

Even  beyond  the  ear  and  within  the  brain  there  is  some  mechanism  of 
which  we  are  still  ignorant.  Professor  Silvanus  Thompson  has  shown  that 
two  sounds  which  beat  with  one  another  will,  when  conveyed  separately  one 
to  each  ear,  produce  beats  which  appear  to  jar  the  vertex  of  the  brain ;  and 
further,  that  two  sounds  of  the  same  pitch  and  phase,  arriving  separately 
in  the  ears,  appear  to  be  heard  in  the  ears,  while,  if  they  arrive  in  opposite 
phases,  the  effect  is  as  if  the  sound  were  heard  not  in  the  ears  at  all,  but 
within  the  vertex  of  the  cranium.  The  former  phenomenon  may  be  roughly 
shown  by  a  tuning-fork  held  to  each  ear ;  the  latter  by  a  pair  of  telephones, 
one  to  each  ear,  one  being  provided  with  a  commutator,  by  which  the  current 
in  that  telephone  can  be  reversed  at  will. 

Direction  of  sound  can  hardly  be  determined  if  the  head  be  held  fixed; 
we  turn  the  head  slightly  while  listening,  and  interpret  unconsciously  the 
consequent  variations  of  intensity  in  the  two  ears. 

HARMONY  AND  DISSONANCE. 

Just  as  it  is  disagreeable  to  the  eye  to  be  exposed  to  flicker- 
ing light,  so  it  is  painful  to  the  ear  to  be  exposed  to  audible 
flickering  such  as  that  produced  by  two  sounds  which  beat  when 
sounded  together.  The  climax  of  unpleasantness  or  Discord  is 
reached  when  the  beats  amount  to  about  32  per  second.  When 
the  beats  are  still  more  numerous  than  this,  the  two  notes  which 
are  sounded  together  become  more  distinctly  separable  by  the 
ear,  and  the  beats  are  less  prominent  to  the  sense  of  hearing. 


472  ON  SOUND.  [CHAP. 

Were  this  the  only  cause  in  operation,  the  intervals 


which  all  differ  by  32  vibrations  per  second,  would  be  equally 
painful  to  the  ear.  To  some  extent  it  is  the  case  that  an  inter- 
val concordant  between  high  notes  is  painful  in  the  lower  parts 
of  the  scale,  as  may  be  found  on  playing  a  major  third  in 
different  octaves  on  a  harmonium :  but  another  cause  affects 
the  degree  of  painfulness  of  a  discordant  interval.  When  two 
contiguous  sounds  affect  the  basilar  membrane  simultaneously, 
the  vibration  of  the  basilar  membrane  is  not  limited  to  those 
fibres  or  narrow  strips  which  exactly  correspond  to  the  two  con- 
tiguous sounds  heard ;  the  result  is  a  confused  vibration  of  the 
membrane  in  a  wider  strip,  and  a  sound  is  heard,  compounded 
of  all  the  notes  within  the  region  of  that  strip ;  this  is  inter- 
preted by  the  ear  either  —  where  the  two  component  notes  are 
very  close  —  as  a  single  note  of  average  pitch,  or,  where  they 
are  farther  apart,  as  two  separate  notes  coupled  with  a  painful 
sensation.  Discord  is  thus  due  partly  to  beats,  partly  to  diffi- 
culty in  identifying  pitch. 

The  beats  produced  by  mis-tuned  consonant-intervals  correspond   to 
curious  vibrational  curves,  for  which  see  Bosanquet,  Phil.  Mag.  1881. 

When  two  notes  are  produced  by  separate  sources  of  sound, 
the  upper  harmonics  may  possibly  clash  and  beat  with  one 
another.  This  may  be  specially  observed  in  the  roughness  of 
full  chords  of  a  brass  band  within  an  enclosed  space.  If,  for 
instance,  C  and  G  be  sounded  together,  the  aggregate  harmonics 
are  the  following  :  — 


of  which  only  #,#',  and  g"  are  coincident. 

If  we  eliminate  all  the  harmonics  beyond  the  sixth  of  the 
lower  notes,  the  respective  coincident  and  non-coincident  har- 
monics for  various  intervals  become  the  following  :  — 


Fifth. 


Fourth.  Major  Third.          Tenth. 


Minor  Third. 


xiv.]  HARMONY  AND   DISSONANCE.  473 

The  most  concordant  interval  is  that  in  whjch  the  harmonics 
soonest  coincide,  and  in  which  those  non-coincident  harmonics 
which  discord  with  one  another  are  as  remote  as  possible  from 
the  fundamental  tones. 

This  is  not  the  only  element  to  be  taken  into  account  in 
explaining  the  relative  harmoniousness  of  certain  intervals. 
When  two  notes  are  sounded  together,  there  are  produced 
other  tones  called  Differential  and  Summational  Tones, 
or,  generally,  combinational  tones,  which  are  heard  along  with 
the  former. 

Differential  Tones.  —When  c'  (  =  256)  and  g'  (  =  384)  are 
sounded  together  loudly  and  firmly,  the  listening  ear  can  dis- 
tinguish the  sound  c  (=  128)  humming  at  the  same  time.  With 
c'  (-256)  and  e'  (=320)  the  differential  tone  is  C  (=64). 
The  vibrational  number  of  the  differential  tone  is  equal  to  the 
number  of  beats  which  tend  to  be  formed  by  the  two  prime 
tones.  For  this  reason  it  was  long  thought  that  the  combina- 
tional tones  were  produced  by  the  blending,  into  a  continuous 
sound,  of  Beats  too  numerous  to  reckon.  These  tones  are,  on 
the  contrary,  distinct  from  the  beats.  If  two  very  high  notes, 
differing  by  64  vibrations  per  second,  be  sounded  together,  the 
discordant  shiver  due  to  the  beats  can  be  distinctly  perceived, 
though  the  beats  themselves  cannot  be  reckoned,  while  at  the 
same  time  the  corresponding  differential  tone  can  be  heard 
humming.  If  the  sounds  be  separately  conveyed,  one  to  each 
ear,  the  beats  may  be  distinctly  felt  within  the  head,  but  no 
differential  tone  is  heard.  And  further,  if  the  two  sources  of 
sound  be  markedly  distant  from  one  another,  the  beats  may  be 
apparent  while  the  differential  tone  is  feeble. 

There  are  two  causes  for  the  formation  of  these  tones.  When  a  con- 
siderable disturbance  has  been  produced  in  the  air  by  one  source  of  sound, 
the  disturbance  of  the  air  produced  by  another  source  near  the  former  is 
not  simply  added  to  the  disturbance  already  existing,  for  Hooke's  Law  —  as 
the  force  applied,  so  is  the  distortion  produced  —  is  not  true  except  for  very 
small  disturbances :  the  amplitude  of  the  compound  oscillation  falls  short 
of  the  sum  of  the  amplitudes  of  the  components,  and  this  is  equivalent  to 
the  introduction  of  a  new  vibration  whose  fundamental  period  is  that  of  the 
differential  tone.  Differential  tones  so  produced  —  as  by  a  harmonium  — 
can  be  reinforced  and  investigated  by  means  of  resonators.  The  more 
effective  cause  of  the  production  of  combinational  tones  is,  however,  their 
origination  in  the  drum  of  the  ear :  the  drum  of  the  ear  moves  more  freely 
outwards  than  inwards ;  and  von  Helmholtz  showed  that  if  a  transmitter 
possessing  this  peculiarity  be  acted  upon  by  a  disturbance  which  is  the  sum 
of  two  disturbances  of  frequencies  p  and  </,  the  energy  imparted  to  it  is  in 
part  expended  in  producing  new  vibrations  of  frequencies  p  —  q  and  p  +  q; 


474 


ON  SOUND. 


[CHAP. 


these  corresponding"  respectively  to  differential  tones  and  to  summational 
tones.     Sounds  so  produced  cannot  be  reinforced  by  resonators. 

Summational  Tones.  —  These  tones  were  discovered  by 
von  Helmholtz.  The  notes  c1  and  gf  (256  and  384)  will,  when 
sounded  together,  produce  a  faintly  audible  tone  e"  (  =  640 
=  256  -h  384).  Tones  of  this  kind  are  only  to  be  heard  with 
difficulty  —  a  fortunate  circumstance,  since  they  mostly  discord 
with  their  prime  tones. 

The  summational  and  differential  tones  for  some  consonant 
intervals  are  represented  in  the  following  table,  in  which 
minims  (p  )  denote  the  prime  tones  and  crotchets  (j  )  the  com- 
binational tones. 

I 


Harmony.  —  The  chord  c\  e',  g',  ffi— g— ,  has  a  differential 
tone  between  c'  and  e',  the  tone  C ;  between  e'  and  #'  the  same; 
between  c'  and  g*  the  tone  c ;  together  — =£=.  Again,  c'  has 

a  first  harmonic  c" ;  with  e'  and  g1  this  makes  the  differential 
tones  g  and  c.  The  note  e'  has  a  first  harmonic  e"  ;  with  c'  and 
fff  the  difference-tones  are  respectively  g1  and  c1 ;  and  similarly 
the  first  harmonic  of  g1  produces,  with  c'  and  e',  the  tones  c"  and 
b*b — .  Altogether,  these  form  the  series 


all  the  tones  of  which  occur  as  partials  in  the  note  C,  and  there- 
fore blend  smoothly  together.  The  minor  chord  c\  e'b,  g'  in  a 
similar  way  has  the  primary  differential  tones  A^,  Et>,  c,  and  the 
secondary  differential  tones  c,  bb,  d',  g'b—,Vb,  c" ;  and  the 
primary  chord  is  thus  embedded  in  a  mass  of  combination-tones 
comprising,  among  others,  the  discordant  series  — 


xiv.]  HARMONY  AND  DISSONANCE,  475 

The  consonance  of  c1,  e'b,  g'  is  therefore  necessarily  much 
harsher  than  that  of  c\  e\  and  g'. 

For  further  developments  of  this  subject  the  reader  must  be  referred  to 
von  Helmholtz's  Sensations  of  Tone,  translated  by  Mr.  Ellis. 

When  just  intonation  is  possible,  as  among  glee-singers  or 
quartette-players,  each  listens  to  his  fellow-performers  as  well 
as  to  his  own  voice  or  instrument,  and  gives  out  that  note  which 
he  feels  to  belong  to  the  key  in  which  the  party  is  performing, 
and  learns  to  do  so  in  such  a  way  as  to  avoid  beats :  thus,  as  is 
said,  the  performers  rub  off  one  another's  asperities.  In  this 
way  mathematically-exact  ratios  of  considerable  complexity  are 
accurately  attained  without  necessary  knowledge  of  them  on 
the  part  of  the  performers. 

VOICE — VOWELS, 

The  voice  is  produced  by  vibrations  of  the  larynx,  especially 
of  the  vocal  chords,  in  whole  or  in  part.  Above  these  is  placed 
the  mouth-cavity,  which  may  assume  various  forms  under  the 
action  of  the  various  muscles  which  regulate  the  position  of  the 
tongue,  the  soft  palate,  the  floor,  and  the  sides  of  the  mouth. 
This  mouth-cavity  acts  as  a  resonator  and  reflector.  According 
to  the  number  of  upper  harmonics  which  are  reinforced,  and 
the  extent  to  which  they  are  severally  reinforced,  will  vary  the 
quality  of  the  sound  emitted.  Upon  this  quality,  and  upon 
nothing  else,  depends  that  Character  which  we  recognise  as 
some  particular  Vowel ;  for  every  vowel  is  a  particular  Quality 
of  Sound. 

An  elementary  example  of  this  is  furnished  by  a  common  pocket  tuning- 
fork  ;  when  set  in  vibration  and  the  broad  face  of  one  of  the  prongs  pre- 
sented to  the  ear,  the  fork  seems  to  emit  the  vowel  u  or  oo ;  when  its  shank 
is  pressed  against  a  table  the  fork  seems  to  say  0 ;  now  the  octave  becomes 
prominent.  The  reason  is  that  the  fork  swings  in  circular  arcs,  and  not 
in  transverse  straight  lines ;  it  consequently  presses  against  the  table  at 
the  end  of  each  half -oscillation,  and  causes  it  to  emit  the  octave  as  well 
as  the  fundamental  tone:  A  tone  almost  pure  gives  the  hollow  sound  of 
the  vowel  u ;  one  accompanied  by  its  octave  gives  the  brighter  sound  of.  the 
vowel  o.  Each  vowel  gives  a  particular  form  of  indentation  in  a  phonograph. 

Vowel  sounds  can  be  analysed  by  means  of  resonators ;  and 
when  a  particular  vowel  is  sung  in  presence  of  an  open  piano 
(loud  pedal  down)  that  vowel  is  repeated  by  the  strings :  each 
component  of  the  complex  vibration  is  taken  up  by  that  string 
which  is  in  unison  with  it.  On  the  other  hand,  von  Helmholtz 


476  ON   SOUND.  [CHAP. 

showed  that  by  causing  a  number  of  resonators  of  a  series  whose 
frequencies  were  as  1:2:3:4:5:6:7,  etc.,  to  vibrate  with 
independent  intensities,  he  could  at  will  produce  by  synthesis 
not  only  a  great  number  of  qualities  of  tone  widely  differing 
from  one  another,  such  as  clarionet- tone,  etc.,  but  could  also 
build  up  the  different  vowels  themselves. 

To  each  vowel  corresponds  a  different  form  of  the  resonating 
mouth-cavity  ;  to  each  such  form  corresponds  a  different  natural 
pitch  of  vibration.  When  the  larynx  emits  a  complex  sound 
containing  as  one  of  its  components  a  tone  of  this  natural  pitch, 
this  tone  is  strongly  reinforced,  and  the  quality  of  tone  some- 
what affected. 

TRANSFORMATIONS  OF  THE  ENERGY  OF  SOUND. 

Sound  being  in  its  physical  aspect  a  kind  of  motion,  in  the 
course  of  which  work  is  done  against  elasticity  and  inertia,  it  is 
superfluous  to  speak  of  the  conversion  of  the  energy  of  sound 
into  that  of  mechanical  work.  The  transmission  of  sound  is  a 
transmission  of  energy,  and  the  sound  produced  by  a  sounding- 
body  is  mechanically  equivalent  to  a  definite  amount  of  work. 
When  a  heavy  tuning-fork  is  attached  to  the  piston  of  a  little 
pump,  as  in  Edison's  harmonic  engine,  it  can  be  made  to  do 
work;  but  then  it  produces  somewhat  less  sound  than  when 
vibrating  freely.  The  mechanical  equivalent  of  sound  may  be 
estimated,  as  it  has  been  by  Mayer,  by  comparing  the  sound  pro- 
duced by  a  free  tuning-fork  with  the  sound  produced  by  the 
same  fork  on  equal  excitation  when  its  prongs  are  connected  by 
a  thin  strip  of  indiarubber,  and  by  finding  the  amount  of  heat 
developed  in  the  rubber  in  the  latter  case. 

Work  may  be,  on  the  other  hand,  converted  into  sound.  In 
general  there  are  two  methods  of  accomplishing  this  transforma- 
tion, —  firstly,  by  storing  potential  energy  in  an  elastic  body, 
which  is  then  liberated ;  secondly,  by  transforming  uniform  into 
intermittent  motion  through  the  agency  of  friction.  We  have 
already  studied  the  mode  of  excitation  of  a  violin  string.  A 
pointed  slate-pencil  pushed  across  a  slate  at  a  certain  angle  pro- 
duces a  well-known  shrill  scream,  and  the  mark  produced  by  'it 
will  be  found  on  close  examination  to  consist  of  a  train  of  sepa- 
rate dots ;  the  action  is  not  unlike  that  of  the  violin  string. 
The  scream  of  unoiled  bearings  in  a  machine  may  be  accounted 
for  in  the  same  way,  and  in  such  a  case  much  of  the  energy  of 
rotation  of  the  machine  is  wasted  in  the  form  of  sound. 


xiv.]  TRANSFORMATIONS   OF   SOUNP-F^NERGY.  477 

Heat  may  be,  in  some  cases,  transformed  into  the  energy  of 
Sound.  Trevelyan's  rocker  and  singing-flames  we  have  already 
studied ;  the  singing  of  a  kettle  is  due  to  the  rhythmical  agita- 
tion produced  by  the  formation  and  collapse  of  bubbles;  the 
roar  of  steam  issuing  from  a  boiler  is  produced  by  the  disturb- 
ance of  the  surrounding  air  by  steam  which,  after  thrusting 
aside  the  surrounding  air,  collapses  into  water-drops ;  the  roar 
of  a  chimney  is  due  to  the  oscillation  to-and-fro,  within  the 
chimney,  of  heated  columns  of  air  or  smoke  which  set  the  air 
within  the  chimney  in  vibration,  of  which  the  deep  roar  heard 
by  us  is  generally  a  high  harmonic  :  in  all  such  cases  the  energy 
of  the  sound  produced  is  obtained  at  the  expense  of  the  Heat 
supplied. 

But,  like  other  forms  of  energy,  that  of  Sound  is  ultimately 
dissipated.  When  sound  is  produced  in  a  room,  every  particle 
of  the  walls  and  contents  of  the  room  is  set  in  vibration ;  there 
is,  indeed,  no  way  of  protecting  bodies  surrounding  a  source  of 
sound  from  this  influence,  except  perhaps  by  placing  them  upon 
several  alternate  layers  of  caoutchouc  and  soft  putty  within  a 
vacuum.  At  last  the  sound  degenerates,  after  repeated  reflexion 
within  each  object,  into  irregular  molecular  motion,  and  its 
energy  is  converted  into  Heat.  So  when  a  tuning-fork  is  set  in 
motion  and  sounded  in  the  open  air,  part  of  the  energy  which 
was  initially  communicated  to  the  tuning-fork  when  it  was  first 
set  in  vibration  is  lost,  in  consequence  of  the  viscosity  of  the 
fork,  which  becomes  slightly  warmer ;  while  part  of  that  energy 
is  expended  upon  the  external  air,  which,  by  reason  of  its  own 
viscosity,  gradually  extinguishes  the  sound,  beginning  with  the 
highest  components,  and  the  whole  at  length  dies  away,  the 
energy  of  sound-motion  becoming  converted  into  the  degener- 
ated form  of  Heat,  which  ultimately  becomes  diffused  through- 
out the  entire  Universe. 


CHAPTER   XV. 

OF   ETHER-WAVES.  « 

IN  this  chapter  a  variety  of  phenomena  fall  to  be  considered 
which  can  be  explained  as  phenomena  of  undulation  in  the 
all-pervading  Ether,  and  may  thus  be  said  to  be  due  to  Ether- 
Waves. 

It  is  necessary,  however,  to  make  a  reservation  of  opinion, 
and  to  point  out  that  all  we  are  really  entitled  to  affirm  is  that 
the  phenomena  in  question  are  transferences  of  energy  through 
the  Ether,  accompanied  by  variable  disturbances  of  that  medium 
—  disturbances  whose  variations  follow  the  same  laws  as  those 
of  wave-motion,  but  which  may  in  themselves  be  due  to  changes 
not  necessarily  of  position  within  the  Ether,  but  possibly  of  its 
stress,  of  its  electric  condition,  or  of  some  other  property  of  the 
interstellar  medium  as  yet  unknown  to  us.  Their  theory  has 
been  chiefly  developed  by  those  who  considered  the  phenomena 
of  Light,  Radiant  Heat,  etc.,  as  phenomena  of  Wave-Motion  in 
the  Ether;  and,  with  this  preliminary  explanation,  we  shall  in 
the  sequel  speak  unreservedly  of  these  phenomena  as  due  to 
Ether-Waves. 

NATURE  OF  RADIATION. 

The  all-pervading  Ether  can  be  set  in  vibration  by  the  vibra- 
tion of  the  molecules  of  ordinary  matter.  This  local  disturbance 
sets  up  waves;  and  by  these  waves  energy  may  be  transferred 
from  one  place  to  another.  This  process  of  transference  of 
energy  by  Ether-waves  is  the  process  of  Radiation. 

The  Radiation  of  Energy  by  the  Sun  amounts,  according  to  the  results 
of  Prof.  Langley's  experiments  on  Mount  Whitney,  Southern  California,  to 
about  16500  horse-power  per  square  foot  of  the  sun's  surface.  Of  this  about 
one  2125,000000th  part  meets  the  earth ;  this,  at  the  earth,  amounts  to  about 
150  ft.-lbs.  per  sq.  ft.  per  sec.,  or  3  ca  per  sq.  cm.  per  minute.  Of  this  about 
one-third  is  always  spent  in  heating  the  atmosphere ;  the  rest  may  be  more 
or  less  cut  off  from  access  to  the  earth's  surface  by  clouds,  dust,  water- 

478 


CHAP,  xv.]  NATURE   OF  RADIATION.  479 

vapour,  etc.  The  energy  of  the  waves  comprised  within  a  cubic  mile  of  the 
Ether  near  the  earth's  surface,  or,  to  use  Lord  Kelvin's  phrase,  the  "  mechani- 
cal value  of  a  cubic  mile  of  sunlight,"  is  accordingly  about  23000  ft.-lbs. 

Ether- waves  can  also  be  produced  by  electric  methods,  for 
which  see  p.  741.  These  waves  are,  as  yet,  longer  and  of  less 
frequency  than  those  produced  by  the  vibration  of  molecules. 

Heat-waves  and  light-waves  in  Ether  are  not  waves  of  com- 
pression and  rarefaction,  like  those  of  sound  in  air.  The  propa- 
gation of  an  ether-wave  is  effected  after  a  different  fashion,  some- 
what difficult  to  realise.  The  analogy  of  a  transverse  vibration 
running  along  a  cord,  or  of  a  wave  of  up-and-down  oscillation 
running  over  the  surface  of  water  or  over  a  thin  membrane, 
must  be  extended  to  the  Ether,  with  its  three  dimensions  in 
space.  At  any  point  where  the  movement  of  the  Ether  is 
examined,  it  is  found  to  be  an  oscillation  at  right  angles  to  the 
direction  in  which  the  wave  is  being  propagated,  and  therefore 
parallel  to  the  wave-front. 

The  vibration  of  the  Ether,  when  due  to  molecular  vibration, 
is  initially  of  the  nature  of  a  forced  vibration;  it  is  probably 
excited  by  the  oscillation  of  a  part  of  the  Ether,  which  is  in 
some  way  entangled  within,  or  which  envelopes,  the  vibrating 
molecule. 

The  molecular  vibration  which  excites  the  ether-waves  is  a  true  vibra- 
tion of  the  molecule,  not  a  translational  oscillation  from  place  to  place. 

The  molecules  of  ordinary  matter  must  be  supposed,  in  virtue  of  their 
small  size,  to  vibrate  very  rapidly.  We  have  already  stated  that  the  average 
diameter  of  molecules  is  perhaps  the  2>5010000  th  part  of  a  millimetre,  and  that 
they  may  perhaps  consist  of  ether  rolling  within  ether  in  vibratile  vortices. 
A  steel  tuning-fork  2  inches  (50  mm.)  long  may,  if  it  be  of  the  proper  form, 
vibrate  240  times  a  second ;  if  it  were  2(5010000  mm.  long,  and  of  the  same 
shape,  it  would  vibrate  30,000,000000  times  per  second  ;  if  made  not  of  steel, 
but  of  ether,  its  frequency  would  be  greater  in  the  ratio  of  the  velocity  of 
propagation  in  ether  to  that  in  steel,  and  would  therefore  amount  to  about 
2600,000000,000000  oscillations  per  second.  The  vibration  of  a  molecule 
may  be  more  like  that  of  a  disc  than  that  of  a  tuning-fork ;  but  the  rough 
analogy  just  mentioned  may  serve  to  show  that  it  is,  even  a  priori,  possible 
that  some  such  number  might  denote  the  average  frequency  of  molecular 
oscillations,  —  an  average  modified  in  the  direction  of  retardation  by  the 
formation  of  heavier  molecules  through  the  coalescence  of  smaller  molecules, 
or  perhaps  by  the  reaction  of  the  Ether  which  is  set  in  forced  vibration,  or 
modified,  on  the  other  hand,  in  the  direction  of  acceleration  by  the  forma- 
tion of  higher-pitched  vibrations,  which  may  be,  to  use  the  musical  analogy, 
dissonant  with  one  another  when  the  structural  arrangement  of  the  molecules 
is  unsym metrical.  The  molecule  of  sodium-vapour  acts  somewhat  like  a  disc 
which  is  slightly  unsymmetrical :  such  a  disc  would  give  out^two  tones  very 
near  one  another  in  pitch :  and  a  vibrating  sodium-molecule  gives  rise  to 
two  sets  of  ether-waves  which  differ  only  slightly  in  frequency. 


480  OF   ETHER-WAVES.  [CHAP. 

Frequency.  —  The  ether-waves  which  are  produced  by  the 
mechanical  vibrations  of  molecules  have  frequencies  which 
range  between  about  20,000000,000000  (Langley)  and  about 
4000,000000,000000  oscillations  per  second  —  a  range,  to  use 
a  musical  analogy,  of  about  eleven  octaves  :  but  of  these  our 
eyes  are  sensitive  to  scarcely  one  octave  —  to  those,  namely, 
which  range  between  about  392,000000,000000  per  second 
(extreme  red  of  the  spectrum),  and  about  757,000000,000000 
per  second  (extreme  violet).  Those  ether-waves  which  have 
been  produced  by  electric  methods  have  frequencies  ranging 
between  about  500  per  second  and  7500,000000  per  second. 

Velocity  and  Wave-Length.  —  These  waves  all  travel 
through  the  Ether  of  space  at  the  same  rate,  namely,  about 
30057,400000  cm.  (186,680  miles)  per  second.  Ether-waves 
while  traversing  the  Ether  present  no  essential  differences, 
except  in.  respect  of  their  wave-lengths;  the  wave-length  X 
is  equal  to  v/n,  where  v  is  the  velocity  of  propagation  and 
n  the  frequency. 

Those  ether-waves  which  are  produced  by  the  oscillations  of  molecules 
vary  accordingly  in  length,  in  a  vacuum,  from  about  -fa  cm.  to  about 
lt go^ooo  cm.,  and  those  waves  to  which  our  unassisted  eyes  are  sensitive, 
the  waves  of  Light,  have  wave-lengths  ranging  between  TJIJTS  cm>  an(l 
syinf  cm.  These  wave-lengths  are  usually  specified  in  terms  of  "tenth- 
metres  " ;  a  tenth-metre  being  1  metre  -=- 1010,  or  0-000000,01  cm.  Extreme 
red  and  extreme  violet  have  thus,  in  a  vacuum,  the  respective  wave-lengths 
of  7667  and  3970  tenth-metres.  Those  ether-waves  which  have  been  pro- 
duced by  electric  methods  have  wave-lengths  ranging  between  60,000000  cm. 
(373  miles)  and  4  cm. 

Ether-waves  do  not  traverse  all  substances  with  equal  speed: 
hence  their  wave-lengths  in  different  substances  vary;  if  any 
particular  kind  of  radiation  have  to  be  spoken  of,  it  may  be 
denned  by  specifying  its  wave-length  in  some  specified  medium, 
but  it  is  better  to  state  its  numerical  frequency.  To  do  the  lat- 
ter implies,  however,  that  we  assume  —  and  we  are  apparently 
justified  in  assuming  —  that  all  kinds  of  radiation  pass  through 
a  vacuum  —  that  is,  through  the  ether  of  space  —  with  equal 
speed. 

Kinds  of  Radiation.  —  When  a  succession  of  ether-waves 
impinges  on  a  mass  of  ordinary  matter,  the  effect  varies  accord- 
ing to  the  nature  and  the  condition  of  the  body  which  receives 
their  shock  ;  if  it  be  an  ordinary  opaque  mass,  that  mass  may  be 
warmed,  the  energy  of  wave-motion  being  transformed  into  heat, 
and  the  waves  which  have  impinged  upon  the  opaque  mass  are 


xv.]  KINDS  OF   RADIATION.  481 

ex  post  facto  called  a  beam  of  Radiant  Heat ;  if  they  fall  upon 
the  eye,  they  may  produce  a  sensation  of  light,  and  the  wave- 
system  is  then  called  a  beam  of  Light :  falling  upon  a  sensitised 
photographic  plate,  or  a  living  green  leaf,  they  may  operate 
chemical  decomposition,  and  the  wave-system  is  then  called  a 
beam  of  Actinic  rays.  The  word  "rays"  in  the  last  phrase 
may  be  understood  to  mean,  not  imaginary  lines  at  right  angles 
to  the  wave-front,  but  kinds  of  radiation;  and  hence  we 
speak  of  Heat  rays,  of  Light  rays,  of  Chemical  or  Actinic  rays ; 
these  names  being  given  to  one  and  the  same  train  of  waves 
according  to  the  effects  which  it  is  found  competent  to  pro- 
duce. But  while  ether-waves  are  in  course  of  traversing  the 
ether,  there  is  neither  Heat,  Light,  nor  Chemical  Decomposition  ; 
merely  wave-motion,  and  transference  of  energy  by  wave-motion. 
Hence  none  of  these  names  can  in  strictness  be  applied  to  a 
train  of  waves  while  these  waves  are  actually  travelling  through 
the  Ether. 

Ether-waves  which  differ  in  their  frequency  differ  to  some 
extent  in  their  degree  of  power  of  producing  the  motion  of  heat 
or  the  sensation  of  light,  or  of  doing  the  work  of  chemical  decom- 
position. All  ether-waves  can  produce  heat,  for  their  energy  is 
converted  into  heat  when  they  fall  upon  and  are  absorbed  by 
such  a  substance  as  a  thick  layer  of  lampblack,  which  for  the 
most  part  arrests  and  extinguishes  them. 

The  long  and  slow  ether-waves  which  have  been  produced 
by  electric  methods  are  not  so  readily  arrested  and  extin- 
guished as  those  produced  by  molecular  vibration;  they  can, 
therefore,  to  a  large  extent  traverse  such  an  obstacle  as  a 
brick  partition  or  a  deal  door ;  but  the  difference  in  this  respect 
between  them  and  ordinary  radiation-waves  is  a  question  of 
degree,  and  by  a  sufficient  obstacle  their  energy  can  be  reduced 
to  Heat. 

Those  ether-waves  which  take  their  origin  in  vibration  of 
ordinary  matter,  and  whose  frequency  is  less  than  392,000000,- 
000000  per  second,  are  too  slow  either  to  affect  the  eye  with  the 
sensation  of  light,  or,  in  the  ordinary  case,  to  impart  to  mole- 
cules an  agitation  brisk  enough  to  shake  them  to  pieces,  and 
thus  to  operate  chemical  decomposition.  Such  slow  waves, 
whose  presence  can  only  be  recognised  after  their  impact,  by  the 
conversion  of  their  energy  into  Heat,  are  called  Dark-Heat- 
Waves.  If  they  fall  upon  an  ordinary  photographic  plate  they 
do  not  operate  chemical  decomposition;  but  if  the  molecules 

2i 


482  OF   ETHER-WAVES.  [CHAP. 

upon  which  they  impinge  be  specially  heavy  and  complex,  even 
these  slow  heat-waves  may  be  found  to  toss  and  shake  them  with 
briskness  sufficient  to  break  them  up. 

Radiant  Heat  of  sufficient  intensity  is  thus  found  to  operate  chemical 
decompositions  of  a  different  order  from  those  brought  about  by  contact- 
heating;  e.g.,  the  formation  of  olefines  from  paraffin-vapours  (W.  Young), 
instead  of  that  of  aromatic  hydrocarbons  with  deposition  of  carbon. 

A  dark  nebula  in  the  Pleiades  was  first  photographed  and  then  with 
difficulty  seen.  Major  Abney  has  been  able  to  photograph  with  heat-rays 
down  to  a  frequency  of  about  160,000000,000000. 

The  waves  may,  on  the  other  hand,  be  so  rapid  —  above 
757,000000,000000  per  second — as  to  produce  no  visual  effect 
on  the  eye ;  the  eye  is  normally,  physiologically,  blind  to  them, 
and  is  unable  to  feel  their  impact ;  but  they  may  effect  chemical 
decomposition;  their  successive  impulses  may  aid  the  natural 
free  vibrations  of  the  molecule,  which  thus  become  increasingly 
ample :  and  just  as  a  resonant  tumbler  into  which  its  own  note  is 
steadily  sung  vibrates,  shivers,  and  breaks  into  fragments,  so 
a  molecule,  quivering  under  the  steady,  regular,  and  continu- 
ously well-timed  blows  of  the  rapid  ether-waves,  may  yield  and 
break  up  into  its  constituent  atoms,  or  into  groups  of  atoms, 
which  constitute  simpler  molecules.  Such  rapid  waves  are 
called  Invisible  or  Ultra-violet  Chemical  Rays. 

According  to  Lubbock,  ants  and  Daphnia  seem  to  be  able  to  see  ultra- 
violet rays. 

The  power  of  operating  chemical  decomposition  possessed 
by  the  more  rapid  waves  depends  more  upon  their  frequency 
than  upon  their  intensity. 

The  rays  which  are  most  active  in  decomposing  carbonic  acid  by  chloro- 
phyll are  visible,  being  the  yellow  and  the  red. 

For  waves  of  given  length  but  different  intensities,  the  quantity  of 
chemical  decomposition  is,  within  wide  ranges  of  intensity,  the  same  when 
the  product  (Time  of  Exposure  x  Intensity  of  Light)  is  the  same.  For  very 
feeble  intensities  this  product  must  be  increased ;  in  some  cases,  as  in  star- 
photography,  very  considerably.  This  law  is,  however,  the  basis  of  calcu- 
lations of  "  exposure  "  in  photography. 

The  slower  waves  may  thus  produce  heat,  or  perhaps 
chemical  decomposition  of  heavy  complex-molecules ;  waves 
of  medium  rapidity  may  produce  heat,  the  sensation  of  light, 
or  chemical  effect;  the  more  rapid  ones  may  produce  heat  or 
chemical  effect  according  to  the  substance  upon  which  they  fall. 

The  invisible  chemical  rays,  though  they  can  operate  chem- 
ical decomposition,  are  yet  of  very  feeble  physical  intensity; 


xv.]  KINDS   OF   KADIATION.  483 

their  aggregate  kinetic  energy  is,  in  the  radiations  from  the 
sun,  as  we  receive  them  filtered  through  our  atmosphere,  mil- 
lions of  times  less  than  that  of  the  slower  red  or  dark  heat 
rays:  even  those  rays  which  are  visible  are  effective  not  so 
much  in  virtue  of  their  intensity,  which  is  but  small,  as  in 
virtue  of  the  extraordinary  sensitiveness  of  the  eye  to  light  — 
that  is,  to  the  impact  of  ether-waves  of  a  certain  range  of 
frequency. 

Colour.  —  Within  the  limits  of  visibility  —  392  billions  to 
757  billions  —  there  is  an  indefinite  variety  of  integral  and  frac- 
tional numbers,  each  of  which  represents  the  frequency  of  a 
particular  kind  of  radiation,  a  particular  kind  of  light.  Physi- 
cally there  are  as  many  kinds  of  light  as  there  are  possible 
frequencies  between  the  limits  mentioned.  These  kinds  of 
light,  each  physically  characterised  by  the  number  of  waves 
which  strike  the  eye  during  a  second,  are  recognised  by  the 
normal  eye  as  being  distinct,  not  as  the  result  of  any  conscious 
process  of  counting  the  number  of  impulses  suffered  by  the  eye 
during  a  second,  which  would  be  absolutely  impossible,  but  in 
consequence  of  the  distinct  and  peculiar  Sensation  attending 
the  reception,  in  the  eye,  of  wave-motion  of  each  particular  fre- 
quency—  a  sensation  known  in  each  case  as  that  of  a  particular 
Colour.  Thus,  when  we  look  at  a  Bunsen  burner,  the  flame  of 
which  is  caused  to  emit  a  dingy-yellow  light  by  contact  with 
common  salt,  we  recognise  the  sensation  as  one  of  yellow  light. 
Colour  is  a  sensation :  it  is  not  a  material  existence ;  but  the 
physical  basis  and  cause  of  the  special  sensation  of  yellow  light 
is,  in  this  case,  the  joint  simultaneous  impact  on  the  eye  of  two 
kinds  of  ether-waves,  which  have  the  respective  frequencies  of 
508,905810,000000  and  510,604000,000000  per  second,  or  the  re- 
spective wave-lengths  in  air  of  5895  and  5889-04  tenth-metres. 

Either  of  these  trains  of  waves  impinging  singly  on  the  eye 
would  produce  (see,  however,  p.  575)  a  sensation  of  yellow,  the 
slower  one  giving  a  yellow  very  slightly  more  orange  in  its  tint 
than  the  other  does,  The  term  "  yellow  light,"  which  means 
primarily  a  certain  sensation,  means,  secondarily,  the  physical 
cause  of  this  sensation  —  that  is,  a  train  of  ether-waves  of  a  par- 
ticular frequency.  Any  particular  colour  is  best  specified  by  a 
statement  of  the  frequency  of  the  single  wave-motion,  which 
can  produce  that  colour  when  it  enters  the  eye  ;  and  the  analogy 
between  light  of  any  given  Colour  and  a  sound  of  any  given 
Pitch  is  obvious. 


484  OF  ETHER-WAVES.  [CHAP. 

When  there  fall  successively,  upon  the  normal  eye,  trains  of 
light-waves  which  differ  only  slightly  in  their  frequency,  the 
respective  colour-sensations  produced  by  them  may  resemble 
one  another  generically,  though  not  precisely.  When,  in  grad- 
ual succession,  luminous  waves  of  all  possible  frequencies  are 
caused  to  strike  the  eye,  we  obtain  in  successive  gradation  the 
sensations  of  all  the  colours  of  the  spectrum.  The  slowest 
waves  which  can  affect  the  eye  produce  a  sensation  of  red, 
those  somewhat  more  rapid  a  sensation  of  scarlet;  then  in 
succession  we  find,  as  the  frequencies  increase,  that  the  sensa- 
tions produced  are  those  of  orange-red,  reddish-orange,  orange, 
yellow-orange,  orange-yellow,  yellow,  greenish-yellow,  yellowish- 
green,  green,  bluish-green,  greenish-blue,  blue,  blue-violet,  vio- 
let. Waves  of  still  greater  rapidity  than  those  which  produce 
the  sensation  of  violet  are  practically  invisible ;  but  it  must  be 
admitted  that  they  are  not  perfectly  so. 

Even  beyond  the  ordinary  range  of  visibility  some  eyes  are  affected 
by  ultra-violet  ether-waves ;  a  sensation  of  lavender-gray  colour  results :  a 
spectrum  is  often  seen,  especially  if  the  dispersion  be  small,  to  contain  three 
bright  bands  of  lavender-gray  in  the  ultra-violet  region.  This  light  is,  in 
intensity,  about  l-1200th  part  of  that  which  shines  in  the  same  region  of  the 
spectrum  when  it  is  rendered  visible  by  fluorescence.  Beyond  the  red  there 
is,  similarly,  a  crimson. 

The  table,  page  485,  modified  from  Ogden  Rood's  Modern 
Chromatics  and  Lord  Kelvin's  Royal  Instit.  Lecture,  Feb.  2, 
1883,  gives  the  frequencies  and  the  wave-lengths  in  air  of  the 
several  undulations  which  correspond  to  the  several  leading 
colours  of  the  spectrum,  and  to  some  of  the  so-called  Fraun- 
hofer  Lines. 

When  a  source  of  light  is  receding  from  the  eye,  fewer  waves  per  second 
strike  the  eye ;  the  light  approximates  towards  red.  Conversely,  the  light 
of  an  approaching  luminous  object  is,  as  it  were,  sharpened  in  pitch.  The 
characteristic  lines  in  the  spectrum  are  thus  somewhat  displaced ;  and  by 
this  application  of  Db'ppler's  principle,  the  speed  of  relative  approach 
or  recession  of  the  earth  and  many  fixed  stars  has  been  estimated. 

That  which  we  call  white  light  is,  in  the  state  in  which  we 
receive  it  from  such  a  body  as  a  white-hot  bar  of  iron  or,  per- 
haps in  its  purest  form,  from  the  crater  of  the  positive  pole  of 
the -electric  arc,  a  mixture  of  long  and  short  waves;  waves  of 
all  periods  within  the  range  of  visibility  are  either  continuously 
present,  or,  if  absent  for  a  time,  are  absent  in  such  feeble  propor- 
tions or  for  such  short  intervals  that  they  are  not  appreciably 
missed  by  the  eye.  White  light  of  this  kind  is  comparable  to  an 


XV.] 


COLOUR. 


485 


utterly-discordant  chaos  of  sound  of  every  audible  pitch ;  such 
a  noise  would  produce  no  distinct  impression  of  pitch  of  any 
kind;  and  so  white  light  is  un coloured. 


Frequencies. 

Wave-lengths 
in  centimetres. 
(Angstrom.) 

Line  A     
Centre  of  red         ..... 

395,000000,000000 
429,400000,000000 

•00007604 
•00007000 

Line  B 

437,300000,000000 

•00006867 

457,700000,000000 

•00006562 

Centre  of  orange-red     .... 
Centre  of  orange  
Line  D1     

484,000000,000000 
503,300000,000000 
508,905810,000000 

•00006208 
•00005972 
•00005895 

Line  D2     
Centre  of  orange-yellow 

510,604000,000000 
511,200000,000000 
517,500000,000000 

•00005889 
•00005879 
•00005808 

Centre  of  green     ..... 
Line  E      

570,200000,000000 
570,500000,000000 
580,000000,000000 

•00005271 
•00005269 
•00005183 

Centre  of  blue-green      .... 
Centre  of  cyan-blue       .... 
Line  F      

591,400000,000000 
606,000000,000000 
617,900000,000000 
635,200000,000000 

•00005082 
•00004960 
•00004861 
•00004732 

Centre  of  violet-blue      .... 
Line  G      ...... 
Centre  of  puce-violet     . 
Line  H1     
Line  H2    ...... 

685,800000,000000 
697,300000,000000 
740,500000,000000 
756,900000,000000 
763,600000,000000 

•00004383 
•00004307 
•00004059 
•00003968 
•00003933 

If  a  parallel  beam  of  light  of  one  kind,  one  wave-length, 
one  colour,  —  homogeneous  or  monochromatic  light,  —  be 
caused  to  pass  through  a  slit  in  an  opaque  screen,  it  may  be 
received  upon  a  white  screen,  and  it  will  cast  upon  that  screen 
a  coloured  image  of  the  slit.  If  the  light,  on  issuing  from  the 
slit,  instead  of  being  received  directly  upon  a  screen,  be  made  to 
pass  through  a  glass  prism,  the  narrow  edge  of  which  is  held 
parallel  to  the  slit,  it  will  be  refracted  by  that  prism,  and  the 
image  of  the  slit  will  now  be  found  in  a  new  position  on  the 
screen.  If  a  beam  of  white  light  be  so  dealt  with,  a  number  of 
coloured  images  of  the  slit  will  be  formed,  each  in  its  proper 
place  on  the  screen,  each  image  overlapping  its  neighbour  if  the 
slit  be  of  appreciable  width;  there  will  thus  be  formed  a  many- 
coloured  band  of  light,  in  which  the  colours  are  marshalled  in 
the  order  of  the  frequency  of  their  waves,  —  the  slpwest  waves, 
the  red,  being  least  refracted  by  the  glass  prism ;  the  quickest 


486  OF  ETHER-WAVES.  [CHAP. 

waves,  the  violet,  being  most  refracted.  This  is  the  spectrum : 
every  component  of  the  original  white-light  is  displayed  in  the 
spectrum,  each  in  its  distinct  place  ;  and  thus  the  prism  fur- 
nishes us  with  a  means  of  analysing  light  —  that  is,  of  finding 
what  its  components  are. 

But  the  spectrum  extends  be}^ond  the  visible  part  of  it ; 
the  more  rapid  invisible  rays,  being  more  refrangible  than  the 
violet,  form  an  invisible  part  —  an  ultra-violet  region  —  which 
we  detect  by  the  phenomena  of  fluorescence  (p.  504),  or  by 
casting  the  whole  spectrum  upon  a  sensitive  photographic-plate, 
upon  which  we  afterwards  find  a  record  of  a  region  of  the  spec- 
trum invisible  to  the  eye ;  and  the  slower  dark-heat  rays  form 
an  invisible  part  of  the  spectrum  beyond  the  red,  the  heat 
spectrum  or  ultra-red  region,  not  visible,  but  demonstrable 
by  means  of  any  apparatus,  such  as  a  thermometer  or  a  thermo- 
pile (Fig.  212),  which  is  sensitive  to  heat.  If  the  prism  used 
be  made  of  quartz,  or  if  the  spectrum  be  produced  by  reflexion 
from  a  diffraction-grating  (p.  549),  it  will  be  found  that  the 
ultra-violet  region  is,  if  the  light  analysed  be  that  of  the  elec- 
tric arc,  from  six  to  eight  times  as  long  as  the  whole  of  the 
visible  part  of  the  spectrum;  while  if  the  prism  used  be  of 
glass,  it  absorbs  to  a  remarkable  degree  these  rapid  ultra-violet 
waves.  If  the  light  analysed  be  that  of  the  sun,  the  ultra-violet 
part  of  the  spectrum  is  comparatively  very  short,  on  account  of 
absorption  by  the  atmosphere. 

This  effect  of  the  atmosphere  is  of  extreme  importance.  Sunlight  is 
originally  bright  blue,  and  is  extremely  rich  in  the  more  refrangible  rays, 
but  filtration  through  two  absorbent  atmospheres  —  that  of  the  sun  and  that 
of  the  earth  —  renders  it  a  yellowish-white  (Langley).  The  ultra-violet  part 
of  the  spectrum  is  enormously  brighter  at  high  altitudes. 

Compound  Coloured-Light.  —  Let  us  now  cast  a  beam  of 
sunshine  or  of  electric  light,  shining  through  a  slit  in  an  opaque 
screen,  upon  a  piece  of  greenish-blue  glass,  and  receive  upon  a 
white  screen  the  light  which  passes  through  this  coloured  glass: 
by  the  aid  of  a  lens  we  may  obtain  a  greenish-blue  image  of  the 
slit  upon  the  screen.  So  far  as  we  have  yet  learned,  such 
coloured  light,  whatever  be  the  mechanism  of  its  production,  is 
a  single  kind  of  light  —  perhaps  due  to  waves  of  only  a  single 
frequency:  whether  this  be  so  in  the  particular  case  may  be 
tested  by  interposing  a  prism  in  the  path  of  the  coloured  beam 
of  light:  if  the  greenish-blue  light  be  homogeneous,  we  shall 
again  have  on  the  screen  an  image  of  the  slit,  altered  in  position, 


xv.]  COLOUR.  487 

but  not  in  colour.  This  is  not  what  we  find :  a  short  and  imper- 
fect spectrum  is  produced ;  the  transmitted  greenish-blue  light 
is  analysed  by  the  prism  into  green  light,  blue  light,  yellow 
light,  with  perhaps  some  other  colours,  more  or  less  faintly 
represented. 

This  phenomenon  is  very  singular.  It  shows  that  two  widely-differing 
physical  causes  are  capable  of  producing  exactly  the  same  colour-sensation  : 
the  one  being,  as  we  have  already  seen,  the  impact  of  ether-waves  of  a  single 
definite  frequency,  the  other  being  the  joint  impact  on  the  retina  of  a 
number  of  wave-systems,  each  of  which  is  capable,  if  it  were  to  act  inde- 
pendently, of  producing  a  distinct  sensation ;  and  the  colour-sensation  which 
is  produced  by  the  joint  action  of  these  wave-systems  may  differ  from  that 
which  characterises  any  one  of  them.  It  is  as  if  a  listener  to  concerted 
music  were  to  hear  the  strains  of  an  orchestra  compounded  into  some  sort 
of  loud  melody  of  average  pitch,  he  being  wholly  unable,  by  his  unaided 
ear,  to  recognise  the  really  compound  nature  of  the  sound  heard  by  him. 
Then,  whether  the  instruments  all  played  in  unison  or  diverged  into  pre- 
calculated  harmony,  the  effect  on  his  ear  might  remain  the  same. 

Further,  many  such  mixtures  may  produce  the  same  apparently  simple 
sensation ;  and,  accordingly,  such  a  phrase  as  "  green  light "  or  "  orange 
light "  is  perfectly  vague,  unless  it  be  accompanied  by  a  specification  of  its 
physical  cause. 

Complementary  Colours.  —  The  greenish-blue  glass,  in  the 
instance  just  alluded  to,  has  in  whole  or  in  part  prevented 
the  transmission  of  violet  light,  of  red,  of  orange,  and  of  other 
kinds  of  light  which  are  present  in  white  sunlight ;  the  complex 
of  undulations  thus  denied  transmission  would,  if  collectively 
allowed  to  impinge  on  the  eye,  have  produced  a  single  sensation 
of  red  light.  If  this  compound  red-light  had  not  been 
obstructed  by  the  coloured  glass,  the  transmitted  beam  would 
have  been  white ;  this  compound  red-light  thus  obstructed  by 
the  greenish  glass,  and  the  compound  greenish-light  transmitted 
by  it,  will  pass  together  through  a  piece  of  clear  glass,  and  will 
together  produce  the  sensation  of  white  light.  To  the  eye  it  is 
a  matter  of  indifference  whether  the  red  or  the  greenish  light  be 
monochromatic  or  compound ;  monochromatic  red-light  and 
monochromatic  greenish-blue  light,  allowed  to  fall  upon  the 
same  spot  in  the  eye',  will  mingle,  and,  if  they  be  of  the  proper 
hue,  will  produce  the  compound  sensation  of  white  light.  These 
colours,  red  and  greenish-blue,  each  of  the  proper  hue,  are  thus 
complementary  to  one  another ;  together  they  make  up  white 
light. 

The  following  pairs  of  colours  are,  among  others,  thus  complementary 
to  one  another :  —  Red  and  a  very  greenish  blue,  orange  and  cyan-blue  (a 
rather  greenish  blue),  yellow  and  ultramarine  blue,  greenish-yellow  and 


488  OF  ETHER-WAVES.  [CHAP. 

violet,    green  and  "  purple,"  the  latter  being  a  colour  not  in  the  spectrum, 
but  formed  by  the  superposition  of  blue  and  red. 

The  expression  "  white  light,"  standing  alone,  is  thus  also 
wholly  vague  ;  physiologically  it  means  light  which  produces  the 
sensation  of  white ;  physically  it  may  mean  (1)  a  mixture  of  all 
possible  light-waves,  long  and  short,  in  certain  proportions ;  or 
(2)  a  mixture  of  two  complementary  simple  colours  ;  (3)  a  mix- 
ture of  two  complementary  compound  colours ;  or  (4)  a  simple 
colour  blended  with  a  complementary  compound  one  of  any 
degree  of  complexity. 

The  white  light  of  sunlight  at  sea-level  is  made  up  (Vierordt  and  Rood) 
by  a  mixture  (  =  1000)  of  the  following  coloured  lights  :  —  Red,  54  ;  Orange- 
red,  140;  Orange,  80;  Orange-yellow,  114;  Yellow,  54;  Greenish-yellow, 
206 ;  Yellowish-green,  121 ;  Green  and  blue-green,  134 ;  Cyan-blue,  32 ; 
Blue,  40  ;  Ultramarine  and  blue-violet,  20  ;  Violet,  5. 


RADIATIONS  OF  A  HOT  BODY. 

The  hotter  a  body,  the  greater  the  intensity  of  the  aggre- 
gate disturbance  which  it  sets  up  in  the  Ether ;  and  further,  the 
greater  the  frequency  of  the  most  rapid  components  of  that  dis- 
turbance. A  white-hot  iron  ball  is  visible  in  a  dark  room ;  it 
emits  dark  heat-rays,  light-rays,  and  also  the  rapid  ultra-violet 
rays :  it  can  be  seen  and  photographed,  and  its  warmth  can  be 
felt  at  a  distance.  If  it  be  intensely  hot  it  may  emit  so  great  a 
proportion  of  violet  and  blue  light  that  it  appears  bluish ;  it  is 
"  blue-hot." 

As  it  cools  down,  the  more  rapid  vibrations  die  away ;  the 
ultra-violet  waves  cease  to  be  formed ;  the  mass  becomes  some- 
what less  easy  to  photograph  by  its  own  light.  Gradually  the 
violet  rays  cease  to  be  emitted;  the  light  radiated  is  now 
apparently  tinged  with  yellow :  the  apparent  colour  becomes 
orange,  then  red ;  a  body  at  a  red-heat  is  difficult  to  photograph, 
though  it  continues  perfectly  visible  in  the  dark.  When  its 
temperature  sinks  to  a  point  below  525°  C.,  it  ceases  to  radiate 
light  and  becomes  invisible  in  the  dark ;  it  continues,  however, 
to  radiate  heat,  as  may  be  felt  for  some  time  by  the  cooler  hand 
placed  near  it. 

H.  F.  Weber  points  out  that  platinum  at  390°  C.,  gold  at  417°  C.,  and 
iron  (not  quite  free  from  rust)  at  377°  C.,  become  faintly  visible,  first  fog- 
gray,  then  ash-gray,  then  yellowish-gray,  then  faintly  red,  then  red-hot,  and 
so  on. 


xv.]  RADIATIONS   OF  A  HOT  BODY.  489 

The  luminous  radiations  of  an  Argand  oil-lamp  are  \  %  of  the  whole : 
of  a  gas-flame,  0-3  to  1;  a  Welsbach  incandescent  gas-lamp,  1£;  an  electric 
glow-lamp,  5-6 ;  a  small  electric  arc,  5-10 ;  a  5000-candle  arc,  at  3000°  C., 
25%.  Of  the  solar  radiation,  25  %  is  luminous  (Sir  C.  W.  Siemens). 

It  never  ceases  to  radiate  heat;  it  could  not  cease  to 
do  so  unless  it  were  cooled  down  to  absolute  zero.  Since  the 
molecules  of  all  bodies  are  in  repeated  collision  with  their  fel- 
low-molecules, as  they  rebound  at  each  collision,  they  shiver 
and  they  vibrate.  They  must  therefore  continuously  originate 
ether-waves  —  waves  which,  when  the  temperature  of  the  body 
is  below  525°  C.,  are  too  slow  to  affect  the  eye. 

Exchange  of  Radiations.  —  Two  bodies  placed  opposite  to 
one  another,  with  intervening  Ether,  of  which  we  cannot  get  rid, 
and  with  or  without  intervening  air,  may  present  the  two  fol- 
lowing cases : — 

1.  Both  may  be  of  the  same  temperature,  in  which  case  the 
one  loses  by  imparting  to  the  other  exactly  as  much  energy  as  it 
takes  up  from  those  ether- waves  which  strike  it,  having  been  origi- 
nated by  the  other  hot  body;  whence  two  bodies  equally  hot 
exchange  their  energies  by  radiation,  but  do  this  to  an  equal  ex- 
tent, and  there  is  thus  no  change  in  their  relative  temperatures. 

2.  The  one  may,  on  the  other  hand,  be  hotter  than  the 
other.     The  hotter  body  sets  up  a  more  vehement  system  of 
ether-waves  than  the  colder  one  can ;  in  doing  this  it  expends 
its  energy  to  a  greater  extent  than  the  colder  one  does ;  the 
hotter  loses  more  energy  than  it  gains ;  the  colder  gains  more 
than  it  loses ;   in  course  of  time  their  energies,  and  therefore 
their  temperatures,  become  equal :  when  the  temperatures  have 
become   equal,   though  the   two   bodies  still  go   on  imparting 
energy  to  each  other,  neither  profits  by  the  exchange,  and  their 
temperatures  remain  relatively  equal. 

The  absolute  amount  of  radiation  of  energy  from  a  body 
does  not  depend  on  the  condition  or  even  on  the  presence  of 
surrounding  objects,  but  solely  on  the  condition  of  the  body 
itself.  It  is  easy  to  see  that  the  absolute  physical  brightness 
of  the  sun  or  of  a  candle  is  at  any  moment  independent  of  the 
presence  of  illuminated  objects ;  it  is  not,  however,  at  first  sight 
so  clear  that  not  only  does  a  fire  warm  the  walls  of  a  room,  but 
these  walls  also  warm  the  fire ;  that  the  sun  warms  the  earth  while 
the  earth  —  to  a  lesser  extent,  it  is  true  —  warms  the  sun  ;  and 
that  the  warming  of  a  colder  body  by  a  hotter  one  depends  upon 
the  difference  of  two  similar  but  unequally-opposed  actions. 


490 


OF   ETHER-WAVES. 


[CHAP. 


When  a  lump  of  ice  is  placed  near  an  object  at  the  ordinary  temperature, 
that  object  is  cooled  ;  it  loses  to  the  ice  more  heat  than  it  gets  from  the  ice : 
the  ice  apparently  radiates  cold. 

When  one  body  is  surrounded  by  another,  the  body  enclosed 
and  the  walls  of  the  enclosure  come  to  have  the  same  temper- 
ature, if  they  be  relatively  at  rest.  A  thermometer  whose  bulb 
is  immersed  in  a  cavity  will  come  to  indicate  the  temperature  of 
the  walls  of  the  cavity,  whether  it  be  in  contact  with  them  or 
not.  This  equalisation  of  temperature  by  radiation  is  quite  inde- 
pendent of  the  form  of  the  walls  of 
the  cavity;  a  cavity  of  any  form 
acts  in  the  same  way  as  a  spherical 
cavity  would  do.  In  Fig.  154  the 
irregular  hollow  body  ABC  sur- 
rounds a  body  E ;  both  E  and 
ABC  assume  after  some  time  a 
common  temperature,  and  remain 
at  an  equal  temperature. 

The  irregular  hollow  body  ABC  might 
be  replaced  by  the  hollow  spherical- 
body  FGH,  or  by  the  hollow  sphere  KLM, 

or  any  other  hollow  sphere  concentric  with  these.      From  this  ensue  the 

following  propositions. 

1.  The  amount  of  energy  received  by  a  receiving  surface  per  unit  of  its 
area  —  the  amount  of  heat  received,  the  brightness  of  light  there  —  varies 
inversely  as  the  square  of  the  distance  from  the  source  of  radiation.     The 
advantage  of  extensive  surface  possessed  by  the  larger  sphere  KLM  is  ex- 
actly neutralised  by  its  disadvantage  of  distance ;  its  surface  is  greater,  the 
radiation  received  by  it  per  unit  of  area  is  less,  both  in  the  ratio  of  the 
squares  of  the  radii,  and  the  total  radiation  received  by  it  is  the  same, 
whatever  be  its  radius. 

A  candle  at  a  distance  of  1  foot  can  illuminate  a  printed  page  as 
brightly  as  a  25-candle  gas-burner  at  a  distance  of  5  feet. 

A  bright  wall  is  equally  bright  at  all  distances  when  looked  at  through 
a  narrow  conical  tube.  Close  at  hand  it  appears  brighter,  area  for  area,  but 
less  of  it  can  be  seen;  at  a  distance  it  appears  dimmer,  but  more  of  its 
surface  can  be  seen ;  in  all  cases  the  amount  of  light  falling  on  a  given  area 
of  the  retina  is  the  same. 

2.  When  a  plane  wave  whose  area  is  AB  strikes  squarely  and  simul- 
taneously all  parts  of  a  surface  whose  area  is  also  AB  —  the  normals  to  the 
wave  being  also  normals  to  the  receiving  surface  —  the  receiving  surface 
receives  a  certain  number  of  units  of  energy  per  second.    In  Fig.  155  AB  is 
a  hot  or  bright  body  radiating  ether-waves  towards  CD ;  CD  receives  e  units 
of  energy  per  second  per  unit  of  its  area. 

If  the  receiving  surface  be  tilted,  say  into  the  position  DE,  the  wave- 
front,  striking  obliquely,  is  now  able  to  cover  the  larger  surface  DE ;  no 
more  of  the  wave  can  now  reach  DE  than  would  previously  have  reached 


XV.] 


EXCHANGE   OF  RADIATIONS. 


491 


CD.     The  same  quantity  of  energy  is  thus  distributed  over  a  larger  surface  : 
the  quantity  of  energy  communicated  to  it  per  unit  of  its  area  is  dimin- 


ished in  the  ratio 


or  is  equal  to  (e  x  cos  CDE)  units. 

In  accordance  with  this,  the  intensity  of  sunlight  at  noon  is  greater 
than  during  the  earlier  and  later  portions  of  the  day,  when  the  surface  of 
the  earth  is  presented  obliquely  to  the  sun's  radiation. 

•          Fig.155. 

c 


3.  Let  AB  receive  energy  from  CD  or  DE ;  then,  whether  the  surface 
be  the  smaller  CD  vertically  facing  it,  or  the  larger  DE  arranged  obliquely, 
is  a  matter  of  indifference ;  in  either  case  there  will  be  radiated  towards 
AB  the  same  amount  of  energy.     DE  therefore  radiates  towards  AB,  in  the 
direction  DB,  less  energy  per  unit  of  its  surface  than  CD  does  in  that  direc- 
tion when  equally  heated,  and  that  in  the  ratio  of  cos  CDE :  1. 

Were  this  not  so,  and  did  a  hot  surface  radiate  equally  in  all  directions, 
then  a  body  placed  within  a  hot  enclosure  might  become  hotter  than  the 
walls  of  that  enclosure. 

This  principle  explains  the  apparent  uniformity  of  brightness  of  the 
sun's  disc.  Towards  the  margin  of  the  sun's  apparent  disc,  areas  which 
seem  equal  to  similar  areas  near  the  centre  are  in  reality  much  larger ;  but 
we  see  them  obliquely ;  their  larger  superficial  area  exactly  compensates 
the  effect  of  their  oblique  aspect. 

4.  Radiation  reflected  from  a  mirror  to  a  focus  can  never  make  an 
object  placed  at  the  focus  radiate  more  energy  per  sq.  cm.  than  the  Source 
does ;  the  temperature  of  the  object  cannot  exceed  that  of  the  Source  ;  but 
the  object  may,  if  sufficiently  small,  come  to  the  same  temperature  as  the 
source,  after  which  there  is  between  it  and  the  source  an  equilibrium  of 
radiation.     Whence  a  thin  wire  in  the  focus  of  a  very  large  mirror  in  sun- 
light ought  (atmospheric  absorption,  etc.,  apart)  to  come  up  to  the  Sun's 
Temperature  (3000°  C.,  Siemens,  but  this  is  apparently  too  low  a  figure), 
but  not  to  exceed  it.     Ericsson's  Sun-motor  is  practically  a  huge  parabolic 
mirror,  in  whose  focus  a  high  temperature  is  attained,  which  is  utilised  by 
an  engine. 

The  law  just  stated  —  that  bodies  are  always  radiating  and 
receiving  energy  —  that  the  amount  of  radiation  depends  on  the 
temperature  of  the  radiating  body  —  that  at  constant  tempera- 
tures bodies  radiate  as  much  energy  as  they  receive  —  is  known 
as  Pre vest's  Law  of  Exchanges.  From  this  it  follows  that 
good  radiators  are  good  absorbents;  and  conversely,  good 
absorbents  are  good  radiators. 

If  a  hot- water  vessel  be  intended  to  retain  its  heat  for  a  comparatively 
long  period  in  the  open  air,  it  must  be  polished  externally ;  a  polished  sur- 
face, being  a  good  reflector,  is  a  bad  absorbent,  and  is  therefore  a  bad 
radiator;  while  a  blackened  surface,  being  a  good  absorbent,  is  a  good 


492  OF  ETHER-WAVES.  [CHAP. 

radiator,  and  heat  is  with  comparative  rapidity  lost  through  a  coating  of 
lampblack,  provided  that  it  be  not  so  thick  as  to  impede  conduction  of  heat 
to  the  surface. 

Prevost's  Law  is  not  only  true  of  the  aggregate  energy 
gained  or  lost  by  a  body  through  radiation ;  it  is  also  true,  as 
Balfour  Stewart  pointed  out,  with  regard  to  each  particular 
form  or  kind  of  radiation  by  means  of  which  energy  may  be 
conveyed  between  neighbouring  objects. 

If  a  piece  of  yellow  glass  be  placed  within  a  hot  shell  of  iron,  the  glass 
and  the  iron  may  both  shine  by  their  own  light,  and  the  glass  may  be  looked 
at  through  a  minute  aperture  in  the  wall  of  the  hollow  shell.  Yellow  glass 
absorbs  ultramarine  light,  and  a  white-hot  object,  looked  at  through  it, 
appears  yellow,  provided  that  the  glass  be  colder  than  the  source  of  the 
white  light ;  but  when  the  yellow  glass  is  itself  as  hot  as  the  source  of 
white  light,  as  it  must  be  in  this  instance,  in  which  we  look  through  the 
white-hot  glass  at  the  white-hot  wall  of  the  iron  shell,  the  glass  seems  per- 
fectly transparent  to  the  whole  white  light,  —  a  phenomenon  which  may  be 
interpreted  as  showing  that  while  the  glass  only  transmits  yellow,  it  itself 
radiates  blue  light ;  the  aggregate  radiations,  the  transmitted  yellow  and  the 
radiated  blue,  produce  in  the  eye  an  aggregate  effect  of  pure  white.  If 
the  yellow  glass  be  hotter  than  the  source  of  light  behind  it,  it  seems  rela- 
tively blue.  The  conclusion  is,  that  as  yellow  glass  absorbs  blue  light,  so 
when  itself  heated  it  radiates  blue  light. 

Stokes' s  Law.  —  A  body  which  absorbs  any  particular  kind 
of  radiation  will  in  general,  when  heated,  become  a  source  of 
radiation  of  the  same  kind ;  just  as  a  resonator  will,  when  it 
vibrates,  impart  to  the  air  the  same  kind  of  sound  of  which  it 
may  rob  the  air  when  it,  the  resonator,  is  relatively  at  rest. 

If  a  screen  of  strings  tuned,  say  to  the  note  of  a,  be  arranged 
between  a  sounding  a  organ-pipe  and  a  listener,  the  latter  will 
hear  comparatively  little  of  the  sound  produced  by  the  pipe  ;  by 
resonance  the  strings  have  taken  up  the  energy,  and  have  con- 
verted part  of  it  into  Heat.  If  a  mixed  sound  were  produced  on 
the  farther  side  of  such  a  screen,  the  sound  of  a  would  not  be 
transmitted  to  the  listener ;  the  rest  of  the  mixed  sound  would 
be  heard  by  him. 

When  mixed  ether-waves  strike  a  system  of  molecules  of 
which  some  are  tuned  to  particular  frequencies,  those  molecules 
will  take  up  the  energy  of  vibrations  of  those  frequencies :  the 
body  will  appear  to  be  opaque  to  the  corresponding  waves. 

From  the  reciprocity  of  absorption  and  radiation  it  follows 
that  if  a  given  substance  be  divided  into  portions,  of  which  the 
one,  A,  is  hot,  while  the  other,  B,  is  comparatively  cool,  radia- 
tions from  A  will  be  absorbed  by  B ;  the  cooler  portion,  B,  is 


xv.]  STOKES'S   LAW.  493 

opaque  to  radiations  from  the  hotter  portion,  A.  Thus,  if  car- 
bonic oxide  be  burned,  its  flame  contains  hot  carbonic-acid;  the 
radiations  from  such  a  flame  cannot  pass  through  pure,  com- 
paratively cool  carbonic-acid,  and  are  checked  in  very  large  pro- 
portion by  air  containing  even  a  very  small  percentage  of  that 
gas  or,  curiously,  of  CS2-vapour. 

A  hydrogen  flame  contains  hot  aqueous-vapour ;  the  heat  radiated  from 
this  —  very  slow  dark  heat-waves  —  cannot  pass  through  comparatively  cool 
aqueous-vapour :  the  result  is,  as  Prof.  Tyndall  showed,  that  while  the  sun's 
light  and  heat  can  reach  the  earth's  surface  through  the  humid  atmosphere, 
their  effect  is  to  warm  the  earth  and  cause  it  to  produce  slow  waves  of  dark 
heat;  these  resemble  in  frequency  the  waves  produced  by  hot  aqueous- 
vapour  in  a  hydrogen  flame,  and  they  cannot  pass  away  through  the  aque- 
ous vapour  of  the  atmosphere.  The  atmosphere  thus  acts  as  a  kind  of 
heat-trap,  and  the  surface  of  the  earth  is  preserved  from  extremes  of  cold 
produced  by  excessive  radiation.  Prof.  S.  P.  Langley  has  shown  that, 
besides  this,  the  atmosphere  is  itself,  independently  of  aqueous  vapour, 
remarkably  opaque  to  certain  heat-waves  of  great  length,  which  are  radi- 
ated outwards  from  the  soil,  but  which,  being  absorbed  by  the  atmosphere 
and  spent  in  warming  it,  are  trapped  by  it ;  that  these  same  heat-rays,  on 
their  way  from  the  sun,  are  absorbed  by  our  atmosphere  and  never  actu- 
ally reach  the  earth ;  and  that  were  it  not  for  the  atmosphere  the  earth's 
temperature  would  be  below  —  200°  C.,  even  under  the  vertical  rays  of  a 
tropical  sun.  Thick  glass  has  also  a  remarkable  effect  of  this  kind. 

Burning  sodium-vapour  emits  a  particular  yellow  light ;  if 
looked  at  through  a  mass  of  sodium-vapour,  it  can  hardly  be 
seen ;  sodium-vapour  absorbs  the  light  given  out  by  hotter 
sodium-vapour.  Even  though  light,  of  that  particular  kind,  do 
not  happen,  in  any  particular  instance,  to  have  been  emitted  by 
burning  sodium,  if  the  attempt  be  made  to  transmit  it  through 
sodium-vapour  the  sodium-vapour  will  be  found  opaque  to  that 
kind  of  light.  If  an  electric  lamp  produce  a  beam  of  light  which 
contains  amongst  others  this  particular  kind  of  light,  and  if  a 
spirit  lamp  have  salt  (NaCl)  or,  better,  NaBr,  placed  in  its  wick 
so  that  it  gives  out  this  particular  yellow  light  (this  denoting 
that  the  spirit-lamp  flame  contains  incandescent  sodium-vapour)  ; 
and  if  the  electric  arc  be  looked  at  through  the  spirit-lamp  flame, 
then  the  colour  of  its  light  would  appear,  if  the  eye  were  suffi- 
ciently sensitive,  to  be  altered ;  it  is  bluer ;  the  sodium-yellow 
light  of  the  electric  arc  is  absorbed  as  it  passes  through  the  com- 
paratively cool  spirit-lamp  flame,  which,  by  its  own  compara- 
tively-feeble radiation,  does  not  repair  the  damage  done  by  it, 
and  the  light  which  has  passed  through  the  spirit-flame  is  com- 
paratively (not  absolutely)  wanting  in  that  particular  kind  of 
yellow.  The  beam  may  be  made,  after  passing  through  the 


494  OF   ETHER-WAVES.  [CHAP. 

sodium-vapour,  to  traverse  a  slit  and  a  prism,  and  thus  to  form 
a  spectrum  on  a  screen.  It  will  be  found,  if  this  be  done,  that 
the  spectrum  is  discontinuous  ;  at  the  place  where  the  particular 
yellow  light  ought  to  have  been  found,  and  would  have  been 
found  had  no  spirit-lamp  flame  intervened,  we  find  a  dark  line 
—  a  dark  image  of  the  slit,  which,  if  the  slit  be  fine  and  the 
focussing  accurate,  is  found  to  be  a  double  line  ;  a  line  not  abso- 
lutely lightless,  but  shining  with  the  comparatively -feeble  rays  of 
the  spirit-lamp,  and  therefore  dark  in  comparison  with  its  envi- 
ronment. If  the  temperature  of  the  spirit-flame  be  increased,  the 
dark  line  brightens  up;  if  the  temperature  of  the  absorber  be 
equal  to  that  of  the  source,  there  is  no  dark  line ;  if  the  temper- 
ature of  the  absorber  be  higher  than  that  of  the  source,  more  of 
the  particular  light  is  emitted  than  is  absorbed  by  it,  and  the 
line  is  relatively  bright.  The  prism,  which  resolves  any  com- 
pound light  into  differently-coloured  linear  images  of  a  slit,  — 
images  which  stand  side  by  side  so  closely  as  to  blend  into  one 
another,  but  any  defect  or  redundancy  of  brightness  in  any  one 
or  in  any  group  of  which  can  be  at  once  detected,  —  offers  a 
more  delicate  means  of  investigation  than  the  eye  can  afford.  In 
Spectrum  Analysis  a  prism  or  a  diffraction-grating  is  used,  to 
disperse  into  a  spectrum  the  light  which  passes  through  a  nar- 
row slit  from  a  luminous  body;  by  inspection  of  the  spectrum 
we  can  at  once  see  what  kinds  of  light  are  emitted,  and  what 
kinds  are  not  emitted,  by  a  luminous  body.  But  the  kinds  of 
light  emitted  by  incandescent  substances  are  generally  (since 
they  depend  on  the  vibrational  frequencies  of  the  molecules  of 
the  substances)  distinctively  characteristic  of  each  chemical 
element,  and,  to  a  certain  extent,  of  each  physical  state  —  of 
each  degree  of  temperature  —  and  even  of  the  chemical  consti- 
tution of  the  incandescent  substance. 

The  spectrum  of  the  limelight  is  continuous ;  that  of  the 
sun  is  not.  It  presents  dark  lines;  among  others,  the  double 
sodium-line:  the' presence  of  this  indicates  a  bright  central 
source  of  light,  a  hot  region  of  the  sun's  atmosphere,  containing 
incandescent  sodium-vapour,  the  light  from  which  is  absorbed  by 
the  cooler  sodium-vapour  in  the  upper  and  cooler  regions  of 
the  same  atmosphere.  These  lines,  discovered,  by  Fraun- 
hof  er,  and  named  after  him,  are  distinguished  by  letters;  and 
the  best-marked  of  the  numerous  Fraunhofer-lines  are  known  as 
A,  B,  and  C  in  the  red,  D  (a  double  line)  on  the  orange  side  of 
yellow,  E  in  the  green,  F  in  the  blue,  G  at  the  beginning  and 


xv.]  SPECTRUM   ANALYSIS.  495 

Hj  and  H2  near  the  end  of  the  violet.  The  position  of  any  col- 
our is  often  roughly  specified  by  stating  its  proximity  to  one  or 
other  of  these  Fraunhofer-lines. 

The  lines  C,  F,  and  G  pertain  to  hydrogen,  the  double  line  D  to  sodium, 
and  E  to  iron. 

The  vapour  of  Helium,  an  element  found  in  solar  eruptions,  and  whose 
spectrum  is  a  single  line  D3,  a  little  to  the  violet  side  of  the  sodium-lines 
Dj  and  D2,  seems  to  have  no  absorptive  power ;  this  is  perfectly  exceptional. 

Particular  wave-lengths  do  not  occur  among  the  radiations  from  ice; 
whence  ice  presents  a  dark-heat  spectrum  with  invisible  dark  lines. 

When  the  body  radiating  energy  consists  of  a  gas,  each  mole- 
cule, as  it  proceeds  in  its  free  path,  executes  free  vibrations,  like 
a  vibrating  tuning-fork  thrown  through  the  air ;  and  the  mass 
thus  vibrating  may  impress  upon  the  Ether  only  one  kind  of 
vibration,  or  perhaps,  if  the  structure  of  the  vibrating  molecule 
be  complex,  a  large  though  not  an  indefinite  number  of  simul- 
taneous oscillations  whose  frequencies  may  or  may  not  be  com- 
mensurable. Thus  a  white-hot  vapour  may  emit  only  a  few 
distinct  kinds  of  light,  and  may  produce  a  line-spectrum  —  a 
spectrum  consisting  of  a  few  isolated,  linear,  diversely-coloured 
images  of  the  slit. 

A  very  rare  gas  may  emit  very  little  heat  or  light,  even  at  such  temper- 
atures as  1500°  C. :  gases  are  bad  radiators.  The  outer  shell  of  a  flame  is 
non-luminous,  and  may  sometimes  cause  the  formation  of  dark  axes  in  the 
bright  lines  of  the  spectrum.  Vapours  are  nearer  their  points  of  liquefac- 
tion than  gases  are,  and  are  better  radiators. 

When  the  particles  are  so  close  together  as  to  have  no 
free  path,  or  but  a  small  one,  they  very  frequently  collide  and 
rebound,  and  thus  vibrate  in  an  irregular  manner ;  no  rate  of 
vibration  is  long  enough  absent  for  the  eye  to  detect  its  absence. 
From  the  radiations  of  an  incandescent  solid  or  liquid,  no  kind 
of  radiation  appears  to  be  absent,  up  to  the  most  rapid  which  is 
given  out  by  the  incandescent  body ;  and  the  spectrum  of  such 
a  body  is  continuous,  so  far  as  it  extends.  It  is  not,  however, 
necessarily  equally  bright  throughout ;  didymium  and  erbium 
oxides  give  well-marked  bright  bands  in  the  spectrum  of  the 
light  which  they  emit  while  incandescent. 

If  a  heated  gas  or  vapour  be  compressed,  the  shocks  between 
its  molecules  become  proportionately  more  numerous  :  if  its  tem- 
perature be  increased,  the  energy  of  each  shock  becomes  greater ; 
in  either  of  these  cases  the  vibrations  of  the  molecules  tend 
towards  irregularity  and  complexity;  and  there  may,  in  addi- 
tion to  the  main  free-vibration  of  the  molecules  —  which  is  well- 


496  OF   ETHER-WAVES.  [CHAP. 

marked  if  there  be  any  appreciable  free  path  —  be  a  number  of 
additional  vibrations  of  all  or  of  many  frequencies :  a  condition 
which  is  indicated  by  the  broadening  of  the  lines  in  a  linear 
spectrum  of  a  gas  into  the  bands  of  a  band-spectrum.  As 
pressure  is  relieved,  the  spectrum  merges  into  that  of  an  ordi- 
nary incandescent  gas,  or,  on  the  other  hand,  as  the  pressure  is 
increased,  into  the  continuous  spectrum  of  an  incandescent 
liquid  or  solid. 

Even  the  flame  of  hydrogen,  at  high  pressure,  is  luminous, 
with  a  continuous  spectrum  (Lockyer). 

Continuity  between  the  gaseous  or  vapourous  and  the  liquid 
states  is  thus  indicated  on  an  independent  ground. 

Light  from  incandescent  solids  or  liquids  travels  from  some  distance 
within  the  surface ;  for  it  is  polarised  at  right  angles  to  the  plane  of  inci- 
dence ;  this  shows  that  it  has  been  refracted  on  its  outward  passage  through 
the  surface  of  the  incandescent  body  and  into  the  rarer  surrounding  medium. 
Light  from  incandescent  gases  is  not  polarised ;  sunlight  is  not  polarised ; 
hence  sunlight  is  due  to  incandescent  gas  or  vapour. 

Variations  in  the  light  emitted  by  one  and  the  same  sub- 
stance under  different  conditions,  and  therefore  in  the  spectrum 
of  that  light,  serve  to  indicate  molecular  changes  in  the  substance 
which  radiates  light.  Salts,  if  undissociated,  have  a  different 
mode  of  vibration,  and  therefore  a  different  spectrum,  from  their 
component  elements  ;  heat,  or,  if  ordinary  heat  fail,  the  extremely 
high  temperature  produced  by  a  discharge  of  high-tension  elec- 
tricity will  break  them  up  into  their  elements.  Even  the  ele- 
ments are  reduced  to  comparatively-simple  forms  of  aggregation 
by  high  temperatures ;  their  continuous  spectrum  breaks  up 
into  one  of  bands ;  a  still  higher  temperature,  such  as  that  of 
a  high-tension  electric  spark  if  other  means  fail,  converts  the 
spectrum  into  a  line  spectrum,  —  the  line  spectra  being  perhaps 
due  to  atoms,  the  band  spectra  to  molecules.  The  spectra  of 
the  same  substance  at  different  temperatures  are  often  remark- 
ably dissimilar. 

At  temperatures  beyond  our  reach,  such  as  those  of  some  of 
the  fixed  stars,  or  the  lower  levels  of  the  sun's  atmosphere  — 
the  high  temperature  of  which  may  be  inferred  from  the  great 
amount  of  the  highly-refrangible  rays  emitted  by  them  —  the 
elements  themselves  appear  to  be  broken  up  and  reduced  to 
simpler  forms  of  matter.  This  lends  probability  to  the  belief 
that  the  various  elements  are  modifications  of  one  kind  of  mat- 
ter—  a  belief  somewhat  strengthened  by  numerous  coincidences 
between  the  lines  of  the  spectra  of  different  elements. 


xv.]        TRANSMISSION,  REFLEXION,  AND  ABSORPTION.        497 

TRANSMISSION,  REFLEXION,  AND  ABSORPTION. 

When  ether-waves  fall  upon  a  transparent  body,  they  pass 
through  it:  they  are  propagated  through  the  ether  which  lies 
between  the  molecules.  When  a  body  is  thus  pervious  to  light 
it  is  specially  said  to  be  transparent;  when  pervious  to  dark 
heat,  as  rock-salt  is,  it  is  said  to  be  diathermanous,  —  ho 
special  term  being  used  to  denote  transparency  to  actinic  radia- 
tion. A  body  impervious  to  light  is  opaque;  one  impervious 
to  dark  heat  is  adiathermanous. 

A  perfectly-transparent  body  is  invisible.  Colourless  thin 
glass,  with  a  dustless,  polished,  clean  surface,  approaches  this 
character :  objects  are  seen  beyond  it,  and,  as  we  say,  through  it : 
they  appear,  if  the  glass  be  thin,  inappreciably  distorted.  Light 
may  be  reflected  from  the  polished  surface  of  glass,  and  the  pres- 
ence of  the  glass  may  thus  be  rendered  manifest  to  one  standing 
in  a  particular  position ;  the  sun  shining  on  the  windows  of  a 
distant  house  makes  the  window-glass  visible. 

Glass  is  almost  invisible  in  a  mixture  of  6  vols.  essence  of  cloves  and 
1  vol.  essence  of  turpentine. 

If  glass  be  roughened  at  its  surface,  it  presents  numerous 
facets  which  reflect  light  so  as  to  make  the  glass  visible  in  all 
directions  ;  and  light  passing  through  it  is  irregularly  turned  out 
of  its  path  in  all  directions  ;  objects  beyond  cannot  be  seen  dis- 
tinctly, though  light  can  pass  through  the  whole  mass,  and  rough- 
ened glass,  though  not  perfectly  transparent,  is  translucent. 

When  glass  is  powdered,  the  powder  presents  so  many  facets 
and  reflects  so  often  the  light  which  falls  upon  it  that  the  whole 
is  practically  opaque:  it  is  a  powder  which  reflects  in  every 
direction  the  light  incident  upon  it  —  in  white  light  a  white 
powder,  in  red  light  a  red  powder. 

When  ether-waves  of  any  kind  impinge  upon  a  body  impervi- 
ous to  them  their  progress  is  arrested ;  in  part  they  are  reflected 
or  scattered;  in  part  they  are  absorbed  by  the  impervious 
body ;  the  Ether  loses  energy,  ordinary  matter  gains  it,  and  the 
impervious  body  is  heated  to  an  extent  corresponding  with 
the  amount  of  energy  absorbed  —  this  heat  being  first  communi- 
cated to  the  superficial  layer  of  the  body. 

If  ether-waves  impinge  upon  a  body  which  is  transparent 
and  diathermanous,  that  body  is  not  heated,  for  the  ether- 
waves  pass  through  it  and  are  not  absorbed.  Thus,  clear  moun- 

2K 


498  OF   ETHER-WAVES.  [CHAP. 

tain  air  is  not  heated  by  the  sunshine  which  streams  through  it ; 
in  the  shade  it  may  be  very  cold.  Sunshine  may  stream  through 
clear  ice,  or  even  through  hoar-frost,  without  melting  it.  If 
there  be  any  particles  of  dust  in  the  air  or  in  the  ice,  these, 
being  opaque,  will  become  heated,  and  the  air  is  then,  by  con- 
duction, rendered  warm,  or  the  ice  is  melted. 

Some  bodies  are  impervious  to  all  kinds  of  radiation ; 
others,  having  a  power  of  Selective  Absorption,  are 
impervious  to  some  kinds  only. 

Thus  radiant  heat  can  pass,  while  the  more  rapid  light-waves  cannot 
pass,  through  a  thin  piece  of  black  vulcanite,  or  through  a  strong  solution 
of  iodine  in  bisulphide  of  carbon :  while  a  crystal  of  alum  is,  on  the  other 
hand,  transparent  to  light,  but  is  almost  adiathermanous,  impervious  to 
heat-rays.  Lampblack,  again,  is  very  transparent  to  the  slowest  heat-waves, 
and  air  very  opaque  to  some  of  them. 

A  soap-bubble  film  is  remarkably  adiathermanous,  cutting  off  about 
half  the  heat  of  an  incident  beam. 

Glass  is  transparent  and  diathermanous,  but  is  somewhat  opaque  to  the 
ultra-violet  rapid  ether-waves ;  a  quartz  prism  or  lens  allows  a  great  amount 
of  ultra-violet  radiation  to  pass  through  it  which  a  glass  prism  or  lens  would 
extinguish.  On  the  other  hand,  very  long  ether-waves  go  readily  through 
a  stone  wall;  and  silver-leaf,  just  thick  enough  to  be  opaque,  transmits 
ultra-violet  rays. 

The  absorptive  power  of  a  substance  may  not  be  so  extensive 
as  to  enable  it  to  absorb  and  extinguish  light-rays  or  heat-rays  of 
all  kinds ;  it  may  arrest  some  only.  A  piece  of  green  glass  can 
only  allow  a  certain  number  of  kinds  of  light  to  pass  through  it ; 
by  their  joint  impact  on  the  retina  these  produce  the  sensation 
of  green.  Sunlight  contains  other  waves  than  these  ;  they  have 
been  absorbed;  the  green  glass  is  opaque  to  them.  These  waves 
would  together  have  produced  a  sensation  of  purple-coloured 
light.  If  this  purple  light  had  alone  fallen  upon  the  green 
glass,  it  would  riot  have  been  transmitted  ;  the  glass  would  have 
appeared  to  be  opaque.  When  sunlight  is  directed  first  through 
purple  glass  and  then  through  green,  the  eye  perceives  black- 
ness :  the  two  pieces  of  glass  are  together  opaque,  though  each 
of  them  is  transparent  to  its  own  kind  of  light. 

Very  dark-red  glass  and  green  glass  together  produce  a  similar  effect 
of  blackness :  pale-red  glass  allows  some  green  light  to  traverse  it,  and  so, 
when  it  is  combined  with  green  glass,  the  result  is  dark-green  light. 

Nickel  nitrate  absorbs  red  and  violet,  and  is  therefore  green  when  in 
solution.  Cobalt  solution  is  red.  A  mixture  of  strong  solutions  of  the  two 
metals  is  black :  diluted  it  becomes,  however,  almost  colourless. 

Copper,  when  it  receives  the  impact  of  white  light,  emits  orange  light, 
together  with  superficially-reflected  white  light.  Electrically-deposited  cop- 


xv.]  ABSORPTION.  499 

per,  while  immersed  in  a  solution  of  sulphate  of  copper,  which  does  not 
allow  the  transmission  of  orange  light,  looks  as  white  as  plaster-of-Paris  does 
in  the  same  liquid. 

The  colour  of  a  coloured  object,  as  seen  by  transmitted 
light,  is  produced  by  subtraction  of  the  light  absorbed  from 
the  light  incident  upon  the  object. 

The  kind  of  light  transmitted  may  vary  with  the  thick- 
ness of  the  absorbing  medium.  In  such  a  case  the  medium  is 
said  to  be  dichromatic,  or  dichroic.  A  solution  of  chloride  of 
chromium,  in  a  thin  layer,  absorbs  much  yellow,  orange,  and 
yellowish-green  light ;  in  a  thicker  layer  it  absorbs  all  but  the 
red  and  some  green  and  blue ;  in  a  still  thicker  layer  the  only 
colour  transmitted  is  red.  Thus  a  wedge-shaped  layer  of  this 
solution  appears  to  vary  in  colour,  according  to  the  thickness, 
from  a  greenish-blue,  through  purple,  to  red.  Chlorophyll 
appears  green  in  thin  layers,  red  in  thick.  Iodine  vapour  trans- 
mits a  blue  group  and  a  red  group,  as  also  ultra-violet  rays ; 
together  these  produce  an  impression  of  purple :  in  thicker 
layers  the  blue  rays  alone  are  transmitted,  and  the  vapour 
appears  blue. 

Each  wave-length  has  its  own  Coefficient  of  Transmission  through  each 
transparent  substance ;  if  half  the  intensity  be  lost  by  transmission  through 
a  layer  1  cm.  in  thickness,  the  proportion  actually  transmitted  is  50  per 
cent.,  and  the  coefficient  of  transmission  is  0-50.  The  next  cm.  of  thick- 
ness will  transmit  0-50  x  0-50  =  0-25 ;  the  tenth  cm.  will  only  transmit 
0-5010  =  0-00098  times  the  original  intensity.  Small  differences  in  the  coeffi- 
cients for  the  several  wave-lengths  make  considerable  differences  in  the 
composition  of  the  aggregate  light  transmitted  through  a  thick  layer  of  a 
selectively-absorbent  substance. 

When  a  strong  solution  of  blood  is  interposed  in  the  path  of  a  beam  of 
light,  no  light  but  red  is  transmitted;  dilute  the  solution  gradually,  and 
successively  the  solution  appears  more  and  more  yellowish,  and  of  increas- 
ingly paler  hue. 

The  special  absorptions,  of  absorbent  bodies  are  most 
thoroughly  studied,  not  by  means  of  their  visible  colours,  but  by 
the  prismatic  analysis  of  the  light  which  passes  through  them. 
It  is  then  found  that  some  substances  absorb  several  distinct 
kinds  of  light,  belonging  to  different  regions  of  the  spectrum. 

Transparent  coloured-objects,  through  which  light  is  filtered, 
give  dark  bands  across  the  spectrum  —  the  so-called  "  Absorp- 
tion-bands, "  which  indicate  what  kind  of  light  has  been  stopped 
and  extinguished  by  the  absorbent  object.  These  bands  vary  in 
breadth  with  the  degree  of  concentration  of  the  absorbent  solu- 
tion employed,  and  they  vary  in  position  with  its  nature. 


500  OF  ETHER-WAVES.  [CHAP. 

When  a  strong  solution  of  blood  is  interposed  in  the  path  of  a  beam  of 
light  which  is  on  its  way  to  form  a  spectrum  on  a  screen,  all  the  spectrum, 
with  the  exception  of  the  red  part  of  it,  disappears.  As  the  liquid  is  diluted 
the  spectrum  lengthens  out :  orange,  yellows,  greens,  blues,  are  successively 
added ;  but  there  always  remain  two  relatively-dark  absorption-bands  in  the 
spectrum,  in  the  yellow  and  in  the  green,  between  the  Fraunhofer-lines  known 
as  D  and  E. 

If  the  blood  be  treated  with  sulphide  of  ammonium,  it  will  be  reduced ; 
its  oxyhsemoglobin  will  become  reduced  haemoglobin ;  the  chemical  constitu- 
tion changes,  and  with  it  the  absorbent  power ;  the  absorption-band  is  now 
a  single  band  placed  between  the  two  preceding. 

If  absorption-bands  be  numerous  and  pretty  uniformly  distributed 
throughout  the  spectrum,  or  if  they  be  in  complementary  regions,  the 
absorbent  substance  may  present  no  distinctive  colour,  e.g.,  benzene. 

On  the  evidence  of  absorption-bands  Major  Abney  has 
brought  to  light  the  existence  of  traces  of  benzene  vapour 
between  the  earth  and  the  sun,  and  Prof.  Langley  has  shown 
that  there  are  very  peculiar  gaps  in  the  heat-spectrum,  which  are 
probably  due  to  absorption  by  the  upper  regions  of  the  solar 
atmosphere. 

The  kind  of  light  absorbed  by  a  body  may  also  vary  with 
its  molecular  constitution. 

It  is  supposed  (von  Helmholtz)  that  each  absorption  depends  on  the  pres- 
ence of  a  particular  kind  of  molecule,  differing  from  the  simple  chemical 
molecule.  Chlorine  has  many  absorption-bands  in  its  spectrum,  and  it  must 
either  arrange  itself  in  many  kinds  of  molecules,  or  else  its  ordinary  mole- 
cules, considered  as  vibrating  bodies,  must  be  extremely  complex,  and  have 
many  free  periods  of  vibration. 

By  changes  in  the  absorption-bands  we  may  learn  that  sub- 
stances change  their  molecular  constitution  when  heated.  Iodine 
vapour  gives  an  extensive  absorption ;  when  highly  heated,  the 
absorption-spectrum  becomes  reduced  to  a  few  bands  ;  when  the 
vapour  is  still  more  highly  heated,  some  of  the  absorption-bands 
disappear,  and  one  of  them  is  replaced  by  a  group  of  fine  lines. 

Sulphur-vapour  changes  its  absorption-spectrum  when  its  density  changes 
at  1000°  C. ;  N2O4,  when  it  becomes  NO2,  changes  its  spectrum,  though  it 
does  not  do  so  when  it  becomes  a  liquid ;  iodine,  on  the  other  hand,  when 
dissolved  in  carbon  disulphide,  has  the  same  absorption-spectrum  as  when  it 
is  in  the  state  of  vapour. 

In  a  red  solution  of  cobalt  —  the  chloride,  for  example  —  when  heat  is 
applied  to  it,  the  salt  enters  into  a  different  state  of  hydration ;  its  molecular 
structure  is  changed ;  the  solution  becomes  blue. 

If  there  be  no  molecular  difference  between  a  substance 
incandescent  and  the  same  substance  absorptive  of  light  from 
a  hotter  object,  a  condition  probably  realised  in  the  case  of  didy- 


xv.J  ABSORPTION.  501 

mium,  erbium,  and  terbium  compounds,  the  incandescence-  and 
the  absorption-spectra  will  be  mutually  complementary ;  the  one 
presenting  bright  lines  where  the  other  presents  dark. 

The  Colour  of  a  coloured  object  seen  by  reflected  light 
is  also  generally  due  to  absorption.  An  object  seen  by 
reflected  sunlight  does  not  appear  to  be  coloured  in  any  degree 
unless  there  have  been  absorption  of  some  of  the  components 
of  the  incident  white-light,  and  the  colour  of  a  coloured  object 
is  complementary  to  the  colour  which  would  have  been  pro- 
duced by  these  absorbed  components  had  they  jointly  impinged 
on  the  eye. 

Some  of  the  light  incident  on  a  piece  of  coloured  glass  is 
reflected  at  its  surface ;  there  is  no  absorption ;  if  the  incident 
light  be  white,  the  light  reflected  is  also  white.  If  a  piece  of 
green  glass  be  laid  upon  black  paper,  and  if  it  be  looked  at  in 
such  a  direction  that  daylight  is  not  directly  reflected  from  it 
into  the  eye,  it  will  be  nearly  invisible,  and  will  be  devoid  of 
colour ;  it  will  appear  black.  If  coloured  glass  be  ground  to 
powder,  the  powder  is  white  ;  white  light  is  reflected  at  every 
facet,  while  the  light  reflected  from  the  lower  surfaces  of  the 
fragments,  and  again  issuing  into  the  air,  has  nowhere  traversed 
a  layer  of  sufficient  thickness  to  cause  the  extinction  of  all  the 
absorbable  components  of  the  incident  sunlight.  The  finer  the 
powder,  the  whiter  it  is ;  the  coarser  it  is,  or  the  more  energetic 
its  absorption,  the  more  marked  is  its  colour.  For  the  same 
reason,  froth  is  white.  If  the  upper  surface  of  a  sheet  of  green 
glass  be  ground,  it  will  appear  almost  white ;  if  the  ground  sur- 
face be  looked  at  through  the  glass,  it  will  appear  green,  for  the 
light  issuing  from  the  glass  is  white  light,  which  has  undergone 
a  certain  amount  of  absorption. 

If  the  green  powder  be  immersed  in  water  or  oil,  there  is  less  superficial 
reflexion  at  the  several  facets ;  there  is  deeper  penetration  of  the  light  into 
the  mass,  and  consequently  more  absorption ;  the  colour  appears  to  deepen. 
Hence  the  value  of  oil  as  a  medium  in  painting. 

A  solution  of  chloride  of  copper  placed  in  a  deep  black- 
walled  vessel  will  not  appear  to  have  any  colour;  it  will  seem 
black ;  it  reflects  no  light  except  from  its  surface.  If  powdered 
chalk  be  mixed  with  it,  light  is  now  reflected  from  the  white 
particles  of  chalk,  and  passes  out  in  every  direction,  through 
every  part  of  the  surface ;  so  much  of  the  reflected  light  is 
absorbed  that  it  appears  green  when  it  reaches  the  eye,  —  the 
milky  mass  appears  green.  In  a  similar  \vay  a  piece  of  malachite 


502  OF  ETHER-WAVES.  [CHAP. 

is  penetrated  by  light  to  a  very  small  depth ;  internal  reflexion 
occurs ;  absorption  of  all  the  outpassing  light  takes  place,  with 
the  exception  of  certain  kinds,  which  jointly  appear  green ;  the 
malachite  is  green.  A  piece  of  polished  gold  reflects  white  light 
at  its  surface ;  it  also  reflects  interiorly,  and  from  within  the  sub- 
stance of  the  gold  at  a  very  small  depth  there  is  reflected  in  all 
directions  a  quantity  of  light  which,  by  absorption  before  reach- 
ing the  surface,  has  become  of  an  orange  colour. 

If  the  layer  of  gold  be  very  thin,  that  part  of  the  light 
which  would  be  absorbed  by  a  thicker  layer  may,  in  part,  pass 
through  and  issue  into  transparent  media  before  its  energy  is 
wholly  converted  into  heat.  A  thin  piece  of  gold-leaf  thus 
appears  transparent  and  allows  a  greenish-blue  kind  of  light 
to  pass  through  it,  which,  if  the  leaf  be  rendered  very  thin  by 
the  action  upon  it  of  a  solution  of  cyanide  of  potassium,  may 
become  violet,  for  both  green  and  violet  light  then  find  their 
way  through. 

The  object-glass  of  an  astronomical  telescope  may  be  covered  with  a  thin 
layer  of  silver,  which  will  reflect  the  heat  and  some  of  the  light,  allowing  a 
pleasant  greenish  light,  and  also  some  actinic  rays,  to  pass. 

When  a  beam  of  light  enters  the  eye  after  undergoing  repeated  reflexion 
from  gold  to  gold,  it  is  of  a  deep-orange  colour ;  this  is  the  true  colour  of 
gold.  As  we  ordinarily  see  gold,  the  orange  light  coming  from  its  deeper 
particles  is  mixed  with  much  white  light  irregularly  reflected  from  its  sur- 
face. The  true  colour  of  copper  is  scarlet,  of  silver  a  yellowish-bronze 
colour,  of  brass  a  rich  golden-red.  By  reason  of  such  repeated  reflexion,  a 
deep  metal-vase,  equally  polished  within  and  without,  appears  to  be  of  a 
much  richer  colour  internally  than  it  is  externally,  and  silk-velvets  appear 
of  a  richer  colour  than  silks,  for  light  undergoes  repeated  reflexions  between 
the  vertical  fibres  which  constitute  the  outer  aspect  of  the  former. 

When  there  is  little  opportunity  for  reflexion  from  the  inner  particles  of 
a  body,  as  where  light  falls  exceedingly  obliquely  upon  a  gold  mirror  from 
a  white  object  and  is  reflected  into  the  eye,  the  image  of  the  white  object  in 
the  polished  gold-mirror  appears  not  gold-coloured,  but  white. 

Some  metals  can  be  rendered  transparent,  not  by  being  reduced  to  thin 
films,  but  by  being  reduced  to  the  liquid  state  :  potassium  and  sodium  can 
be  dissolved  in  anhydrous  liquid-ammonia;  the  solution  is  blue,  and  the 
true  colour  of  these  metals  is  therefore  a  copper  colour. 

If  the  incident  light  be  already  coloured,  it  may  be  that  the 
whole  of  it  is  absorbed.  An  object,  blue  or  red  in  daylight,  if 
illuminated  by  a  sodium-flame,  may  absorb  all  the  light  that  falls 
upon  it ;  if  it  do  so,  it  appears  black ;  a  bunch  of  flowers,  looked 
at  in  such  a  light,  where  it  is  riot  yellow  appears  black ;  it  must 
either  reflect  some  or  none  of  the  light  which  falls  upon  it.  A 
piece  of  red  cloth  illuminated  by  the  red  regions  of  the  spec- 


xv.]  ABSORPTION.  503 

trum  glows  with  a  bright  red ;  when  moved  into  other  regions 
it  becomes  black,  for  it  absorbs  the  incident  light. 

The  blue  colour  of  opalescent  bodies,  which  in  general  present  a 
multitude  of  reflecting  particles  embedded  in  a  uniform  matrix,  and  of  which 
we  may  take  as  a  type  the  sky-blue  liquid  obtained  by  adding  to  water  a 
very  small  proportion  of  milk,  is  not  primarily  due  to  absorption.  The 
principle  is  an  established  one  (p.  523,  near  top),  that  where  there  is  most 
refraction  of  light  there  is  the  greatest  proportion  of  reflected  light.  A 
beam  of  mixed  light  falls  upon  a  colourless  transparent-body :  all  the  rays 
are  both  refracted  and  reflected ;  the  blue  and  violet  are  the  more  sharply 
refracted,  and  a  greater  proportion  of  them  is  reflected  than  of  the  less- 
refrangible  rays.  Even  after  one  reflexion  the  image  of  an  object  in  a  mirror 
is  bluer  than  the  object  itself.  After  multiple  reflexion,  light  may  become 
distinctly  blue.  Multiplicity  of  reflexion  is  favoured  by  smallness  of  the 
individual  particles.  The  light  which  is  not  reflected  is  wholly,  or  in  part, 
absorbed  ;  the  sun,  looked  at  through  a  thin  layer  of  dilute  milk,  appears 
yellow ;  through  a  thicker  layer,  orange  or  red  ;  through  a  still  thicker  layer 
it  cannot  be  seen.  Similar  phenomena  are  presented  by  water  into  which  a 
little  very  dilute  alcoholic-solution  of  resin  or  mastic  has  been  dropped  with 
stirring,  by  salt  water  into  which  a  few  drops  of  a  very  dilute  solution  of 
nitrate  of  silver  have  been  stirred,  by  a  thin  haze,  by  smoke ;  all  these 
appear  blue  by  reflected,  yellow  or  red  by  transmitted  light.  Even  the  Sky 
itself  is  a  haze  of  this  kind,  the  air  being  rendered  visible,  against  the  dark 
background  of  black  space,  by  sunlight  reflected  from  its  fine  suspended 
dust-  or  water-particles  ;  while  the  light  transmitted  is  always  more  or  less 
yellowish,  and,  in  the  afternoon  and  evening,  when  sunlight  comes  to  us 
through  a  greater  thickness  of  the  more  dusty  layers,  verges  towards  orange 
or  even  red.  Such  a  dust-haze  is  more  opaque  than  adiathermanous. 

When  the  particles  of  a  haze  increase  in  size  they  jointly  offer  a  greater 
resistance  to  the  entry  of  light  into  the  fog :  light  is  reflected  more  promptly, 
and  the  reflected  light  presents  a  large  proportion  of  white  light.  This  phe- 
nomenon is  familiar  to  the  smoker ;  the  thick  clouds  of  smoke  produced  by 
vigorous  smoking  are  obviously  different  from  the  thin  fine  blue  columns 
which  ascend  from  a  cigar  laid  aside  for  a  moment. 

The  colours  of  metals  may  be  partly  accounted  for  in  a  similar  way. 
Steel  and  zinc  have  a  normal  refraction ;  the  violet  is  most  refrangible ;  they 
appear  blue.  Bell-metal,  brass,  Au,  Cu,  Ag,  have  abnormal  dispersion;  the 
red  end  is  most  refrangible  and  most  reflected ;  they  appear  red  or  reddish. 
Speculum-metal  refracts  red  more  than  green,  but  also  violet  more  than 
green;  on  the  whole  it  is  reddish  (Jamin). 

Those  rays  which  are  absorbed  in  the  greatest  pro- 
portion by  any  substance  are  reflected  by  it  in  the  least; 
when  a  beam  of  sunshine  falls  on  a  green  leaf,  the  actinic  rays 
are  absorbed  and  spent  in  doing  chemical  work ;  the  light 
reflected  from  such  a  leaf  is  feeble  in  actinic  ra}~s,  and  foli- 
age is  consequently  not  easy  to  photograph.  Light  which  is 
absorbed  is  generally  converted  into  Heat ;  this  may  presently 
be  radiated  away  ;  shorter,  quicker  light-waves  strike  the  body ; 
longer,  slower  waves  of  dark  heat  leave  it. 


504  OF   ETHER-WAVES.  [CHAP. 

FLUORESCENCE,  PHOSPHORESCENCE,  AND  CALORESCENCE. 

Fluorescence  and  Phosphorescence.  —  The  molecular  dis- 
turbances of  the  interior  particles  of  a  body  impinged  upon  by 
light  may,  however,  give  rise  to  other  waves  which  are  not  so 
slow  as  to  be  invisible ;  the  ether-waves  absorbed  may  thus  give 
rise  to  Light.  In  this  case  the  body  may  not  only  reflect  light, 
but  it  may  also  seem  to  emit  light  from  within ;  it  is  fluo- 
rescent. The  particles  down  to  a  very  small  depth,  being  set  in 
agitation,  originate  a  new  set  of  ether-waves,  which  are  propa- 
gated from  each  particle  in  every  direction. 

The  phenomena  of  Fluorescence  maybe  shown  by  a  solution  containing 
sesculin  and  fraxin,  which  may  be  very  simply  prepared  by  stirring  some 
horse-chestnut  twigs  in  water ;  a  beam  of  light  is  caused  to  pass  through  this 
solution,  and  then  for  some  distance  within  the  solution  the  liquid  seems 
self-luminous  and  shines  in  a  dark  room  with  an  opalescent  shimmer  along 
the  track  of  the  beam  of  light.  This  effect  is  partly  due  to  the  impact  of 
the  light  rays,  but  is  principally  due  to  the  rapid  invisible  ultra-violet  waves. 
If  a  piece  of  paper  be  wetted  with  this  solution,  and  if  this  paper  be  then 
used  as  a  screen  on  which  the  image  of  a  slit  is  thrown  through  a  quartz 
prism,  the  ultra-violet  part  of  the  spectrum  is  rendered  visible  ;  a  compound 
blue  light  radiates  from  the  paper  over  an  area  six  or  eight  times  as  long  as 
the  ordinary  visible  coloured  spectrum ;  the  light  refracted  by  a  prism  may, 
with  the  same  effect,  fall  on  the  walls  of  a  glass  vessel  containing  the  fluo- 
rescent solution.  Quinine  chloride  or  disulphate,  on  paper  or  in  solution, 
gives  a  blue  light  —  that  blue  which  is  seen  about  the  edge  of  the  upper 
surface  of  a  solution  of  quinine  in  a  phial ;  petroleum  or  shale  oil  a  green ; 
turmeric  solution  in  alcohol,  or  much  better  in  castor  oil,  a  green;  uranium 
compounds,  especially  uranium  glass,  a  green  light ;  chlorophyll  in  solution, 
or  lying  undissolved  in  the  cells  of  leaves,  a  red ;  an  alcoholic  solution  of 
soot  or  one  of  datura  stramonium,  a  greenish  blue.  Among  fluorescent  sub- 
stances we  find  also  such  compounds  as  eosin  (tetrabromofluorescein),  fluo- 
rescein  (resorcin-phthalein),  anthracene,  fluor-spar  (especially  chlorophane, 
which,  when  heated  by  conduction  or  by  radiant  heat,  shines  with  an  emer- 
ald-green light),  many  sulphides,  especially  those  of  barium  and  calcium, 
and,  to  a  slight  degree,  the  cornea  and  the  crystalline  lens,  and  the  rods  and 
cones  of  the  retina. 

Very  frequently  a  body  goes  on  vibrating  for  some  time 
after  ether-waves  have  ceased  to  strike  it ;  this  is  familiar  when 
the  waves  given  out  by  it  are  Heat-waves.  Sometimes,  how- 
ever, the  body  thus  vibrating  produces  Light,  and  such  a  body 
—  Balmain's  luminous  paint,  for  example  —  which  goes  on  visi- 
bly shining  or  fluorescing  for  some  time  after  ether-waves  have 
ceased  to  impinge  upon  it,  is  said  to  be  phosphorescent. 

Among  such  bodies  we  find  barium  and  calcium  sulphides,  diamonds, 
chlorophane,  dry  paper,  silk,  sugar,  teeth,  the  alkalies  and  alkaline  earths 
and  their  salts  in  general,  and  compounds  of  uranium. 


xv.]  PHOSPHORESCENCE.  505 

These  substances  may  be  placed  in  a  Geissler  tube  in  a  dark  room  ;  an 
electric  current  passes  ;  the  solids  commence  to  fluoresce  in  the  light  pro- 
duced by  the  discharge,  but  the  observer's  eyes  are  kept  shut  ;  the  current 
is  stopped,  and  the  eyes  are  at  once  opened  to  look  at  the  tubes  ;  the  solids 
are  seen  shining  in  the  dark  room. 

For  substances  the  duration  of  whose  phosphorescence  is  very  small 
Becquerel's  Phosphoroscope  may  be  employed.  In  rapid  succession  a  phos- 
phorescent body  is  exposed  to  bright  light  and  brought  against  a  dark  back- 
ground before  the  eye  of  an  observer  situated  in  darkness.  Most  objects  are 
found  by  this  means  to  be  to  some  extent  phosphorescent  ;  and  apparently 
all  are  markedly  so  when  extremely  cold. 

The  compound  nature  of  the  light  produced  by  fluorescence  or  by  phos- 
phorescence can  be  ascertained  by  means  of  a  slit  and  a  prism. 

It  is  a  very  singular  fact  that  the  red  rays  of  the  spectrum  and  the 
invisible  heat-rays  have  the  effect  of  accelerating  the  exhaustion  of  a  phos- 
phorescing body.  If  a  body,  phosphorescing  after  exposure  to  white  light, 
or  better,  to  violet  and  ultra-violet  rays,  have  a  spectrum  instantaneously 
thrown  upon  it,  the  body  thereafter  phosphoresces  more  brightly  in  the  area 
occupied  by  the  ultra-red  part  of  the  spectrum  ;  if  the  exposure  to  the  spectral 
image  be  relatively  prolonged,  the  phosphorescence  becomes  exhausted  in 
those  regions  on  which  heat-rays  had  fallen,  and  now  the  Fraunhofer  dark 
lines  in  the  invisible  part  of  the  spectrum  are  rendered  manifest  by  the  sur- 
vival of  local  phosphorescences  in  those  parts  of  the  screen  which  have  not 
been  affected  by  the  impact  of  heat-waves  (Becquerel). 

A  similar  action  of  these  rays  has  been  long  known  :  they  often  reverse 
the  chemical  action  of  the  actinic  rays. 

As  a  rule  a  fluorescent  or  phosphorescent  body  emits  for  a 
longer  or  shorter  time,  on  exposure  to  light,  or,  specially,  on 
exposure  to  actinic  rays,  the  same  kind  of  light  which,  when 
light  falls  upon  it,  it  absorbs  ;  and  thus,  in  some  instances,  the 
light  emitted  by  fluorescent  and  phosphorescent  bodies  presents 
bright  bands  where  the  absorption-spectrum  of  the  same  sub- 
stance presents  dark  bands  ;  but  the  whole  series  of  phenomena 
of  fluorescence  is  one  full  of  anomalies  ;  we  do  not  fully  know 
the  laws  of  the  molecular  groupings  of  different  substances, 
simple  and  compound,  their  necessary  modes  of  vibration,  or 
their  relations  to  the  Ether. 

A  mixed  beam  of  sunlight  which  has  passed  through  a  fluo- 
rescent solution  cannot  affect  another  solution  of  the  same  kind  ; 
fluorescent  solutions  rapidly  absorb  those  rays  which  are  the 
effective  cause  of  their  luminosity. 

We  sometimes  find  transformation  of  slower  waves  into 
more  rapid  ones.  When  a  solution  of  naphthaline-red  has 
been  shone  upon  by  a  beam  of  deep-red  light,  it  emits  by  fluo- 
rescence an  orange-yellow  light.  Chlorophyll  presents  an  analo- 
gous phenomenon  ;  it  fluoresces  with  a  red  light,  even  though  it 


TOT 

TJITIVBRSITY 


506  OF  ETHER-WAVES.  [CHAP. 

be  shone  upon  by  a  slower  red-light.  In  the  case  of  chloro- 
phane,  the  impact  of  slow  radiant-heat  waves  is  competent  to 
set  up  an  emerald-green  light. 

Calorescence.  —  When  a  beam  of  light  is  filtered  through 
a  solution  of  iodine  in  bisulphide  of  carbon,  so  that  dark  heat- 
rays  can  alone  pass  through,  these  heat-rays  may  be  brought  to 
a  focus  by  a  lens,  and  absorbed  by  a  piece  of  platinum  placed 
at  the  focus ;  this  will  become  luminous  and  give  rise  to  ether- 
waves  of  all  kinds  ;  if  its  light  be  examined  by  a  prism  it  will 
be  found  to  give  a  continuous  spectrum.  This  phenomenon  was 
called  by  Tyndall  the  Calorescence  of  heat-rays. 

SOURCES  OF  ETHER-WAVES. 

Vibrations  of  Molecules.  —  Light,  Heat,  and  Chemical 
Radiation  being  primarily  due  to  the  vibration  of  particles  of 
ordinary  Matter  in  the  midst  of  Ether,  the  energy  of  ether- 
waves  is  derived  from  the  kinetic  energy  of  vibrating  particles ; 
and  whatever  increases  the  Kinetic  Energy  of  these  vibrat- 
ing particles  increases  their  vibratory  movement,  and  gives  rise 
to  increased  radiation.  When  by  any  action  a  given  amount  of 
energy  is  liberated  in  or  communicated  to  a  system  of  material 
particles,  the  rapidity  of  their  resultant  vibration,  and  therefore 
that  of  the  ether- waves  set  up  by  them,  depends  on  the  rapidity 
with  which  that  action  occurs.  When  energy  is  slowly  imparted 
to  or  liberated  among  them,  the  vibrations  of  the  particles  may 
remain  relatively  slow,  and  radiant  heat  may  alone  be  the  result ; 
while  if  the  particles  be  suddenly  set  in  violent  commotion,  their 
vibration  will  be  complex  and  irregular,  the  particles  will  become 
incandescent,  and  they  will  at  once  originate  not  only  heat,  but 
also  light  or  even  actinic  waves. 

When  a  flash  of  lightning  or  an  electric  spark  rushes  through  the  air  it 
jars  the  particles  of  air,  and  renders  the  air  incandescent  and  luminous  ;  and 
it  even  originates  actinic  waves,  for  an  electric  spark  can  be  photographed  as 
well  as  seen.  When  the  electric  discharge  through  the  air  is  continuous  or 
rapidly  intermittent,  its  light  is,  to  the  eye,  apparently  continuous,  and  we 
have  the  Electric  Light.  When  a  flint  and  steel  are  struck  together,  the  con- 
cussion agitates  the  molecules  of  those  particles  of  steel  which  are  knocked 
off,  and  a  luminous  spark  is  produced ;  so  also  when  a  b'ullet  strikes  a  target 
there  is  a  flash  of  light.  Within  a  gas-flame,  molecules  of  a  hydro-carbon  are 
robbed  of  part  of  their  hydrogen  by  a  process  of  destructive  distillation  ;  the 
residues  are  heavy,  almost  purely-carbonaceous  molecules,  and  these,  in  virtue 
of  the  energy  supplied  by  the  combustion  of  the  hydrogen,  become  strongly 
agitated  and  incandescent,  oscillating  within  the  gas-flame,  and  therein  act- 


xv.]  VIBRATIONS  OF  MOLECULES.  507 

ing  as  sources  of  light  until  the  current  takes  them  into  the  zone  of  perfect 
combustion  in  the  outer  region  of  the  flame  ;  there  they  become  completely 
oxidised  into  gaseous  carbonic  acid,  and  thereupon  lose  in  great  part  their 
radiative  power.  The  brightness  of  a  gas-flame  is  favoured  by  external 
pressure,  or  by  a  relatively  small  internal  pressure  and  velocity  of  outflow, 
by  the  long  continuance  of  carbon  particles  or  other  solid  particles  (which 
in  a  candle-flame  can  reflect  light  and  cast  a  shadow  in  sunlight)  within  the 
flame  in  which  they  are  incandescent,  and  by  heating  the  gas  before  it  reaches 
the  flame. 

When  a  crystal  is  cleft  it  often  emits  a  flash  of  light; 
work  is  done  in  splitting  the  crystal :  the  energy  of  part  of  this 
work  appears  as  that  of  ether-waves. 

When  salts  suddenly  crystallise  out  of  a  liquid  men- 
struum it  not  unfrequently  happens  that  the  formation  of  crys- 
tals is  attended  with  a  flash  of  light ;  the  salt  leaves  the  water 
and  coheres  with  particles  of  its  own  substance ;  the  agitation 
attending  this  process  causes  ether-waves  to  be  set  up. 

Even  the  application  of 'moderate  heat,  falling  far  short 
of  such  a  temperature  as  might  produce  incandescence,  may 
cause  a  body  to  become  luminous,  as  in  the  case  of  the  fluo- 
rescence of  fluor-spar  (chlorophane)  and  the  diamond,  which 
shine  when  heat  is  imparted  to  them  by  conduction. 

We  have  already  seen  that  light  may  result  from  the 
impact  of  ether- waves  upon  a  body. 

Chemical  union  is  often  attended  with  both  heat  and 
light:  as  when  we  drop  copper  filings  or  powdered  antimony 
into  chlorine  gas,  or  in  the  ordinary  phenomena  of  combustion. 
Even  slow  combustion,  such  as  that  of  eremacausis  or  decay, 
may  cause  light,  as  in  the  luminosity  of  decaying  wood  ;  or  the 
green  luminosity  visible  on  the  surface  of  some  fish  when  in  a 
state  of  incipient  decay;  or  the  slow  oxidation  of  apiece 
of  phosphorus  in  the  air  at  ordinary  temperatures,  or  of  sulphur 
or  the  metal  arsenic  at  higher  temperatures.  Even  during  the 
life  of  organisms  they  may  become  luminous  either  abnor- 
mally, as  when  the  skin  of  the  human  body  evolves  phosphu- 
retted  hydrogen  ;  or. normally,  as  in  the  glow-worm,  in  the  nocti- 
luca,  in  medusoids,  and  in  many  other  invertebrate  animals ; 
and  the  production  of  light  may  even  be  under  the  control  of  the 
animal,  as  in  the  fish  photichthys,  which  can  temporarily  illumi- 
nate its  prey.  Light  is  in  these  cases  produced  at  the  expense 
of  the  animal  heat  which  might  otherwise  have  been  evolved. 

Vibrations  communicated  to  the  Ether.  —  Jn  the  cases 
discussed,  the  origin  of  the  light  plainly  is  in  the  agitation  of 


508  OF   ETHER-WAVES.  [CHAP. 

ordinary  matter,  but  there  is  a  certain  deficiency  of  knowledge  in 
respect  of  the  next  step  in  the  transference  of  energy.  How 
is  any  ether-wave  set  up  in  the  Ether  by  the  motion  of  any 
particle  of  ordinary  matter  within  it  ?  A  full  answer  to  this 
question  would  involve  a  full  knowledge  of  the  constitution  of 
the  Ether,  and  of  the  relation  of  the  Ether  to  the  particles  of 
ordinary  matter  which  are  embedded  in  it  —  a  question  still 
under  discussion. 

Some  hold  that  the  Ether  is  entirely  independent  of  ordi- 
nary matter,  being  unaffected  in  density  by  its  presence ;  others 
hold  that  it  is  of  various  densities  in  various  substances,  these 
densities  being  in  different  transparent  substances  inversely 
proportional  to  the  squares  of  the  velocities  of  light  within 
them.  Some  hold  that  it  is  so  independent  of  ordinary  matter 
that  a  moving  solid  body  moves  freely  through  ether  like  an 
ideal  net  through  ideally-frictionless  water ;  in  which  case  it 
would  be  difficult  to  understand  how  a  vibrating  molecule  could 
set  up  vibrations  in  it.  If  this  were  so,  the  most  rapidly-moving 
solid  transparent  object  would  allow  the  transmission  of  light 
through  the  ether  which  permeates  it,  as  if  it  were  itself  at  rest. 
The  contrary  view  seems  probable ;  a  ray  of  light  is  said  to  be 
retarded  a  little  by  being  made  to  pass  up  a  running  stream  of 
water ;  the  effect,  quite  perceptible  in  the  case  of  water  circu- 
lating at  the  comparatively-slow  rate  of  two  metres  per  second, 
is,  however,  imperceptible  in  a  current  of  air. 

A  beam  of  light  was  found  by  Fizeau  to  be  retarded  when  made  to 
pass  through  a  cylinder  of  glass,  rotating  in  such  a  direction  that  the  rota- 
tion of  the  glass  tended  to  carry  back  the  light  while  in  the  act  of  passing 
through  it. 

The  consequence  of  such  an  adhesion  between  the  Ether  and  the  matter 
embedded  in  it  is,  that  the  earth  must  to  some  extent  drag  the  Ether  with 
it  as  it  rolls  through  space ;  yet  Aberration  (p.  511)  tells  somewhat  against 
this. 

The  whole  subject  is  as  yet  one  of  the  most  recondite  in  Physics. 


PROPAGATION  OF  WAVES  THROUGH  THE  ETHER. 

At  present  it  is  usual,  in  discussing  the  propagation  of  ether- 
waves,  to  assume  the  wave  to  have  been  effectually  set  up ;  the 
wave-motion  is  studied  as  it  diverges  from  a  small  wave-front 
formed  in  the  immediate  neighbourhood  of  the  vibrating  mole- 
cule ;  and  in  discussing  the  transmission  of  ether-waves  of 
different  wave-lengths  through  different  transparent  bodies,  we 


xv.]  PROPAGATION   THROUGH   THE  ETHER,  5Q9 

shall  have  to  take  for  granted  that  the  interaction  of  the  Ether 
and  the  ordinary  Matter  —  an  action  which  cannot  be  very  great, 
for,  if  it  were,  Transparence  would  be  impossible  —  is  such  as, 
in  different  media,  unequally  to  retard  ether- waves  of  different 
wave-lengths.  This  retarding  effect  depends  somehow  upon  the 
nature  of  the  transparent  body ;  and  this  holds  good  not  only 
with  regard  to  light  in  general  —  as  where  a  diamond  is  found 
to  transmit  light  much  more  slowly  than  water  does  —  but  also 
with  reference  to  each  particular  kind  of  light.  Each  trans- 
parent substance  has  its  own  rate  of  transmission  for 
ether- waves  of  each  particular  frequency;  and  this  is  found 
for  each  case  only  by  experiment.  A  denser  substance  may  some- 
times transmit  ether-waves  more  rapidly  than  a  rarer  one  does : 
light  passes  more  rapidly  through  water,  for  example,  than  through 
alcohol  or  oil  of  turpentine.  A  substance  through  which  light 
travels  more  slowly  is  said,  however,  to  be  optically  denser. 

On  the  assumption  that  the  density  of  the  Ether  is  different  in  different 
substances,  it  would  follow  that  all  wave-lengths  must  be  diminished  or 
increased  in  equal  proportions,  that  all  kinds  of  waves  must  be  equally 
retarded  or  accelerated,  and  all  kinds  of  light,  heat,  or  chemical  rays  there- 
fore equally  refracted,  on  passing  from  one  medium  into  another  —  a  con- 
clusion contradicted  by  the  simplest  experiment  with  a  prism.  Cauchy,  on 
the  arbitrary  assumption  that  the  Ether  consisted  of  separate  particles  of 
an  average  size  extremely  minute  as  compared  with  the  average  distance 
between  them,  found  that  the  amount  of  retardation  was  affected  by  the  fre- 
quency of  undulation  (much  in  the  same  way  as  the  speed  of  propagation 
of  a  wave  along  a  cord  is  modified  by  stiffness  of  the  cord),  and  that  thus 
prismatic  dispersion  became  explicable ;  an  assumption  which  more  modern 
writers  —  unwilling  to  admit  that  the  Ether,  which  is  not  found  to  be  capable 
of  having  waves  of  compression  and  rarefaction  set  up  in  it,  and  whose  parts 
yet  preserve  or  tend  to  preserve  their  mean  positions,  can  be  a  fluid  com- 
posed of  separate  molecules  —  have  converted  by  interpretation  Into  the 
following,  namely,  that  there  is  some  kind  of  discontinuity  in  the  relations 
between  the  Ether  and  the  ordinary  matter  which  it  permeates ;  a  discon- 
tinuity which  is  held  to  show  that  while  the  Ether  may  be  considered  to  be  a 
homogeneous  jelly-like  solid,  which  can  yield  to  powerful  stresses  after  the 
manner  of  a  fluid,  the  matter,  apparently  homogeneous,  which  is  embedded 
in  it,  is  not  truly  homogeneous  throughout. 

The  index  of  refraction,  /8,  varies  with  the  wave-length,  A,  being  con- 
nected with  it  by  the  law  ft  =  k\?  +  A  +  B/A2  +  C/A.4  (Briot),  where  k,  A, 
B,  and  C  are  constants  to  be  determined  by  experiment. 

The  Ether  is  analogous  to  a  very  weak  solution  of  gelatine  :  to  relatively- 
great  momenta  it  acts  as  a  fluid,  and  it  closes  up  behind  moving  particles; 
to  small  stresses  it  acts  as  a  solid,  and  it  suffers  tangential  strain,  without 
change  of  volume,  under  the  influence  of  a  tangential  stress. 

Ether-vibration  transverse.  —  When  any  part/>f  the  Ether 
is  displaced  by  a  vibrating  molecule,  the  displaced  portion  always 


510  OF   ETHER-WAVES.  [CHAP. 

tends  to  return  to  its  normal  position ;  in  doing  so  it  sets  up 
waves.  These  are  waves  of  transverse  vibration,  like  those  of 
an  elastic  string  or  rod  plucked  laterally. 

Some  have  held  that  the  Ether  is  absolutely  incompressible,  and  that  it 
is  impossible  to  form  waves  of  compression  in  it ;  according  to  others,  waves 
of  compression  are  at  first  formed,  but  very  rapidly  die  out. 

If  it  be  assumed  that  the  Ether  is  analogous  to  an  elastic  solid ;  that 
the  resistance  to  compression,  ft,  is  very  great  in  comparison  with  the  rigid- 
ity, n,  to  transverse  distortion;  then  (Green)  it  can  be  shown  that  the 
compressional  waves  will  travel  with  extreme  velocity,  but  will  die  out 
after  a  few  wave-lengths.  In  explaining  Double  Refraction  on  this  basis, 
Green  found  it  necessary,  however,  to  assume  that  the  vibrations  occur 
parallel  to  the  plane  of  polarisation,  an  assumption  which  is  now  considered 
inadmissible  (p.  522)  :  and  there  are  also  other  difficulties  in  working  out 
this  theory.  If,  on  the  other  hand,  it  be  assumed  (Lord  Rayleigh,  Lord 
Kelvin,  and  Mr.  Glazebrook)  that  the  resistances  of  the  Ether  to  compres- 
sion and  distortion  are  the  same  in  all  media,  but  that  in  crystals  the 
matter  present  acts  so  as  to  make  the  Ether  behave  as  if  it  were  itself  dif- 
ferent in  density  in  different  crystallographic  directions;  that  the  Ether 
has  a  certain  negative*  resistance  to  compression,  which  means  that  it 
would  dilate  on  pressure,  arid  is  infinite  in  extent  or  bounded  by  a 
rigid  boundary;  and  that  the  vibrations  are  at  right  angles  to  the  plane 
of  polarisation  :  —  then  there  can  be  no  compressional  or  dilatational  waves, 
and  the  results  of  mathematical  calculation  agree  in  the  main  with  the  facts, 
so  far  as  these  are  ascertainable  by  experiment  external  to  crystals,  and  also 
agree  with  the  results  deduced  from  Clerk  Maxwell's  theory. 

According  to  Clerk  Maxwell's  view  the  Ether  is  a  homogeneous  body,  a 
non-conductor  of  electricity  :  periodic  electric  stresses  applied  to  this  produce 
waves  which  travel  at  the  rate  of  about  300,000000  metres  per  second ;  these 
waves  are  waves  of  transverse  vibration,  and  there  is  no  vibration  longitu- 
dinal or  normal  to  the  wave-front.  These  waves,  due  to  electric  displacement, 
are  quite  competent  to  explain  the  ordinary  phenomena  of  light,  and  this 
theory  explains  on  mathematical  grounds  that  absence  of  the  normal  or 
compressional  vibration  which  has  been  a  source  of  great  perplexity  in  most 
of  the  mechanical  theories  of  light.  According  to  this  view,  each  particle 
of  a  body  through  which  light  is  shining  is  in  rapid  succession  exposed 
to  alternately-opposite  electric  stresses:  at  each  half-vibration  it  becomes 
oppositely  electrified ;  but  the  ordinary  effects  of  electricity  are  not  generally 
observed  when  light  shines  through  or  on  a  body,  for  the  electrification 
produced  by  any  one  half -vibration  simply  reverses  the  effect  of  that  pro- 
duced by  the  previous  half-vibration. 

The  Velocity  of  propagation  of  ether-waves  through  the 
Ether  of  space  is  found  by  two  astronomical  methods. 

*  Any  compression-and-rarefaction  waves  formed  will  promptly  die  out  if  their 
velocity,  \/ik  ~\~  ttt)  /P,  be  very  great,  or  if  it  be  very  small,  in  comparison  with  the 
velocity,  v'n/P,  of  transverse-distortional  waves.  Green  assumed  their  velocity  to 
be  comparatively  very  great,  whence  ft  would  be  very  great  in  comparison  with  n ; 
the  other  physicists  named  have  assumed  their  velocity  to  be  comparatively  very 
small,  whence  (Jt+stt)  very  nearly  =0,  and  ft  would  be  negative,  and  nearly 
equal  to  —  PL 


xv.]  VELOCITY   OF  LIGHT.  511 

I.Jupiter's  Satellites.  — These  pass  out  of  sight  behind  the  mass 
of  Jupiter  and  again  reappear :  when  the  earth  is  nearest  to  Jupiter  the 
eclipses  and  reappearances  appear  to  take  place  8|  minutes  earlier,  when 
the  earth  has  wheeled  round  to  the  opposite  side  of  its  orbit  and  is  at  its 
farthest  from  Jupiter  8£  minutes  later,  than  they  would  have  appeared  if 
the  earth  had  been  at  the  centre  of  its  orbit.  The  suddenly  commencing  or 
ceasing  light  takes  16  £-  minutes  to  cross  the  earth's  orbit,  a  distance  of 
299,270,000000  metres :  it  therefore  travels  302,300000  metres  per  second. 
According  to  the  latest  determinations,  the  velocity  is  299,336000  metres 
per  second. 

2.  Aberration.  —  No  star  is  seen  in  its  true  place:  every  star  seems 
to  describe  a  little  ellipse  in  the  heavens,  and  seems  to  travel  round  the 
ellipse  once  a  year.  The  reason  is,  that  as  the  earth  wheels  onward  in  its 
orbit,  bearing  the  observing  telescope  with  it,  rays  of  light,  coming  from 
distant  stars,  on  their  way  down  the  telescope  tend,  short  though  the  tele- 
scope tube  be,  to  verge  towards  the  hinder  side  of  that  tube :  for  which 
reason,  in  order  to  see  the  star  in  the  centre  of  the  field,  the  eyepiece  must 
be  tilted  appreciably  backwards  in  a  direction  opposed  to  that  of  the  earth's 
orbital  motion  :  the  telescope,  when  the  star  is  seen  in  the  centre  of  its  field, 
is  therefore  directed  not  towards  the  true  position  of  the  star,  but  towards  a 
point  in  advance  of  it.  In  the  course  of  a  year,  therefore,  as  the  earth  bowls 
round  its  elliptical  orbit,  the  successive  points  to  which  it  is  necessary  to 
direct  the  telescope  are  found  to  have  been  situated  on  the  circumference  of 
an  ellipse.  The  size  of  this  ellipse  indicates  the  amount  of  tilting  of  the 
telescope :  from  this  can  be  inferred  the  proportion  between  the  length  of 
the  telescope  and  the  distance  traversed  by  the  ocular  during  the  time  spent 
by  the  ether-waves  in  passing  down  the  telescope  tube ;  the  speed  of  the 
waves  of  light  can  be  calculated  from  these  data,  and  is  found  to  be 
299,300,000  metres  per  second. 

Such  is  the  simple  theory  of  Aberration :  but  the  amount  of  aberration 
is  the  same  whatever  be  the  transparent  medium  —  e.g.,  water  —  with  which 
the  telescope  is  filled.  Hence  it  would  appear  that  a  diminution  of  the 
relative  motion  of  the  earth  and  the  ether  exists,  and  may  be  explained  by 
assuming  that  the  water  carries  the  contained  ether,  wholly  or  partly,  along 
with  it.  This  is  confirmed  by  Fizeau's  experiments  on  the  bodily  transfer- 
ence of  ether-waves  in  a  stream  of  water,  like  that  of  sound-waves  by  wind ; 
but  Michelson  and  Morley  find  that  the  earth  as  a  whole  drags  the  surround- 
ing ether  with  it  in  a  way  which  is  difficult  to  reconcile  numerically  with 
the  ordinary  theory  of  Aberration. 

Do  waves  of  different  frequencies  travel  through  the  Ether 
of  space  at  the  same  or  at  different  rates  ?  If  their  rates  were 
different,  then  a  suddenly-appearing  satellite  of  Jupiter,  or  a 
suddenly-brightening  variable  star,  would  be  first  rendered  visi- 
ble by  that  light  which  first  arrives  at  and  enters  the  eye,  and  it 
might  consequently  appear  violet  or  blue ;  and  when  it  disap- 
peared it  would  continue  for  the  longest  time  visible  by  that 
component  of  light  which  is  slowest  in  travelling,  and  therefore 
might  appear  red  before  vanishing ;  or  again,  aberration  of  light 
would  necessarily  have  the  effect  of  giving  us  an  image  of 


512  OF   ETHER-WAVES.  [CHAP. 

each  star  drawn  out  into  a  spectrum.  Nothing  of  the  kind  is 
observed  ;  all  kinds  of  ether-waves  must  therefore  travel  through 
the  ether  of  space  at  the  same  rate. 

Terrestrial  experiments  for  ascertaining  the  velocity  of  light  are  based 
upon  one  of  two  principles. 

1.  Fizeau's  principle.  —  A  ray  of  light  is  rendered  intermittent  by 
flashing  between  the  teeth  of  a  rotating  cogwheel.     It  travels  to  a  distant 
mirror ;  each  flash  is  there  reflected  along  its  former  path.     Before  a  flash 
can  again  reach  the  cogwheel,  the  cogwheel  may  have  rotated  so  far  that 
one  of  its  cogs  now  obstructs  the  returning  ray ;    if  a  sufficiently-increased 
speed  be  imparted  to  the  cogwheel,  the  light  is  allowed  again  to  pass  between 
the  teeth  of  the  wheel  through  a  neighbouring  notch,  which  has  now  come 
to  occupy  the  position  at  first  occupied  by  that  notch  through  which  the 
light  had  flashed  on  its  outward  journey.      Given,  then,  that  the  light  has 
travelled  to  a  certain  distance  and  back,  and  that  in  the  meantime  the  cog- 
wheel has  been  rotated  through  a  certain  angle,  it  is,  in  principle,  easy 
to  find  the  speed  of  propagation  of  the  light.     Fizeau  found  this  to  be 
314   million   metres :    Cornu,  by  similar  experiments,  obtained  the  value 
300,400,000  metres  in  vacuo. 

2.  Foucault's  principle.  —  A  beam  of  light  starts  from  a  source  S; 
it  strikes  a  mirror  M,  and  is  reflected  to  a  distant  mirror  R,  on  which  it  is 
focussed  by  a  lens  between  S  and  M :    it  is  there  reflected  and  retraces  its 
journey :  it  is  again  reflected  from  M  and  returns  to  S.      If,  however,  the 
mirror  M  have,  in  the  meantime,  been  rotated  perceptibly  before  the  beam 
of  light  has  had  time  to  return  from  the  distant  R,  the  light  can  no  longer 
be  reflected  from  M  towards  the  original  point  S;  it  illuminates  some  other 
point  T.     The  distance  between  S  and  T  can  be  measured ;  the  amount  of 
rotation  of  the  mirror  M  in  the  time  taken  by  the  light  to  go  from  M  to  R 
and  back  can  be  inferred  from  this ;  the  speed  of  rotation  of  the  mirror  M 
can  be  read  off  on  a  speed-indicator  attached  to  the  rotating  apparatus  :  the 
distance  traversed  by  light  in  one  second  can  be  ascertained  by  calculation 
from  these  data.     There  is  no  need  to  use  instantaneous  flashes  of  light 
from  S ;  the  steady  beam  from  S  reflected  from  the  rotating  mirror  M  only 
encounters  the  small  fixed  mirror  R  for  an  instant  once  in  the  course  of  each 
revolution,  and  is  thus  rendered  practically  instantaneous.     Michelson  put 
the  lens  between  M  and  R,  and  thus  obtained  greater  brightness,  which 
enabled  M  and  R  to  be  much  farther  apart,  and  a  greater  deflection  to  be 
produced. 

By  this  means,  with  a  mirror  rotating  1000  times  in  a  second,  Foucault 
demonstrated  that  light  takes  a  measurable  time  to  pass  through  a  distance 
of  7  or  8  yards. 

Lord  Rayleigh  has  shown  that  these  different  methods  cannot  be 
expected  to  give  the  same  results,  for  it  is  not  precisely  the  same  thing 
which  is  observed  in  all  these  cases.  In  some  (aberration  method)  the 
speed  of  single  waves  is  observed ;  in  others  (Fizeau,  Jupiter's  satellites) 
the  speed  of  propagation  of  groups  of  waves,  which  is  not  the  same  as  that 
of  a  single  wave,  unless  the  velocity  of  the  wave  be  independent  of  the 
wave-length;  in  others  (Foucault)  these  are  blended.  From  the  concord- 
ance of  the  results  obtained  by  the  different  methods  it  would  appear  that 
the  wave-velocity  is,  at  any  rate  for  wave-lengths  between  blue  and  red,  not 
dependent  on  the  wave-length. 


xv.]  VELOCITY   OF  LIGHT.  513 

As  a  mean  result  it  may  be  stated  that  the  velocity  of 
ether-waves  in  a  vacuum  —  that  is,  in  the  Ether  of  space — is 
300,574000  metres,  or  30,057,400000  cm.  per  second  =  186772 
miles  per  second. 

From  this  it  follows  that  tt,  the  rigidity  of  the  Ether,  and  />,  its  density 
in  vacuo,  are  definite  in  amount,  and  bear  to  one  another  the  relation 
n  =  p  x  (30057,400000) 2;  for  v  —  VnTp  centimetres  per  second.  The  mean 
velocity  in  air  is  less  than  that  in  vacuo  in  the  ratio  of  1  to  1-000294. 

Tt  is  generally  believed  that  light  of  all  colours  travels  with 
equal  velocities  through  air,  though  some  doubt  has  been  cast 
on  this  result  by  the  experiments  of  Forbes  and  Young  (Phil. 
Trans.  1882),  who  conclude  that  blue  light  travels  more  rapidly 
in  air  than  red  light  does,  in  the  ratio  of  1018  to  1000.  If  this 
were  so,  however,  Foucault's  experiment  would  give  drawn-out 
coloured  images ;  which  it  does  not  do. 

By  a  modification  of  Foucault's  method,  above  described,  the  relative 
speeds  of  light  in  two  different  transparent  media,  or  in  the  same  medium 
at  different  temperatures  or  under  different  pressures,  may  be  compared. 
The  light  between  M  and  R  has  to  traverse  a  space  in  which  a  certain  thick- 
ness of  the  medium,  whose  retarding  power  is  to  be  examined,  may  be  laid 
in  the  path  of  the  beam :  the  beam  may  be  exposed,  by  having  to  pass 
through  this  medium,  to  a  retardation,  which  is  rendered  manifest  and 
measurable  by  an  alteration  of  the  position  of  the  image  T. 

The  Physical  Intensity  of  light  at  a  place  is  measured  by 
the  energy  transmitted  through  that  place,  Der  unit  of  cross- 
sectional  area,  in  a  second  of  time ;  for  light  of  constant  colour, 
this  intensity  is  also  proportional  to  its  brightness  as  perceived 
by  the  eye. 

Hence  there  are  two  methods  of  measuring  the  intensity  of  a  beam  of 
light:  —  1.  Calorimetrical:  allow  the  beam  to  fall  upon  a  thermopile, 
and  estimate  the  intensity  of  the  light  by  the  amount  of  heat  into  which  it 
is  converted  upon  absorption ;  the  beam  in  this  case  having  undergone  a 
preliminary  sifting  through  some  adiathermanous  medium.  2.  Photo- 
metrical:  two  sources  of  light  are  placed  at  such  distances  from  an 
illuminated  body  that  they  appear  to  produce  the  same  effect,  such  as  equal 
shadows,  or  equal  illumination  of  the  two  sides  of  a  disc ;  but  this  method 
is  only  accurate  when  the  two  lights  to  be  compared  are  of  exactly  the  same 
colour.  The  intensity  of  .actinic  radiation  may  be  estimated  by  observing 
the  depth  of  tint  produced  in  a  piece  of  photographic  paper  exposed  for  a 
given  time.  The  total  radiation  may  be  measured  calorimetrically. 

As  a  unit  of  photometric  intensity  the  Paris  Electrical  Standards 
Committee  recommended  (May  1884)  the  light  emitted  by  1  sq.  cm.  of 
melted  platinum  at  its  solidification-temperature.  The  twentieth  part  of 
this  is  the  normal "  decimal  candle  "  (International  Electrical  Congress,  1889). 

For  many  purposes  of  mathematical  calculation  it  is  more  Advantageous 
to  measure  the  intensity  of  radiation  by  the  average  energy  in  ergs  per  cub. 
cm.  See  p.  142. 

2L 


514  OF   ETHER-WAVES.  [CHAP. 


MODE  OF  PROPAGATION  —  POLARISATION. 

Waves  of  light  have  the  peculiarities  of  propagation  char- 
acterising waves  whose  wave-length  is  generally  small  in  com- 
parison with  the  breadth  of  their  wave-front.  They  do  not 
usually  diverge  laterally  from  the  directions  mapped  out  by  the 
normals  to  their  wave-fronts  ;  or,  as  it  is  commonly  expressed, 
Light  travels  in  straight  lines;  they  can  only  so  diverge 
when  they  are  made  to  pass  through  apertures  or  round  obsta- 
cles not  very  much  greater  in  breadth  than  their  own  wave- 
length. 

The  light  from  a  single  luminous  point  is  propagated  in 
spherical  waves;  that  from  such  an  extended  object  as  a 
candle-flame,  in  waves  which,  at  some  distance  from  the  source, 
are  approximately  spherical.  If  light  from  a  wide  source  be 
made  to  pass  through  a  narrow  tube,  or  successively  to  traverse 
equal  apertures  in  two  opaque  screens,  at  such  a  distance  from 
the  source  that  the  wave  passing  through  the  second  screen  has 
a  plane  front  (see  Fig.  57  a),  then  on  the  farther  side  of  the  sec- 
ond screen  there  may  be  an  unwidening  or  parallel  beam  of 
light.  Such  a  parallel  beam  of  light,  as  it  traverses  space,  may 
be  compared  to  a  bundle  of  vibrating  strings  of  ether,  iso- 
lated in  the  ether,  vibrating  independently,  and  practically 
unaffected  by  the  ether  situated  laterally  with  respect  to  them. 
Each  such  imaginary  individual  cord  may  enter  into  transverse 
vibrations  of  different  kinds,  analogous  to  the  vibrations  of 
strings. 

1.  It  may  transversely  vibrate  simply  up-and-down,  or  from 
side-to-side,  or  in  any  other  single  direction,  —  its  vibrations  are 
restricted  to  one  plane  ;  the  whole  beam  is  then  called  a  beam  of 
Plane-Polarised  Light. 

2.  Its  vibration  may  be  resoluble  into  equal  simultaneous 
transverse-vibrations  in  two  planes  at  right  angles  to  one  another. 

(#.)  These  may  be  of  equal  period,  and  the  vibration  in  one 
plane  may  be  J  period  behind  or  in  front  of  that  in  the  other ; 
looked  at  endwise,  any  part  of  the  ether  in  such  a  beam  would 
necessarily  be  seen  —  if  it  could  be  rendered  visible  like  a 
bright  point  on  a  vibrating  string  —  to  execute  small  circular 
vibrations.  Such  a  beam  of  light  is  said  to  be  Circularly- 
Polarised.  Looked  at  from  one  side  the  vibration  would 
apparently  progress  like  a  screw. 


XV.] 


POLAEISATION. 


515 


Fig.156. 


A  common  corkscrew  is  a  right-handed  spiral.  A  simple  experiment 
with  a  string,  one  end  of  which  is  fixed  to  a  wall,  while  the  other  is  held  in 
the  hand,  will  show  that,  in  order  to  impress  upon  the 
string  the  right-handed  spiral  form,  we  must  rotate 
the  free  end  in  a  direction  opposed  to  that  of  the 
hands  of  a  watch.  Such  is  the  movement  in  a  so- 
called  right-handed  circularly-polarised  beam  of 
light.  When  the  rotation  is  in  the  opposite  sense, 
the  circularly-polarised  ray  is  left-handed.  Fig.  156 
shows  the  direction  of  propagation  and  of  rotation, 
and  the  forms  assumed  by  the  vibrating  ether  in  a 
right  (R)  and  in  a  left-handed  (L)  ray  respectively. 

(6.)  The  circle  may,  by  a  difference  of 
phase  other  than  J  period,  be  converted  into 
an  ellipse.  A  beam,  the  ether  in  which  rotates 
in  ellipses,  is  one  of  Elliptically-Polarised 
light ;  this  again  may  be  right  or  left-handed. 

(<?.)  The  periods  of  vibration  in  the  two 
planes  may  not  be  equal,  but  may  be  commen- 
surable. A  beam  of  light  of  this  kind  would 
present  movements  which,  looked  at  end-on, 
would  present,  for  each  portion  of  the  vibrating  ether,  figures 
like  those  of  Figs.  35-40,  etc. 

(6?.)  The  periods  of  vibration  in  the  two  planes  may  not  be 
commensurable  or  even  constant,  and  further,  the  vibration  in 
each  plane  may  be  variable  in  its  amplitude.  In  such  cases  the 
vibrations  would  rapidly  run  through  a  great  variety  of  figures, 
circles,  ellipses,  figures  of  eight,  and  non-reentrant  complex 
harmonic  curves  of  every  kind.  This  is  the  condition  of  a 
beam  of  Common  Light.  No  single  plane  has  any  advantage. 
If  for  a  moment  the  amplitude  in  any  particular  plane  prepon- 
derate, this  is  but  momentary  :  and  since  the  most  irregular 
transverse-vibration  can  be  resolved  into  a  vibration  up-and- 
down,  and  one  side-to-side,  or  may  be  resolved  in  any  other 
two  planes  arbitrarily  chosen  at  right  angles  to  each  other,  a 
beam  of  common  light  may  be  held  to  be  the  result  of  the 
superposition  of  two  -simultaneous  irregular  transverse-vibra- 
tions, each  plane-polarised,  each  possessed  of  half  the  energy 
of  the  whole  vibration,  and  both  propagated  with  the  same 
velocity  through  the  Ether. 

The  doctrines  of  composition  and  resolution  of  harmonic 
motion  are  applicable  to  each  small  portion  of  the  Ether  within 
such  a  transversely- vibrating  beam,  just  as  they  ar'e  to  trans- 
versely-vibrating strings. 


516  OF   ETHEK-WAVES.  [CHAP. 

A  beam  of  common  light  encountering  an  object  which  is 
selectively  transparent  to  vibrations  in  one  plane,  but  opaque  to 
vibrations  in  a  plane  at  right  angles  to  this,  will  have  the  latter 
vibrations  extinguished ;  it  will  lose  half  its  energy  ;  the  beam 
to  which  the  object  is  transparent  —  the  transmitted  beam  — 
will  have  half  the  energy  of  the  original  beam  ;  and  all  its' 
vibrations  being  executed  in  one  plane,  it  will  be  a  beam  of 
Plane-polarised  Light.  A  body  which  acts  in  this  way  on  a 
beam  of  common  light  is  called  a  Polariser. 

A  beam  of  plane-polarised  light  falling  on  a  polariser  will, 
should  the  plane  in  which  its  vibrations  are  executed  happen  to 
coincide  with  the  plane  of  those  vibrations  to  which  the  polariser 
is  transparent,  be  found  to  pass  freely  through  it :  if  the  plane 
of  vibration  be,  on  the  other  hand,  a  plane  at  right  angles  to  the 
plane  of  the  freely-transmitted  vibrations,  no  vibration  can  get 
through,  no  light  is  transmitted,  and  to  plane-polarised  light 
vibrating  in  such  a  plane  the  selectively-transparent  polariser 
proves  perfectly  opaque.  If  the  condition  be  intermediate  — 
that  is,  if  the  plane  of  the  actual  vibrations  and  the  plane  of 
free  transmission  through  the  polariser  be  neither  coincident 
nor  at  right  angles  to  one  another  —  then  the  actual  vibrations 
of  the  plane-polarised  light  must  be  resolved  into  two  sets  of 
component  plane-polarised  vibrations  at  right  angles  to  one 
another ;  these,  looked  at  end-on,  carry  out  the  principle  of 
Fig.  42 ;  and  of  these  components  —  the  one  in  the  plane  of 
free  transmission,  the  other  at  right  angles  to  that  plane  —  the 
former  is  transmitted,  while  the  latter  is  extinguished  by  absorp- 
tion, its  energy  becoming  converted  into  heat. 

When  ordinary  light  has  had  its  vibrations  in  a  given  plane  quenched 
in  a  certain  proportion,  while  in  the  plane  at  right  angles  to  this  they  are 
not  quenched  at  all,  or  not  quenched  in  equal  proportion,  —  it  is  in  a  state 
of  partial  polarisation,  and  is  called  Partially-Polarised  Light. 

A  plane-polarised  beam  of  light  may  not  only  be  resolved 
into  two  at  right  angles  to  one  another  and  coincident  in  phase, 
but  also  (see  Figs.  46  and  47)  into  two  circularly-polarised 
beams,  the  one  left-handed,  the  other  right-handed.  If  it  be 
supposed  that  a  transparent  body  or  a  region  of  space  is  so 
peculiarly  constituted  or  stressed  that  a  left-'handed  circularly- 
polarised  beam  travels  more  rapidly  through  it  than  a  right- 
handed  one  can,  then,  on  passing  a  plane-polarised  beam  of  light 
through  such  a  region,  the  left-handed  circular  component 
emerges  with  its  phase  less  advanced  than  the  right-handed  one  ; 


xv.]       /  POLARISATION.  517 

but  the  plane-polarised  light  equivalent  to  the  synthesis  of  two 
such  circularly-polarised  beams  can  no  longer  be  due  to  a  vibra- 
tion in  the  original  plane ;  the  plane  has  been  turned  round  a 
longitudinal  axis  in  the  centre  of  the  beam  ;  and  the  farther  a 
plane-polarised  beam  travels  through  such  a  body  or  region  of 
space,  the  greater,  in  a  direct  ratio,  will  be  the  rotation  of  the 
plane  of  polarisation  of  that  beam  —  a  result  observed  in  many 
cases,  and  to  be  described  under  the  head  of  the  so-called  Rota- 
tory Polarisation. 

When  the  left-handed  component  is  relatively  accelerated  in  its  trans- 
mission, or  retarded  in  its  phase  of  emergence,  the  plane  is  rotated  to  the 
left  —  i.e.,  to  an  observer  stationed  at  the  source  of  light  the  plane  of  polari- 
sation of  the  receding  beam  is  seen  to  rotate  in  a  direction  opposed  to  that 
of  the  hands  of  a  watch  :  the  more-rapidly-travelling  left-handed  component 
is  at  any  point  less  advanced  in  phase  than  the  more-slowly-travelling  right- 
handed  component  at  the  same  point ;  the  contrary-to-clock  rotation  of  the 
right-handed  ray  prevails  over  the  less-advanced  clockwise  rotation  of  the  left- 
handed  ray. 

REFLEXION  AND  REFRACTION. 

When  a  ray  of  light,  travelling  in  a  rarer  medium,  strikes  the 
surface  of  an  optically  denser  transparent  medium,  some  light  is 
reflected,  some  refracted ;  and  if  there  had  been  neither  absorp- 
tion nor  scattering,  the  energy  of  the  reflected  ray,  together  with 
that  of  the  refracted  ray,  would  have  been  equal  to  that  of  the 
incident  ray. 

Lord  Rayleigh  finds  that  there  is  always  some  reflexion,  even  when  the 
optical  density  is  the  same  in  the  two  media. 

The  incident  ray  and  the  reflected  ray  are  in  one  plane : 
this  is  called  the  plane  of  incidence.  The  plane  of  incidence  is 
at  right  angles  to  the  reflecting  surface  at  the  point  of  incidence 
and  reflexion. 

The  vibration  of  the  incident  light  may  be  either  at  right 
angles  to  the  plane  of  incidence,  —  i.e.,  parallel  to  the  reflecting 
surface,  —  or  it  may  be  in  that  plane,  in  which  case  the  vibrating 
ether  will  not  brush,  but  will  strike  the  reflecting  surface :  or  it 
may  be  in  any  intermediate  direction  or  sequence  of  directions. 

Fresnel,  in  investigating  this  subject,  made  the  following  assumptions, 
viz.  —  (1)  that  of  the  conservation  of  vires  vivce*  or,  as  we  would  now  say, 
the  Conservation  of  Energy ;  (2)  that  the  movement  of  the  incident  ray 
merges  continuously  into  that  of  the  refracted  ray ;  and  (3)  —  a  very  arbi- 

*  Vis  viva  =  Smw2 ;  energy  =  S(iwv2). 


518  OF  ETHEB-WAVES.  [CHAP. 

trary  assumption  —  that  differences  of  velocity  of  ether-waves  in  different 
substances  are  due  to  differences  of  density  of  the  ether,  whose  elasticity 
remains  unaffected. 

From  these  postulates  he  showed  by  mathematical  reasoning  — 

(1.)  If  the  incident  beam  be  a  plane-polarised  beam,  vibrat- 
ing parallel  to  the  reflecting  surface,  the  refracted  and 
the  reflected  beams  are  also  plane-polarised  beams  whose  vibra- 
tions are  parallel  to  the  original  direction. 

(2.)  The  frequency  of  vibration  is  unaffected  by  reflexion 
and  refraction:  the  colours  of  the  incident,  reflected,  and 
refracted  rays  are  the  same. 

(3.)  The  angle  of  incidence  is  equal  to  the  angle  of  reflexion. 

(4.)  The  sine  of  t,  the  angle  of  incidence,  bears  to  the  sine 
of  g,  the  angle  of  refraction,  a  constant  ratio,  /3,  the  "Index  of 
Refraction;"  the  numerical  value  of  fi  depends  upon  the 
nature  of  the  two  media.  Sin  i  —  /3  sin  g. 

(5.)  The  amplitudes  of  the  three  rays  are  related  to  one 
another  in  the  following  way :  — 

The  angle  of  incidence  is  i ;  that  of  refraction  is  g.     Then  a;,  the  ampli- 
tude of  the  reflected  ray,  is  equal  to  ( ^ »^  ),  while  a ,,,  the  amplitude 

V   sin  (c  +  g)    J 

of  the  refracted  ray,  is  equal  to  (  ~  sm  §  cos  *  ]    times  the  original  ampli- 

V  sin  (t  +  §)  / 
tude  a.     From  the  former  formula  we  learn  — 

(a)  That  the  greater  the  angle  of  incidence  t,  the  greater  is  a,,  the 
amplitude  of  the  reflected  ray. 

(&)  That  when  the  incident  ray  is  so  nearly  parallel  to  the  surface  of  the 
glass  as  simply  to  graze  it,  the  reflected  ray  is  equal  to  the  incident 
one,  and  the  amplitude  of  the  refracted  ray  =  0. 

(c)  That  when  t  is  greater  than  g,  a/  is  negative ;  while  when  t  is  less 
than  g,  at  is  positive :  or  in  words,  when  a  ray  strikes  the  sur- 
face of  a  denser  medium,  the  reflected  ray  is  a  direct  continua- 
tion of  the  incident  ray,  changed  in  direction  ;  while  when  light 
travels  in  a  denser  medium,  half  a  wave-length  is  lost  on  reflexion 
at  the  surface  of  a  rarer  medium.  This  conclusion  is  independent 
of  any  hypothesis  as  to  particles,  such  as  that  by  which  we  have 
already  illustrated  the  same  propositions  on  pages  124  and  125. 

(6.)  The  respective  intensities  of  a  reflected,  a  refracted, 
and  the  incident  ray,  in  ergs  per  cub.  cm.,  are  in  the  ratios  of 

sin2  (t  -  g)  .       sin2  2t        .  ^ 
sin2  (t  +  g)       sin2  (i  +  g) 

(7.)  When  light,  travelling  in  a  denser  medium,  strikes  the 
surface  of  separation  between  the  denser  and  a  rarer  medium  at 
such  an  angle  i  that  sin  L  is  greater  than  /3,  both  reflexion  and 
refraction  are  possible  ;  but  if  the  angle  of  incidence  be  such  that 


xv.]  REFRACTION.  519 

sin  i  =  /3,  then  sin  g  =  1, .  and  the  ray  refracted  into  the  rarer 
medium  grazes  the  reflecting  surface,  for  g  =  90° ;  and  any  light 
falling  still  more  obliquely  upon  the  surface  will  be  totally 
reflected,  the  reflected  ray  possessing  the  whole  energy  of  the 
original  incident-ray.  In  the  last  case  sin  §,  =  sin  *./£,  would 
be  greater  than  1,  and  §,  the  angle  of  refraction,  an  impossible 
angle ;  there  is  therefore  no  refraction. 

In  Fig.  157  the  ray  AO  is  partly  reflected  to  A",  and  partly 
refracted  to  A' ;  the  ray  BO  is  partly  reflected  to  B",  and  partly 
refracted  to  B',  the  refracted  por- 
tion   grazing   the    refracting   sur- 
face;   the    ray    CO    is    wholly          - 
reflected    to    C",    and    is    not       Jj 
refracted    at    all    into    the    rarer     H    •  E(     \ 
medium.  /"'    C\N\\ 

As  examples  of  Total  Reflex-  « 
ion  we  may  take  a  tumbler  of  **» 
water  held  above  the  head;  it 
will  give  a  clear  mirror-image  of 
the  objects  on  the  table  below 
it;  a  bubble  of  air  in  water,  or  a  test  tube  containing  air 
immersed  in  water,  will,  when  looked  at  under  a  certain  angle, 
appear  to  have  as  bright  a  mirror-surface  as  that  of  mercury. 
In  Fig.  158,  light  entering  a  total-reflex- 
ion prism  at  M  is  totally  reflected  at  O,  rig  158 
and  travels  towards  R,  which  it  reaches 
without  refraction.  Total  reflexion  is 
also  exemplified  by  the  silvered  glass 
bars  used  by  surgeons  and  microscopists 
to  transmit  light. 

(8.)  The    energy  of   a   ray   totally 
reflected  is  equal  to  that  of  the  incident  ray ;  there  is,  however, 
a  slight  retardation  of  its  phase. 

These  laws  all  apply  to  vibrations  executed  at  right  angles 
to  the  plane  of  incidence,  and  were  deduced  by  Fresnel  from 
the  fundamental  hypotheses  already  mentioned. 

Let  us  now  turn  to  the  reflexion  and  the  refraction  of  plane- 
polarised  light  whose  vibrations  are  at  right  angles  to  these,  and 
are  thus  executed  in  the  plane  of  incidence. 

In  the  refracted  and  reflected  rays  the  vibrations  will  still  be  in  the 
plane  of  incidence,  but  they  cannot,  after  encountering  the  refracting  sur- 
face, remain  parallel  to  their  original  direction.  The  consequence  deduced 


520 


ETHER-WAVES. 


[CHAP. 


by  Fresnel  from  this  is,  that  while  the  ordinary  laws  of  refraction  and 
reflexion  are  obeyed  by  such  plane-polarised  light  so  far  as  directions  are 
concerned,  tlie  relative  amplitudes  and  intensities  of  the  incident,  the 
reflected  and  the  refracted  light,  are  not  the  same  as  they  were  in  the  pre- 
ceding case,  but  are  now  respectively  in  the  ratio 

tan  (i  —  o)  2  cost  sin  o 

1       :  -          -&     :  -.  —  7  -  r  --  f  -  r 
tan  (i  +  3)       sin  (L  +  g)  cos  (t  -  g) 

tan2  (t  -  g)  .  sin2  2t 

tan*  (i  +       '        '  *-  (^tensities.) 

reflected 


(Amplitudes  0 


8in'(t  +  8)cos*(i-8) 

refracted. 


Fig.  15g 


incident 

According  to  Fresnel's  formulae,  the  intensity  of  the  reflected  light  is, 
in  a  particular  case,  =  0  ;  that  is,  when  {tan2  (t  —  §)  -f-  tan2  (t  +  g)  }  =  0,  or 
when  (t  +  g)  =  90°.  In  fact,  however,  it  does  not  entirely  vanish  :  it  only 
attains  a  minimum. 

In  Fig.  159  the  ray  AO,  whose  vibration  is  in  the  plane  of 
incidence,  falls  at  such  an  angle  i  that  it  is  refracted  along  OA'  ; 

if  it  had  been  reflected  at  all  it  would 
have  been  reflected  along  OA"  ;  at  one 
particular  angle  of  incidence,  AON, 
the  refracted  ray  OA'  and  the  reflected 
ray  OA"  (if  there  had  been  such  a 
ray)  would  have  been  at  right  angles 
to  one  another.  When  the  angle  of 
incidence  and  the  angle  of  refraction 
together  make  up  a  right  angle,  there 
is  no  reflected  ray  ;  no  vibration  effected  in  the  plane  of  inci- 
dence is  reflected  at  all,  and  a  plane-polarised  ray  of  this  kind 
falling  at  the  appropriate  angle  of  incidence,  however  bright  it 
may  be,  will  fail  to  be  reflected  from  a  mirror.  This  is  a  case 
of  Total  Refraction  ;  the  whole  of  the  energy  of  the  incident 
ray  is  in  the  refracted  ray.  When  the  incident  light  grazes  the 
refracting  surface,  the  reflected  beam  also  grazes  it,  and  there  is 
no  refraction. 

When  light  whose  vibration  is  in  the  plane  of  incidence  is  totally  reflected, 
it  undergoes  a  slight  retardation  of  phase,  less  than  that  observed  in  the  case 
of  light  whose  vibration  is  at  right  angles  to  the  plane  of  incidence. 

From  these  results  it  is  easy  to  pass  to  the  case  of  light  whose 
vibration  may  be  considered  to  be  the  result  of  the  composition 
of  vibrations,  parallel  to  the  plane  of  incidence,  with  others  at 
right  angles  to  that  plane.  Such  light  may  be  plane-polarised, 
vibrating  in  some  plane  neither  the  plane  of  incidence  nor  at 
right  angles  to  it  ;  it  may  be  circularly-  or  elliptically-polarised 
light,  or  it  may  be  common  light.  In  all  these  cases  each  com- 
ponent is  reflected  and  refracted  according  to  its  own  laws.  In 


xv.]  REFRACTION.  521 

this  way  the  reflected  and  refracted  rays  may  come  to  differ  in 
character  from  one  another  and  from  the  original  ray.  As  an 
extreme  case,  let  a  beam  of  common  light  fall  upon  a  piece  of 
glass  at  such  an  angle  that  i  -f-  3  =  90°  (Fig.  159) ;  of  that  part 
of  the  incident  beam  which  is  due  to  vibrations  executed  at 
right  angles  to  the  Plane  of  Incidence,  there  is  reflected  a 
certain  proportion;  of  that  part  of  the  incident  ray  which  is 
due  to  vibrations  parallel  to  this  plane,  there  is  reflected 
none.  The  light  reflected  from  glass  at  such  an  angle  has  its 
vibrations  thus  restricted  to  a  plane  at  right  angles  to  the  plane 
of  incidence ;  it  is  plane-polarised  light.  A  piece  of  glass  held 
in  the  course  of  a  beam  of  common  light  at  the  proper  angle 
may  thus  be  used  as  a  simple  means  of  obtaining  a  beam  of 
light  Plane-Polarised  by  Reflexion. 

The  precise  angle  of  incidence  i,  for  which  i  +  g  =  90°,  is  called  the 
Angle  of  Complete  Polarisation.  It  depends  upon  ft,  the  refractive 
index  of  the  refractive  substance,  and  the  angle  t  is  such  that  tan  t  =  (3 ;  for 
sin  i  =  ft  sin  g  =  ft  cos  t.  Polarisation  is,  however,  never  complete  on  one 
reflexion,  except  in  substances  whose  refractive  index  is  146 ;  in  others  it  is 
only  a  maximum  at  the  angle  whose  tangent  is  ft.  In  metals  the  departure 
from  completeness  is  most  marked.  There  is  also  no  angle  of  complete 
polarisation  for  the  refracted  ray. 

Such  are  the  consequences  deduced  by  Fresnel  from  his 
hypothesis,  that  the  Ether  is  condensed  around  the  particles  of 
ordinary  matter  while  its  elasticity  remains  unaffected. 

Neumann  and  MacCullagh,  from  the  contrary  hypothesis  —  that  the 
density  of  the  Ether  is  the  same  in  all  substances,  while  its  elasticity  or 
rigidity  is  different  in  different  substances  —  deduced  a  set  of  conclusions 
precisely  similar  to  those  above  given,  so  far  as  regards  all  the  results  which 
it  is  possible  to  verify  by  experiment,  but  with  this  remarkable  difference, 
that  the  properties  attributed  by  Fresnel  to  plane-polarised  ether-waves 
whose  oscillations  are  effected  at  right  angles  to  the  plane  of  incidence  were, 
by  the  latter  writers,  found  to  be  associated  with  plane-polarised  light  whose 
vibrations  are  parallel  to  that  plane,  and  vice  versa.  The  fundamental  pos- 
tulates of  the  two  theories  are  closely  associated  with  these  consequences. 

We  must  now  turn  to  a  point  of  terminology.  When  a 
beam  falls  upon  a  mirror  at  the  angle  of  complete  polarisation, 
the  reflected  ray,  if  there  be  any,  is  plane-polarised ;  it  is  said 
to  be  polarised  in  the  plane  of  incidence,  and  the  plane  of 
incidence  is  called  its  plane  of  polarisation.  According  to 
Fresnel's  view,  the  vibrations  in  this  beam  are  supposed  to  be 
executed  in  a  direction  at  right  angles  to  the  plaice  in  which 
the  beam  is  thus  said  to  be  polarised. 


522  OF  ETHER-WAVES.  [CHAP. 

The  question  between  the  followers  of  Fresnel  and  Cauchy  on  the  one 
hand,  and  those  of  Neumann  and  MacCullagh  on  the  other,  may  thus  be 
stated :  Are  the  vibrations  of  plane-polarised  light  executed  at  right  angles 
or  parallel  to  the  plane  in  which  the  light  is  said  to  be  polarised  —  a  plane 
which  by  convention  is  called  the  plane  of  polarisation  ?  On  the  one  hand, 
there  appears,  on  Fresnel's  assumptions,  to  be  a  serious  objection  to  his  view; 
this  objection  is  based  on  certain  mathematical  difficulties  arising  from  the 
admission  of  his  second  hypothesis,  that  the  movement  of  the  ether  in  the 
second  medium  is  continuous  with  that  in  the  first ;  for  this  hypothesis  is 
found  to  lead  directly  to  the  conclusion  that  the  density  of  the  ether  in  the 
two  media  must  be  the  same,  and  is  therefore  one  which  is  incompatible 
with  his  third  hypothesis. 

Professor  Stokes,  on  the  other  hand,  reasons  in  favour  of  the  movement 
perpendicular  to  the  plane  of  polarisation,  and  the  following  is  a  sketch  of 
his  argument.  The  vibrations  in  a  beam  of  light  are  admittedly  transverse 
to  the  direction  of  propagation.  Consider  a  polarised  reflected  beam;  the 
vibrations  are, admittedly  symmetrical  with  regard  to  the  plane  of  reflexion  ; 
they  must  be  either  parallel  to  it  or  at  right  angles  to  it.  Now,  suppose  a 
horizontal  beam  to  strike  a  haze  and  then  to  be  reflected  vertically  upwards 
into  the  observer's  eye.  The  reflected  light  is  undoubtedly  polarised  in  a 
plane  passing  through  the  source  of  light,  the  point  of  the  haze  looked  at, 
and  the  observer's  eye ;  that  is,  it  is  polarised  in  the  plane  of  reflexion.  If 
the  vibrations  be  parallel  to  this  plane  in  the  ascending  beam,  they  must 
either  have  been  originally  parallel  to  the  direction  of  propagation  of  the 
incident  beam,  which  is  impossible ;  or  else  they  must  have  been  changed 
in  their  vibratory  direction  by  impact  against  obstacles  smaller  than  their 
own  wave-length,  which  is  improbable.  If,  on  the  other  hand,  the  vibra- 
tions be  at  right  angles  to  the  plane  of  reflexion,  the  general  direction  of 
vibration  is  the  same  after  reflexion  as  before  it.  The  latter  view  is  prefer- 
able :  and  according  to  it,  the  vibrations  are  at  right  angles  to  the  plane  of 
polarisation.  He  has  also  shown  that  certain  phenomena  of  Diffraction 
afford  a  crucial  test,  the  experimental  answer  to  which  is  in  favour  of  the 
proposition  that  the  vibrations  are  perpendicular  to  the  plane  of  polarisation. 

Though  Neumann  and  MacCullagh's  hypothesis  works  out  in  many 
respects  better  than  Fresnel's,  it  is  therefore  inadmissible ;  and  besides,  it 
would  lead  (Lord  Rayleigh)  to  the  conclusion  that  there  are  two  angles 
of  polarisation,  which  is  not  the  case. 

When  ordinary  monochromatic-light  is  reflected  at  any  angle 
other  than  that  of  complete  polarisation,  the  reflected  and  the 
refracted  beam  must  both  be  partially  polarised,  and 
each  will  be  polarised  to  an  equal  extent,  though  in  con- 
trary senses.  In  the  reflected  beam,  light  polarised  in  the  plane 
of  incidence  preponderates  until  the  incidence  is  a  grazing  one : 
in  the  refracted  ray,  light  polarised  at  right  angles  to  that  plane 
preponderates  to  an  exactly  equal  extent,  so  far  as  the  energy 
of  the  vibration  is  concerned. 

When  mixed  light,  such  as  white  light,  falls  upon  a  refracting  surface, 
then,  since  /?,  the  index  of  refraction,  is  different  for  each  kind  of  light,  the 
proportions  of  each  coloured  light  present  in  the  reflected  and  the  refracted 


xv.]  REFLEXION  AND   REFRACTION.  523 

rays  respectively  are  different ;  white  light,  when  reflected  from  a  normally 
refracting  surface,  always  becomes  bluer,  the  refracted  light  redder ;  and  we 
have  seen  this  to  account  for  the  blue  colour  of  haze. 

The  intensity  of  a  reflected  ray  is  represented  by  {sin2  (i-§)  -*-sin2  (i+g)}. 
If  we  pass  to  a  ray  of  greater  refrangibility  we  alter  §  to  g,,  and  our  inten- 
sity becomes  {sin2  (t  -  g,)  H-  sin2  (t  +  g,)},  which  is  always  greater  than  the 
former.  Where  the  actual  refraction  is  greater,  the  corresponding  angle  of 
refraction  is  less ;  wherefore,  in  this  case,  g,  is  a  smaller  angle  than  g. 

Near  the  incidence  of  total  reflexion  some  colours  may  be 
totally  reflected,  others  in  part  refracted ;  near  the  incidence  of 
total  refraction  or  complete  polarisation  an  analogous  result  is 
obtained.  The  slower  waves  of  heat  have  a  lower  refractive 
index,  and  must  therefore  strike  a  refracting  surface  at  an 
angle  somewhat  more  vertical  or  nearer  the  normal  than  those 
of  light,  in  order  to  become  completely  polarised. 

We  thus  learn  that  refraction  and  reflexion  may  materially 
modify  the  character  of  light  which  strikes  on  a  refracting  sur- 
face. If,  however,  we  attend  only  to  the  Direction  of  the 
respective  rays,  and  not  to  their  states  of  polarisation,  or  to 
their  colour,  etc.,  we  may  study  the  effect  of  mirrors  or  lenses 
in  modifying  the  direction  of  an  incident  beam  of  light,  whether 
this  be  plane-fronted,  convergent,  or  divergent.  We  shall  first 
consider  the  case  of  monochromatic  light. 

Mirrors.  —  Plane  mirrors  reflect  light  in  such  a  way  that 
the  reflected  waves  are,  as  regards  their  direction,  precisely  such 
as  might  have  come  from  an  object  or  source  of  light  situated 
behind  the  reflecting  surface,  and  at  a  distance  behind  it  equal 
to  the  distance  between  the  object  and  the  mirror.  This  is 
illustrated  in  Fig.  61,  and  it  is  a  matter  of  familiar  knowledge 
in  the  use  of  looking-glasses,  and  in  the  appearance  presented 
by  the  inverted  image  of  objects  on  the  shore  when  these  are 
seen  reflected  on  a  surface  of  smooth  water.  The  images  then 
seen  are  apparently  at  the  same  horizontal  distance  from  the 
eye  as  the  objects  themselves ;  while 
the  image  of  a  slightly-clouded  sky, 
as  reflected  in  very  smooth  turbid 
water,  appears  extremely  deep. 

The   apparent   inversion    of    an 
image  in  a  mirror  is  a  natural  result 
of  the  fact  that   the  image  of   each 
point  is  apparently  situated  behind  the  mirror.     Fig.  160  ex- 
plains this  result. 

The  reader  may  construct  a  diagram  to  show  how  it  is  that  a  mirror 


524  OF   ETHER-WAVES.  [CHAP. 

about  half  a  man's  height,  and  placed  opposite  the  upper  half  of  his  body, 
will  give  him  a  full-length  image  of  himself. 

The  use  of  mirrors  fixed  at  an  angle  of  about  45°,  in  order 
to  cast  the  light  of  the  sky  into  a  room,  or  in  order,  when  fixed 
outside  a  window,  to  enable  a  person  within  a  room  to  see  the 
passengers  in  the  street  outside,  is  sufficiently  intelligible. 

In  medicine  the  same  principle  is  utilised  in  the  Laryngoscope. 
Light  falling  from  a  lamp  upon  a  concave  mirror'  is  cast  upon  a  small 
plane-mirror,  held  by  means  of  a  long  handle  at  an  angle  of  45°  within  the 
pharynx  of  the  patient ;  the  light  is  reflected  and  passes  down  towards  the 
larynx,  which  is  illuminated  and  becomes  a  source  of  light;  rays  returning 
from  this,  passing  upwards,  strike  the  small  pharyngeal-mirror,  and  are 
diverted  by  it  so  that  they  traverse,  horizontally,  the  cavity  of  the  mouth, 
and  pass  through  a  small  aperture  in  the  centre  of  the  concave  mirror  into 
the  eye  of  the  observer,  who  is  thus  enabled  to  see  the  larynx  and  windpipe. 

In  a  mirror,  the  object  and  the  image  are  interchangeable ; 
so  that,  for  example,  a  person  cannot  look  at  another  in  a 
mirror  without  the  person  looked  at  being  also  able  to  see  him 
in  the  mirror,  unless  indeed  the  first  observer  be  in  darkness. 

The  brightness  and  distinctness  of  an  image  depend  upon 
the  polish  of  the  mirror  and  on  its  not  scattering  the  light 
which  falls  upon  it.  A  smooth  clean  mirror  is,  itself,  almost 
invisible. 

A  piece  of  polished  platinum  reflects  light  as  well  when  it  is  white-hot 
as  when  it  is  cold. 

The  image  of  an  object  in  a  mirror  can  never  be  brighter 
than  the  object  itself,  however  smooth  the  mirror  be.  Hence, 
if  a  candle  be  held  between  two  parallel  mirrors,  the  long  series 
of  images  produced  by  multiple  reflexion  grows  fainter  as  the 
images  seem  to  grow  more  distant. 

When  we  have  multiple  reflexions  of  light  between  two  polished  plates, 
if  the  plates  be  parallel  and  the  incidence  oblique,  the  reflexions  are  more 
numerous  the  nearer  the  plates  are  to  one  another.  If  the  two  plates  be 
inclined  to  one  another  at  an  angle  of  60°,  the  images  of  a  point  lying 
between  them,  the  image  of  which  is  multiplied  by  repeated  reflexion,  are 
so  situated  that  the  first,  second,  .  .  .  sixth,  form  together  a  symmetrical 
Kaleidoscope-image  of  the  point,  a  group  of  images  ranged  round  a 
central  axis,  while  the  seventh  and  further  images  coincide  with  their 
predecessors.  Similarly  for  such  angles  as  90°,  45°,  30°,  24°,  and  other 
aliquot  parts  of  360°. 

When  a  mirror  is  rotated,  a  beam  of  light  reflected 
from  it  is  deflected  through  an  angle  equal  to  twice  that  of  the 
rotation  of  the  mirror  itself.  In  Fig.  161  S  is  a  source  of  light ; 
AB  a  mirror;  SM  an  incident  ray;  MR  a  reflected  ray.  If  the 


xv.]  MIRRORS.  525 

mirror  be  turned  into  the  position  A'B',  the  reflected  beam  is 
now  MR' :  the  reflected  ray  has  swept  through  the  angle  RMR', 
which  is  equal  to  twice  the  angle  AMA'. 
We  have  already  (p.  122)  consid- 
ered some  cases  of  the  reflexion  of  waves 
at  parabolic  and  elliptical  mirrors,  and 
on  segments  of  spheres.  Parabolic 
mirrors  are  used  when  it  is  desired  to 
bring  the  plane-fronted  light  of  a  dis- 
tcint  star  accurately  to  a  focus,  or  to 
produce  a  parallel  beam  of  light :  in  the  B';/  \R' 

latter  case  the  source  of  light  is  placed 
at  the  focus  of  the  paraboloid.  Spherical  mirrors  may  be 
considered,  if  we  restrict  our  attention  to  those  rays  which  fall 
very  near  the  centre  of  the  mirror,  to  have  an  approximate 
focus  at  the  tip  of  their  "caustic  by  reflexion  "  (Fig.  64). 

When  the  beam  is  broad  the  rays  do  not  converge  accurately  to  a  focus, 
and  the  image  of  a  point  is  a  circle,  brightest  towards  its  centre.  This  is 
the  Spherical  Aberration  of  a  mirror,  which  renders  its  definition,  espe- 
cially of  somewhat  broad  objects,  very  bad.  In  consequence  of  this,  it  is 
often  necessary  to  cut  off  the  lateral  rays  by  a  diaphragm,  which  increases 
the  clearness  of  definition,  though  it  diminishes  the  brightness  of  the  image. 

When  light  coming  from  a  source  at  a  positive  distance, 
c?,  is  reflected  by  a  concave  spherical-mirror,  it  is  reflected  back 
to  an  approximate  focus  at  a  positive  distance  d'.  The  distance 
d  of  the  source,  the  distance  d'  of  the  approximate  focus,  and  r, 
the  radius  of  a  spherical  mirror,  all  measured  positively,  are 
connected  by  the  law  l/t£  +  l/t?'  =  2/r. 

Here  r,  the  distance  of  the  centre  of  curvature  of  the  mirror,  is  to  the 
right,  positive. 

When  the  source  is  infinitely  distant,  1/d  =  0,  and  d1  =  ?r;  the  case  of 
Fig.  64,  page  123.  The  focus  to  which  the  light  converges  in  this  case  is 
called  the  Principal  Focus  of  the  mirror,  and  its  distance  d',  =  4r,  is 
written/,  the  Principal  Focal  Distance.  If  the  course  of  the  light  be 
reversed,  and  d  —f  —  \r,  d'  =  oo,  and  the  light  is  reflected  to  a  focus  at  an 
infinite  distance  ;  it  is  approximately  parallel-rayed  or  plane-fronted. 

When  the  source  is  at  a  definite  distance,  beyond  the  centre  of  the 
sphere,  the  focus  is  between  the  principal  focus  and  the  geometrical  centre 
of  the  sphere ;  conversely,  when  a  source  of  light  is  between  the  principal 
focus  and  the  centre  of  the  sphere,  the  reflected  light  converges  upon  a 
point  at  a  definite  point  beyond  the  centre. 

Light  radiating  from  the  centre  of  a  spherical  mirror,  after  reflexion 
again  converges  upon  the  same  point.  When  the  source  is  between  the 
principal  focus  and  the  mirror,  i.e.  when  d  is  less  than  \r,  d',is  negative, 
and  the  reflected  rays  seem  to  diverge  from  a  point  on  the  other  side  of  the 
mirror,  and  the  Image  of  the  point  is  imaginary  or  Virtual. 


526 


OE  ETHER- WAVES. 


[CHAP. 


Pairs  of  points  at  the  respective  distances  d  and  d\  as 
denned  by  this  formula,  are  called  Conjugate  Foci. 

Conjugate  foci  are  found  in  one  of  the  four  following  relations :  — 

(a.)    Coincident,  both  being  at  the  geometrical  centre  of  the  mirror. 

(ft.)  One  between  the  principal  focus  and  the  centre ;  the  other  beyond 
the  centre. 

(c.)  One  at  the  principal  focus ;  the  other  at  an  infinite  distance  beyond 
the  centre. 

(d.)  One  between  the  principal  focus  and  the  mirror  itself;  the  other, 
a  virtual  focus,  apparently  behind  the  mirror. 

If  the  source  of  light  be  not  in  the  axis  of  the  mirror,  the 
light  is  not  brought  to  a  focus  in  the  axis,  but  at  some  point 
situated  laterally.  In  Fig.  162  the  rays  proceeding  from  A  con- 
verge upon  A',  and  an  eye  situated  in  the  direction  of  B  looking 

towards  the  mirror  will  receive 
rays  which  appear  to  diverge  from 
a  Real  Image  at  the  point  A', 
a  point  which  the  waves  really 
traverse.  If  the  source  of  light 
be  an  extended  object  AZ  (Fig. 
163),  there  will  be  formed  an 
inverted  real  image  Z'A'.  After 
the  reflected  rays  from  the  mirror 
have  reached  this  real  image,  they  diverge  from  every  point  of 
it,  as  if  it  were  a  real  object  suspended  in  space.  In  Fig.  163  the 
object  AZ  produces  a  diminished  and  inverted  real  image  at 
Z'A';  or,  conversely,  an  object  at  Z'A'  will  present  an  inverted 

Fig.163. 


Fig. 162. 


and  enlarged  real  image  to  the  eye  placed  beyond  AZ,  an  image 
which  appears  nearer  than  the  object  itself. 

Suppose  that  the  object  AZ  is  the  face  of  a  person  looking  at  a  concave 
mirror,  it  is  plain  that  the  eyes  in  that  face  are  eyes  situated  beyond  Z'A' : 
a  real  image  of  the  face  is  seen,  diminished  and  inverted,  and  apparently 
situated  at  Z'A',  between  the  observer  and  the  mirror.  As  the  observer 
approaches  the  mirror,  the  image  approaches  him  :  it  appears  to  grow  larger. 
When  his  eye  is  at  the  geometrical  centre  of  the  mirror,  the  inverted  image 


XV.] 


MIRRORS. 


527 


coincides  with  his  face,  and  he  sees  nothing;  when  his  eye  is  between  the 
centre  and  the  principal  focus,  the  inverted  image  is  (or  would  be)  behind 
his  eye,  and  again  he  sees  nothing.  When  his  eye  passes  between  the  prin- 
cipal focus  and  the  surface  of  the  mirror,  the  rays  reflected  seem  to  come 
from  a  virtual  image  behind  the  mirror ;  this  virtual  image  is  erect,  and  is 
larger  the  nearer  the  eye  is  to  the  mirror. 

The  student  will  have  little  difficulty  in  drawing  the  diagrams  appropri- 
ate to  each  case,  and  in  verifying  the  results  even  by  means  of  such  a  simple 
concave-mirror  as  the  inside  of  a  watch-case,  or  of  a  common  large  spoon. 

Convex  Spherical-Mirror.  —  In  Fig.  164  we  see  that  a 
beam  of  light,  plane-fronted  or  parallel-rayed,  and  therefore  com- 
ing, apparently  or  really,  from  an  infinite  distance,  is  so  reflected 
that  it  appears,  after  reflexion,  to  diverge  from  the  principal  focus 
of  the  convex  mirror,  this  point  being  the  tip  of  its  caustic ; 
while  a  beam  diverging  from  any  point  A  appears,  after  reflexion, 
to  diverge  similarly  from  some 
point  A'  situated  between  the 
mirror  and  the  principal  focus. 

The  formula  is  1/d  +  l/d'  *-' 


Fig.164. 


This  formula  is  really  the  same 
as  the  preceding ;  but  r  is  now 
negative. 

When  mixed  coloured- 
light  falls  upon  a  mirror,  all  the  reflected  rays,  whatever  be 
their  wave-lengths  or  relative  intensities,  are  reflected  in  the 
same  directions. 

When  a  mirror  is  flexible  and  has  a  variable  curvature,  as  the  form  of 
the  mirror  is  made  to  vary  irregularly,  intermittently,  or  in  accordance  with 
some  harmonic  law  of  simple  or  complex  variation,  so  the  intensity  of  light 
at  any  point  near  the  focus  varies  irregularly,  intermittently,  or  harmonically. 

Refraction  of  Light.  —  Light-rays  are  bent  when  they  reach 
the  surface  of  separation  between  an  optically  rarer  arid  an  opti- 
cally denser  medium.  In  Fig.  165  a  body  situated  at  S  seems 
to  be  situated  approximately  at  S'. 
In  this  way,  if  a  coin  be  placed 
in  a  basin,  and  the  eye  placed  in 
such  a  position  as  just  not  to  see 
it,  water  poured  into  the  basin 
will  bring  the  coin  into  view. 
The  sun  is  seen  before  he  has 
astronomically  "risen,"  and  con-  s^~ 

tinues  to  be  seen  after  the  true 
"  sunset."     A  stick  placed  in  water  appears  bent,  for  the  image 


Fig.165. 


528 


OF  ETHER-WAVES. 


[CHAP. 


Fig.166. 


of  each  point  of  its  surface  appears  raised  within  the  water  by  an 
amount  proportioned  to  its  depth  beneath  the  surface  of  the 
water ;  each  point  of  the  image  appears  indistinct,  being  brought 
only  to  an  approximate  focus,  and  the  image  of  the  whole  is  blue 
on  one  side,  red  on  the  other,  for,  as  shown  in  Fig.  165,  the 
different  colours  of  white  light  travelling  from  any  point  S  are 
unequally  refracted  into  the  air. 

When  light  passes  through  a  number  of  parallel-sided  transparent  plates 
of  different  densities,  the  total  angular-deviation  produced  is  the  same  as  if 
the  last  of  them  had  alone  stood  in  the  course  of  the  transmitted  light. 

Prisms  —  Monochromatic  Light.  —  If  we  confine  our  atten- 
tion to  a  single  ray  of  light  impinging  upon  a  prism  surrounded 
by  a  single  medium,  we  may  trace  the  course  of  the  corresponding 
wave-front,  as  in  Fig.  166.  The  light  travelling  from  S  strikes 

at  the  angle  of  incidence  t; 
it  is  refracted  at  the  angle  § ; 
sin  L  =  ft  sin  g  (i.) .  It  strikes 
the  second  face  at  the  angle 
§' ;  it  leaves  the  glass  at  the 
angle  t';  sin  */=/3sing'  (ii.)- 
Further,  the  angles  g  +  §'  can 
be  proved  together  equal  to 
the  angle  A;*  o_+o/  =  A 

(iii.).  Lastly,  if  the  angle  between  the  incident  ray  SD  and 
the  deviated  ray  ES'  be  S,  then  i  +  i'  =  8  +  A  (iv.).f  From 
these  four  equations,  which  involve  0  together  with  the  six 
angles,  t,  i\  g,  §',  A,  S,  we.  may,  if  we  know  any  three  of  these 
angles  (say  A,  £,  and  S),  determine,  for  the  particular  mono- 
chromatic light  employed,  the  Index  of  Refraction  ft  of  the 
material  of  the  prism ;  this  numerical  quantity  ft  expresses,  as 
compared  with  unity,  the  relative  slowness  of  ether-waves 
in  the  prism  as  compared  with  that  in  the  medium  surrounding 
the  prism. 

Thus,  if  a  particular  glass  prism  have  the  index  1-5  in  air  for  a  mono- 
chromatic yellow  light,  that  light  travels  1-5  times  as  fast  in  air  as  it  does 
in  the  glass  out  of  which  the  prism  has  been  cut. 

*  The  normals  to  the  faces,  drawn  at  the  points  of  entrance  and  exit  of  the  ray, 
cross  one  another  at  X ;  the  quadrilateral  AX  has  its  internal  angles  equal  to  four 
right  angles;  two  of  these  (at  D  and  E)  are  right  angles:  the  remaining  two  (A 
and  X)  are  therefore  equal  to  two  right  angles.  In  the  triangle  DEX,  the  angles 
g  +  g'  +  X  are  again  equal  to  180°;  whence  §  +  §'  =  A. 

t  The  angles  g  +  g'  =  A;  §  +  §'  +  5  =  5  +  A;  5  =  OED  +  ODE ;  /.  g  +  g'  +  OED 
+  ODE  =  5  +  A;  but  §  +  ODE  =  XDO  =  i;  and  g'  4-  OED  =  XEO  =  i  ;  .-.  t  +  t'  = 
5  +  A. 


xv.j  INDEX   OF   REFRACTION.  529 

To  find  tne  Refractive  Index  of  a  Liquid,  the  amount  of 
refraction  must  be  directly  observed.  Two  telescopes  arranged 
radially  on  a  vertical  circle  or  alidade;  light  traverses  the 
one,  and  is  rendered  parallel  by  it ;  it  then  impinges  on  the  level 
surface  of  liquid  in  a  certain  vessel,  penetrates  it,  leaves  the 
vessel,  passing  normally  through  the  glass  bottom  of  the  vessel, 
and  enters  the  second  telescope,  which  it  traverses.  The  rela- 
tive angles  made  by  the  two  telescopes  with  a  fixed  bar  indicate 
the  angles  of  incidence  and  of  refraction,  which  being  known, 
the  index  (3  is  known.  The  angle  of  total  reflexion  may  also  be 
observed,  and  the  index  of  refraction  calculated  from  this; 
sin  i  —  /3. 

The  Refractive  Index  of  a  Gas  is  found  by  processes  which 
depend  on  the  amount  of  retardation  suffered  by  light  in  a  long 
column  of  that  gas,  as  will  be  seen  under  "  Interference,"  and 
as  has  already  been  explained  under  "  Velocity  of  Light  "  (the 
rotating-mirror  method,  page  512). 

If  instead  of  a  prism  filled,  say,  with  water  in  air,  we  use  a 
prism  filled  with  air  and  submerged  in  water,  the  deviation  of  a 
ray  travelling  in  the  water  will  be  equal,  but  of  opposite  sense 
to  that  of  a  ray  travelling  in  air  and  refracted  by  the  water- 
prism. 

The  refractive  index  of  organic  substances  is  found  to  have  a  close  rela- 
tion to  their  chemical  constitution.  If  an  organic  liquid  contain  carbon, 
hydrogen,  and  oxygen,  and  if  its  density  be  p,  it  is  found  that  the  numerical 
quantity  {(/?  —  l)//o  x  molecular  weight]  is  constant,  even  though  the  den- 
sity be  varied  by  changes  of  temperature,  and  it  is  called  the  Molecular 
Refractive  Power  of  the  substance.  Instead  of  /3  —  1,  the  factor 
{(/3^  —  1)  -^  (/?£,  +  -)}  gives  very  good  approximations  to  fact,  applicable  to 
the  same  substance  in  both  the  liquid  and  the  gaseous  state.  The  molecu- 
lar refractive  power  depends  on  the  chemical  composition  and  on  the  chemi- 
cal constitution  of  the  substance ;  it  is  the  sum  of  definite  numbers,  one  for 
each  atom  of  the  element,  these  numbers  (or  Atomic  Refractions)  being 
different,  for  O,  N,  C,  S,  etc.,  according  to  the  chemical  part  which  each 
atom  plays  in  the  molecule. 

If  in  the  last  figure  (166)  light  start  from  S',  it  will  retrace 
the  line  S'EDS ;  and-the  same  result  will  follow  if  a  mirror  -at 
S'  turn  back  the  light  coming  from  S  :  on  its  direction  being 
reversed,  light  will  retrace  its  path  —  a  proposition  appli- 
cable to  waves  in  general. 

Minimum  Deviation.  —  It  is  only  when  the  prism  is  turned 
into  such  a  position  that  *,  the  angle  of  incidence,  becomes  equal 
to  the  angle  t',  that  light  diverging  from  a  source  S  (Fig.  166) 
can  converge  upon  a  single  focus  lying  towards  S';  and  this  is 

2M 


530  OF   ETHER-WAVES.  [CHAP. 

the  position  in  which  the  incident  ray  of  light  is,  on  the  whole, 
least  deviated  by  the  prism. 

If  mixed  coloured-light  be  passed  through  a  prism,  each 
colour  has  its  own  index  of  refraction,  /3,  its  own  path  through 
the  prism ;  the  whole  light  is  broken  up  by  Dispersion  into  a 
bundle  of  monochromatic  rays,  each  travelling  in  a  separate 
direction,  and  diverging  the  more  widely  from  one  another  the 
farther  they  travel;  and,  received  on  a  screen,  these  form  a 
Spectrum. 

Each  kind  of  light  is  differently  deviated  by  the  prism ;  and  for  each 
kind  of  light  the  minimum  deviation  of  the  prism  is  different ;  whence  an 
image  of  a  slit  looked  at  through  a  prism,  or  cast  upon  a  screen,  cannot  be 
sharply  in  focus  in  all  parts  of  the  spectrum  at  once  ;  only  one  colour  can  be 
accurately  in  focus  at  any  one  time ;  and  to  put  any  particular  colour  of 
the  spectrum  into  focus,  the  prism  must  be  rotated  one  way  and  another  in 
the  beam  of  light  until  that  position  is  found  for  it,  which  corresponds  to  the 
greatest  approach  of  the  particular  colour  towards  the  red  end  of  the  spec- 
trum :  this  is,  for  that  colour,  the  position  of  minimum  deviation  and  of  most 
accurate  definition. 

The  Rainbow  is  produced  by  reflexion  and  refraction  of  sunlight  in 
the  drops  of  water  which  make  up  falling  rain.  Parallel  sunlight  falls  from 
behind  the  spectator;  in  each  drop  the  light  is  dispersively  refracted,  and 
then  reflected  from  the  farther  face  of  the  drop ;  it  travels  back  through 
the  drop,  and  emerges  in  a  state  of  chromatic  dispersion.  Drops  which  for 
the  moment  are  situated  at  a  certain  angular  height  send  violet  light 
into  the  eye  of  the  observer,  but  the  red  light  from  them  misses  his  eye, 
for  it  strikes  too  low.  Drops  at  a  greater  angular  height  send  red  light  into 
the  observer's  eye,  but  violet  light  proceeding  from  them  is,  as  it  were, 
aimed  too  high,  and  does  not  enter  his  eye.  Drops  in  intermediate  posi- 
tions send  intermediate-coloured  light  into  his  eye.  Drops  above  or  below 
a  certain  range  of  angular  height  do  not  send  light'into  his  eye  at  all.  The 
whole  phenomenon  is  symmetrical  round  an  axis  containing  the  sun  and 
passing  through  the  observer's  eye,  and  the  bow  is,  according  to  the  height 
of  the  sun  in  the  heavens,  a  greater  or  lesser  portion  of  a  circle  whose  parts 
are  equidistant  from  that  axis.  The  rainbow  as  seen  by  the  one  eye  is  not 
formed  by  the  same  water-drops  as  the  rainbow  seen  by  the  other  eye. 

Repeated  reflexions  and  refractions  in  raindrops  frequently  give  rise  to 
secondary,  tertiary,  etc.,  rainbows,  which,  under  experimental  con- 
ditions, have  been  observed  to  the  number  of  eighteen  :  while  interference 
between  rays  which  have  traversed  different  distances  within  the  drops 
gives  rise  to  spectral  fringes  or  supernumerary  bows,  which  sometimes 
reduplicate  the  tints  of  the  rainbow,  and  whose  breadth  is  greater  the 
smaller  the  drops. 

The  halo  seen  round  the  sun  when  it  shines  through  a  frozen  cloud  is 
due  to  refraction  of  sunlight  through  the  crystals.  Conceive  a  circle  of 
prisms  round  the  sun,  arranged  in  such  a  position  as  to  send  a  maximum  of 
sunlight  into  the  eye :  a  circular  spectrum  would  be  seen,  red  internally ; 
among  the  particles  of  ice  in  the  cloud  some  must  be  in  the  favourable  posi- 
tion of  these  prisms. 


xv.]  PRISMS.  531 

Recomposition  of  WMte  Light.  — If  a  bundle  of  coloured 
rays,  emergent  from  a  prism,  be  Fig  167 

received  on  a  second  prism,  simi- 
lar to  the  first  but  reversed  in 
position,  these  monochromatic 
rays  are  again  recombined,  and 
again  form,  on  emergence,  a  beam 
of  the  original  mixed  coloured- 
light,  parallel  to  its  original  direction;  on  the  whole  there  is 
neither  dispersion  nor  angular  deviation. 

Deviation  without  Dispersion.  —  If  white  light  fall  upon 
a  flint-glass  prism  of  such  an  angle  as  to  make  on  a  screen,  at  a 
distance  of,  say,  10  feet,  a  spectrum  whose  length  between  two 
definite  colours  or  Fraunhofer  lines  is,  say,  3  inches;  and  if 
another  prism  of  crown  glass  which  is  able  to  produce  a  spec- 
trum of  an  exactly-equal  length  between  the  same  definite 
colours  or  lines  be  so  placed  as  to  reverse  the  dispersive  action 
of  the  former  prism,  according  to  the  principles  of  the  last  para- 
graph, the  light  leaving  the  prism  is  approximately  white.  It 
is  not,  however,  parallel  to  its  original  course ;  for  when  we  pass 
from  prisms  of  one  substance  to  prisms  of  another,  we  find  a 
phenomenon  known  as  the  Irrationality  of  Dispersion ;  we 
find  that  the  relative  amounts  of  mean  deviation  and  the  rela- 
tive amounts  of  dispersion  produced  by  two  given  prisms  are 
independent  of  one  another;  and  hence,  to  reverse  dispersion 
is  not  necessarily  to  reverse  deviation,  if  this  be  effected  by  a 
prism  of  a  second,  a  different  substance. 

A  flint-glass  prism  and  a  crown-glass  prism,  thus  combined,  may  produce 
deviation  without  producing  any  dispersion,  and  the  emergent  light  is 
approximately  white ;  and  thus,  for  any  two  kinds  of  light,  a  flint-glass 
prism  may  be  achromatised  by  a  second  prism  of  crown  glass.  The  recom- 
position  of  the  colour  is  not  perfect,  except  for  the  two  colours  (or  lines) 
chosen ;  it  would  have  been  perfect  were  the  spectrum  of  crown  glass  pre- 
cisely similar  to  that  of  flint  glass  in  respect  to  the  proportionate  lengths  of 
the  coloured  areas  in  it ;  but  it  is  not  so ;  in  the  crown-glass  spectrum  the 
orange  and  yellow  are  proportionately  more  refracted,  and  are  spread  over 
a  proportionately  greater  area  than  they  are  in  the  flint-glass  spectrum,  and 
the  blue  and  violet  less  so ;  the  former  are  for  accurate  recomposition  too 
much,  the  latter  too  little,  refracted  by  the  achromatising  crown-glass; 
the  issuing  beam,  white  at  the  centre,  is  yellowish  on  one  side,  bluish  on  the 
other.  Three  prisms  may  be  combined  so  as  to  blend  three  colours  in  the 
emergent  ray ;  and  so  forth. 

Reflexion  at  the  surface  of  a  mirror  may  be  said,  to  furnish 
an  example  of  deviation  without  dispersion.  Sometimes  a 


532  OF  ETHER-WAVES.  [CHAP. 

reflecting  prism  (Fig.  158)  is  preferred  to  a  mirror,  especially 
in  lantern  projection-apparatus.  Light  enters  normally  at  one 
face  of  the  prism,  is  totally  reflected  at  the  second,  and  passes 
normally  through  the  third.  The  image  produced  by  such  a 
prism  is  inverted,  and  if  the  incident  beam  be  parallel  there  is 
no  refraction,  and  therefore  no  chromatic  dispersion,  while  the 
loss  of  light  is  very  small. 

Dispersion  without  Deviation.  —  If  crown-glass  prisms 
and  flint-glass  prisms  be  alternated,  they  can  be  made  to  pro- 
duce dispersion  while  the  issuing  rays  are  parallel  to  the  original 
direction  of  the  entering  light. 

This  principle  is  applied  in  the  Direct- vision  spectro- 
scope, which  simply  consists  of  a  train  of  such  prisms,  to  which 

Fig.168. 


the  light  is  admitted  by  a  slit  at  A  (Fig.  168),  and  from  which 
the  light  issuing  at  B  is  caused  to  pass  through  a  lens  which  can 
be  so  adjusted  as,  for  each  colour,  to  give  the  eye  a  clear  image 
of  the  slit ;  and  the  eye  accordingly  receives  the  impression  of 
a  continuous  spectrum,  situated  at  the  focus  of  the  lens. 

Abnormal  Dispersion.  —  The  amount  of  deviation  of  each  kind  of 
coloured  light  can  be  directly  measured  when  a  spectrum  is  formed ;  so  can 
the  angle  of  the  prism  and  the  angle  of  incidence ;  thus  for  each  transparent 
substance,  and  for  each  wave-frequency  of  incident  ether-waves,  the  index 
of  refraction  may  be  calculated  and  recorded. 

The  spectra  produced  by  similar  prisms  of  different  substances  differ 
not  only  in  their  absolute  lengths,  but  also  in  the  proportionate  length  of 
each  colour,  and  even  in  their  arrangement.  A  hollow  glass  prism  filled 
with  iodine  vapour  refracts  red  light  most,  and  violet  least ;  it  gives  a  spec- 
trum the  order  of  succession  in  which  is  ultra-violet,  violet,  blue,  then  a 
dark  band,  then  red,  the  reverse  of  the  ordinary  order;  the  intermediate 
parts  of  the  spectrum  are  lost  by  absorption.  A  weak  alcoholic-solution  of 
fuchsin  in  a  hollow  glass  prism  refracts  the  blue  and  violet  less  than  it 
refracts  the  yellow  and  red;  the  spectrum  thus  presents  the  following 
order:  —  Fraunhofer  lines,  F  to  H  —  i.e.  green,  blue,  violet  —  then  A  to  D, 
red,  orange,  and  yellow,  not  quite  up  to  the  E  line  :  the  green  from  E, 
nearly  as  far  as  F  (E  inclusive),  being  absent  on  account  of  absorption. 
Aniline  violet,  aniline  blue,  indigo  carmine,  give  a  green-blue-orange  spec- 
trum. A  concentrated  solution  of  cyanin  in  a  hollow  glass  prism  gives  a 
spectrum  consisting  of  —  first,  green-blue,  then  a  dark  band,  then  red,  and 
traces  of  orange. 

In  general  all  bodies  which,  like  many  of  the  aniline  dyes  or  crystals  of 
permanganate  of  potash,  act  upon  some  kinds  of  light  as  metallic  reflectors, 


XV.] 


ABNORMAL   DISPERSION. 


533 


and  present,  when  in  the  solid  .state,  superficial  colours  differing  from  the 
body-colours  (the  colours  best  reflected  being  those  absorbed  on  transmis- 
sion), are  found  to  give,  when  their  solution  occupies  a  hollow  glass  prism, 
an  abnormal-refraction  spectrum,  produced  in  the  same  way  as  any  ordinary 
absorption-spectrum,  but  in  which  the  order  of  the  colours  is  not  that  of  the 
spectrum  as  produced  by  a  prism  of  solid  glass.  On  the  redward  side  of  an 
absorption-band  the  index  of  refraction  is  found  to  increase  so  rapidly  as  to 
throw  the  part  of  the  spectrum  near  the  redward  side  of  the  absorption- 
band  over  towards  the  violet;  and  conversely,  the  part  of  the  spectrum 
situated  near  the  violetward  side  of  an  absorption-band  is  thrown  back 
towards  the  red.  The  effect  may  be  simply  to  render  the  absorption-band 
narrower  than  it  would  have  been  had  there  been  no  effect  of  this  kind,  or 
to  cause  overlapping  of  different  parts  of  the  spectrum ;  or,  as  in  the  case  of 
fuchsin  and  cyanin,  actually  to  throw  over  the  redward  and  the  violetward 
parts  of  the  spectrum  into  each  other's  places,  and  to  make  red  light  more 
refrangible  than  .violet.  These  phenomena,  experimentally  systematised 
by  Kundt,  were  discussed  by  von  Helmholtz  on  the  supposition  that  the 
absorbed  light  being  in  tune  with  the  molecules  of  the  body,  these  molecules 
are  set  in  motion,  which  motion  is  modified  by  intramolecular  friction  so  as 
to  react  on  the  transmitted  light.  The  question  is  still  obscure. 

The  heat  region  of  the  spectrum  is  very  much  shortened  or  compressed 
by  the  use  of  glass  prisms. 

Lenses.  —  Simple  lenses  are  of  two  main  kinds  :  — 

a.  Thin-edged,  thick  in  the  centre  ;  either  convex  on  both 
sides,  plano-convex,  or  convexo-concave,  with  a   shallow  con- 
cavity ;  a  parallel-raj ed  beam,  falling  upon  one  of  these  lenses 
from  either  side,  is  made  to  converge  upon  a  Real  Focus  on  the 
opposite  side  of  the  lens. 

b.  Thick-edged,  thinnest  in  the  centre  ;  biconcave,  plano- 
concave, or  concavo-convex,  with  a  deep  concavity:  a  plane- 
fronted  beam ,  incident  on  either  side,  is  made  to  diverge  so  as 
to  seem  to  come  from  a  Virtual  Focus  on  the  same  side  of  the 
lens  as  the  source  of  radiation  itself. 

Fig.  169  shows  these  different  forms  of  simple  lenses,  and 
their   action   on  a  parallel 
beam    of    light    travelling 
from  right  to  left. 

Every  lens  has  a  Prin- 
cipal Focus ;  this  is  the 
point  to  which  a  parallel 
beam  of  rays  is  caused  to 
converge,  or  from  which 
it  is  apparently  caused  to 
diverge,  as  the  case  may  be. 

The  distance  of  this 
principal  focus  from  the  lens  —  or  rather,  from  a  point  in  or  near 


Fig.169. 


534  OF  ETHER-WAVES.  [CHAP. 

the  lens  which  will  be  more  precisely  defined  afterwards,  —  is 
called  the  Focal  Length,  /,  of  the  lens ;  and  the  parallel  rays 
being,  by  convention,  taken  as  coming  from  the  right,  /  is  nega- 
tive (to  the  left)  in  convergent,  positive  (to  the  right)  in 
divergent  lenses. 

It  is  convenient  for  many  purposes  to  assume  that  a  lens 
has  no  thickness,  and  may  be  replaced  by  a  refracting  film,  the 
proper  position  of  which,  as  replacing  the  lens,  will  depend 
upon  the  form  of  the  lens.  This  convention  enables  approxi- 
mate formulae  to  be  used,  but  the  results  obtained  by  its  aid  are 
only  approximate.  The  focal  length  will  then  be  the  distance 
between  the  principal  focus  and  this  film :  and  we  shall  first 
deal  with  the  matter  on  this  understanding.  The  point  of  this 
imaginary  film  which  corresponds  to  the  axis  of  the  lens  is 
called  the  Optical  Centre  of  the  lens.  In  a  symmetrical 
biconvex  or  biconcave  lens  this  Centre  is  at  the  centre  of  the 
lens  itself. 

If  r  and  r'  be  the  distances  of  the  centres  of  curvature  of  the  right  and 
left  faces  of  the  lens  respectively,  both  distances  being  measured  from  the 
centre  of  the  lens  positively  —  that  is,  towards  the  right  on  a  horizontal  line ; 
and  if  a  parallel  beam  come  from  the  right ;  then,  neglecting  the  thickness 
of  the  lens,  l//=(/3  —  1)  (1/r  —  1/r'),  where  J3  is  the  index  of  refraction 
of  the  lens  in  air,  and  f  the  positive  distance  of  the  focal  point  to  the  right, 
in  air.  Iff  be  found  negative,  then  the  focal  point  is  to  the  left,  beyond 
the  lens,  a  real  focus  for  rays  coming  from  the  right. 

This  general  formula  applies  without  alteration  to  concavo-convex  lenses, 
the  convexity  of  which  is  turned  towards  the  left,  and  both  of  whose  centres 
of  curvature  are  therefore  situated  in  a  positive  direction.  In  a  biconvex 
lens  r  is  negative,  and  !//=  (/?  —  1)  (—  1/r  —  1/r')  ;  in  a  biconcave  one  r' 
is  negative,  and  !//=(/?  —  1)  (1/r  +  1/r7)  ;  in  a  convexo-plane  lens  (con- 
vexity to  the  left)  r  =  GO  ,  and  !//=  (/?  —  1)  (—  1/r')  ;  in  a  concavo-plane 
lens  (concavity  to  the  right)  r'  =  oo  ,  and  !//=  (ft  —  1)  (1/r). 

The  expression  I// measures  the  increase  of  divergence  produced  by  a 
lens ;  it  is  called  its  Power.  In  thin-edged  lenses,  which  are  convergent,  it 
is  negative ;  in  thick-edged,  positive.  Thus  in  an  achromatic  combination, 
consisting  of  a  convergent  lens  (focal  length  =  — /)  with  a  divergent  one 
(/))  in  contact  with  it,  the  power  of  the  combination  is,  neglecting  the  thick- 
ness, 1/F  =  —  1//+  I//,;  whence  F  can  be  found  approximately. 

Reversibility  of  Lenses.  —  Let  us  reverse  a  lens,  still  neglecting  the 
thickness.  Then  the  new  r  =  —  r' ;  the  new  r'  =  —  r  ;  the  value  of  1  /f 
remains  unchanged. 

To  find,  roughly,  the  focal  length  of  a  thin-edged  lens,  find 
by  experiment  the  distance  at  which  it  will  make  a  clear  image 
of  the  sun  upon  a  screen  :  light  coming  from  the  sun  is  practi- 
cally plane-fronted,  and  is  caused  to  converge  upon  the  principal 
focus. 


xv.]  LENSES.  535 

Lenses,  like  mirrors,  have  Conjugate  Foci  at  distances 
d  and  d1,  +  or  —  according  to  their  direction  with  reference  to 
the  "  optical  centre  "  of  the  lens ;  rays  coming  from  an  object 
placed  at  one  conjugate  focus  are  caused  to  produce  an  image, 
real  or  virtual,  at  the  other. 

Let  d  be  the  distance  of  the  Object  from  the  lens  or  from 
its  optical  centre,  measured  as  a  positive  distance  in  front  of  the 
lens  or  its  centre ;  let  d'  be  the  distance  of  the  Image,  similarly 
measured ;  and  let  /  be  the  focal  length  (which  is  —  in  thin- 
edged,  +  in  thick-edged  lenses).  Then  the  general  formula, 
applicable  to  all  lenses,  is  ~\./d'  =  l/d  -f-  1/f. 

Examples.  —  1.  A  convergent  (thin-edged)  lens  of  25  cm.  focus 
(/=-25);  object  at  75  cm.  (d  =  75)  in  front  of  the  lens;  l/d'  =  1/75 
-  1/25;  d'  =  —  37  £ ;  that  is,  the  image  is  37  £  cm.  behind  the  lens;  a  Real 
Image. 

2.  The   same   lens;   object   at  20  cm.  (d  =  20) ;   l/d'  =  1/20  -  1/25; 
d'  =  -f  100 ;  that  is,  the  image  is  100  cm.  in  front  of  the  lens ;  a  Virtual 
Image,  beyond  the  object. 

3.  A  divergent  (thick-edged)  lens  of  25  cm.  focus  (/=  +  25);  object  at 
75  cm.  (d  =  75)  ;  l/d'  =  1/75  +  1/25  ;  d'  =  +  18-75  cm. ;  that  is,  the  image 
is  18-75  cm.  in  front  of  the  lens;  a  Virtual  Image,  between  the  lens  and  the 
object. 

4.  The  same  lens;    object   at  20  cm.  (d  =  20);    \/d'  =  1/20  +  1/25; 
d'  =  +  11^;  that  is,  the  image  is  11|  cm.  in  front  of  the  lens;  a  Virtual 
Image,  between  the  lens  and  the  object. 

Thin-edged  Lenses.  —  Since  /  is  negative,  the  equation  becomes 
l/d'  =  l/d-  I//,  and  from  it  we  find  :  — 

A  beam  comes  from  an  infinite  distance;  d  =  oo  ;  then  d'  =  —f;  light 
converges  really  upon  the  principal  focus  on  the  opposite  side  of  the  lens. 

If  the  source  of  light  be  at  a  distance  less  than  infinity,  but  greater  than 
twice  the  focal  length,  the  image  is  real,  and  is  on  the  other  side  of  the  lens, 
at  a  point  between  the  principal  focus  on  that  side  and  a  point  twice  the 
focal  length  from  the  lens ;  that  is,  if  d  =  +  (2/+  x),  d'  =  -  (f+f*/f+x), 
which  is  greater  than  /,  less  than  If:  and  the  image  is  then  smaller  than 
the  object,  in  the  ratio  d' :  d,  linearly. 

If  the  object  be  at  a  distance  2/,  the  image  is  also  at  a  distance  2/,  on 
the  other  side  of  the  lens.  The  object  and  the  image  are  then  equal  in  size. 
Hence  another  method  of  finding  the  focal  length  of  a  thin-edged  lens. 
Adjust  an  object,  the  lens,  and  a  screen,  so  that  the  image  on  the  screen  is 
equal  in  size  to  the  object :  the  screen  and  the  object  are  now  both  situated 
at  distances  equal  to  2/  from  the  optical  centre  of  the  lens ;  one-fourth  the 
distance  between  them  is  the  focal  length. 

If  the  object  be  at  a  distance  greater  than  /,  but  less  than  2/*,  the  image 
is  real,  at  a  distance  greater  than  2/,  and  is  larger  than  the  object,  in  the 
ratio  d' :  d,  linearly. 

If  the  source  of  light  be  at  a  principal  focus,  d=f;  .-.  d'  =  — oo;  the 
image  is  real,  but  at  an  infinite  distance  beyond  the  lens. 

Hence  the  focal  length  of  a  thin-edged  lens  may  also  be  formed  in  the 
following  way.  Focus  a  telescope  upon  a  very  distant  object ;  it  is  then  in 


536  OF   ETHER- WAVES.  [CHAP. 

focus  for  parallel  rays.  In  front  of  the  telescope,  so  adjusted,  fit  up  the  lens, 
well  centred  with  respect  to  the  telescope.  In  front  of  the  lens  adjust  the 
position  of  a  spider-web  or  other  delicate  object,  until  this  is  seen  sharply 
denned  in  the  telescope.  The  rays  divergent  from  the  object  are  then 
rendered  parallel  by  the  lens,  and  the  object  is,  accordingly,  then  at  the 
principal  focus  of  the  lens. 

If  the  object  be  between  the  lens  and  a  principal  focus,  the  rays  are  not 
made  sufficiently  convergent  to  cross  at  any  place ;  they  seem  to  come  from 
a  virtual  image  beyond  that  principal  focus,  and  farther  from  the  lens  than 
the  object ;  but  the  virtual  image  rapidly  gains  011  the  object  as  the  object 
approaches  the  lens. 

Thick-edged  Lenses.  —  Since  f  is  positive,  the  equation  remains  un- 
changed, and  from  it  we  find  :  — 

When  d  is  of  any  value  >  0,  <  oo,  d'  is  less  than/,  and  also  less  than  d. 

When  d  =  +/,  df  =  +  \f;  Virtual  Image  at  half  focal  distance. 

If  rays  converge  upon  a  point  behind  the  lens,  so  that  d  is  negative,  then, 
so  long  as  d  is  not  numerically  greater  than  /,  the  lens  will  make  them  con- 
verge upon  a  point  at  a  greater  negative  distance.  If  d  =  —  /,  the  converg- 
ent rays  are  rendered  parallel.  If  d  be  negative  and  numerically  greater 
than  /,  the  convergent  rays  are  made  to  diverge  as  if  from  some  positive 
distance  de,  <  oo,  >/.  When  d  =  -2f,d'  =  +  2/. 

The  focus  of  a  thick-edged  lens  is  most  conveniently  found  by  coupling 
it  with  a  thin-edged  one,  found  by  trial  among  a  sufficiently  extensive  series, 
so  that  together  they  shall  produce  no  change  in  the  apparent  size  of  an 
object  seen  through  them. 

As  to  the  inversion  or  erectness  of  the  image  produced  by 
a  thin-edged  lens,  an  object  at  O  (Fig.  170),  at  a  distance 
exceeding  twice  the  focal  length,  produces  a  smaller  inverted 

Fig.170. 


image,  a  real  image,  at  I,  and  an  eye  placed  beyond  I  —  that  is, 
at  a  sufficient  distance  from  the  lens  —  will  perceive  the  real 
image  of  a  distant  object,  inverted,  smaller  than  the  object,  and 
apparently  situated  in  space  between  him  and  the  lens;  the 
contrary  being  the  general  impression.  The  eye  must  be  so  far 
beyond  I  that  the  real  image  in  space  can  be  looked  at  in  the 
same  way  as  any  ordinary  object  of  vision.  In  all  cases,  an 
object  and  its  real  image  are  interchangeable,  so  that  an  object 
at  I  will  produce  a  real  image  at  O. 

If  the  distance  be  greater  than  the  focal  length,  but  less 
than  twice  that  length,  the  image  is  still  real  and  inverted,  but 
is  larger  than  the  object. 


XV.] 


LENSES. 


537 


When,  however,  the  object  is  brought  so  near  the  lens  as  to 
lie  at  a  distance  from  the  lens  less  than  the  focal  length,  then, 
to  an  eye  situated  at  any  distance  on  the  other  side  of  the  lens, 
a  virtual  image  will  be  apparent,  erect,  magnified,  and  more 
distant  than  the  object.  Hence  these  lenses  are  commonly 
used  as  magnifying-glasses  (Fig.  171).  The  relative  linear 
sizes  of  object  and  image  are  in  all  cases  proportional  to  their 
respective  distances  from  the  "  optical  centre  "  of  the  lens. 


Fig.171. 


Fig.172. 


In  divergent  or  thick-edged  lenses,  Fig.  172  shows  that 
the  image  of  a  real  object  is  erect,  diminished,  virtual,  and 
nearer  to  the  lens  than  the  object  itself ;  and,  since  there  is  no 
subsequent  crossing  of  the  rays  beyond  the  lens,  there  is  no 
inversion  of  the  image.  The  virtual  image  of  such  an  object  is 
always  at  a  distance  from  the  lens  less  than  the  focal  length. 
Lenses  of  this  kind  may  be  used  as  diminishing-glasses. 

For  some  purposes  flexible  lenses  may  be  used  in  which  the  curvature 
may  be  slightly  varied.  Cusco's  ophthalmoscopic  lens  consists  of  two  pieces 
of  thin  microscopic  cover-glass  fixed  in  a  frame  :  water  fills  the  cavity  between 
them ;  by  forcing  more  or  less  water  into  the  cavity  the  curvature  may  be 
varied. 

Even  when  the  light  transmitted  through  convergent  lenses 
is  monochromatic,  the  focussing  can  never  be  exact  if  their  sur- 
faces be  spherical ;  each  point  of  an  extended  object  forms  a 
slightly-blurred  image.  This  effect  can  be  reduced  somewhat 
by  the  use  of  Diaphragms,  which  allow  only  the  central 
part  of  a  beam  to  pass  through  the  centre,  and  the  marginal 
rays  to  pass  through  the  marginal  part  of  a  lens ;  they  thus 
diminish  the  Spherical  Aberration  of  the  lens  (as  in  the 
pupil  of  the  eye),  but  this  can  only  be  brought  to  a  minimum  by 
modifying  the  curvatures  of  the  lenses  used.  This  could  be 
done,  for  parallel  incident  rays,  or  for  rays  coming  from  a  pre- 
determined distance,  by  making  the  anterior  face  of  a  single 


538  OF   ETHER- WAVES.  [CHAP. 

lens  ellipsoidal  or  hyperboloidal  and  the  hinder  face  spherical ; 
but  such  surfaces  cannot  well  be  produced.  If  lenses  could  be 
produced  diminishing  in  density  towards  the  centre,  the  same 
effect  might  be  attained.  Both  these  refinements  are  present  in 
the  human  eye.  What  the  optician  does  is  to  combine  lenses, 
which  have  spherical  and  plane  surfaces,  so  as  approximately  to 
bring  about  this  desired  result,  and  then  to  correct  the  curvatures 
by  a  process  of  systematic  trial  and  error.  Some  combinations 
of  lenses  are  so  devised  as  to  bring  all  the  points  of  an  extended 
image  into  the  same  plane,  and  thus  to  produce  a  flat  field; 
others  to  bring  points  differing  in  distance  to  foci  which  differ 
very  little  from  one  another,  and  thus  to  secure  penetration.  The 
calculation  of  the  various  curvatures  necessary  for  these  ends 
often  involves  considerable  mathematical  skill. 

In  general  a  lens  with  spherical  surfaces  is  equivalent  to  a 
series  of  prisms  whose  angles  vary  with  the  varying  distances 
from  the  axis.  In  a  thin-edged  lens  the  marginal  rays  are  gen- 
erally more  refracted  than  the  axial ;  thus  a  square  object  yields 
a  Distorted  Image,  the  corners  of  which  appear  squeezed 
in,  and  the  boundaries  of  which  are  convex.  Similarly,  a  thick- 
edged  lens  draws  out  the  corners  and  produces  concave  bound- 
ing lines.  This  tendency  is  obviated  by  using  lenses  in  pairs, 
symmetrically  disposed,  so  that  the  distortions  produced  by  the 
one  lens  may  be  reversed  by  the  other.  Again,  if  the  screen 
on  which  the  image  is  received  be  not  parallel  to  the  object,  the 
image  is  apparently  so  distorted  that  lines  parallel  in  the  object 
appear  to  diverge  or  converge  in  the  image ;  whence  the  use  of 
the  Swing-back,  maintained  vertical,  in  photographic  cameras. 

The  general  principle  underlying  calculations  relating  to 
systems  of  lenses  is  that  the  image  formed  or  tending  to  be 
formed  by  the  first  lens  is  taken  as  the  object  (real  or  virtual) 
of  the  second,  and  so  on.  The  upshot  is,  that  for  every  arrange- 
ment of  any  lens-system,  there  is  always  an  image  formed  some- 
where, corresponding  in  size  (but  not  in  its  position)  with  that 
which  might  have  been  produced  by  an  Equivalent  single 
Lens.  The  adjustment  of  a  system  of  lenses  (e.g.,  the  focus- 
sing of  a  telescope  or  microscope)  is  for  the  purpose  of  causing 
the  image  to  be  formed  in  a  place  where  it  will  be  convenient 
to  inspect  or  to  use  it. 

The  action  of  a  system  of  lenses  is,  approximately,  equivalent  to  the  for- 
mation of  an  image  by  a  simple  lens  plus  a  determinate  shifting  of  the 
image  formed.  This  shifting  being  allowed  for,  it  is  often  convenient  to 


xv.]  LENSES.  539 

represent  a  system  of  lenses  by  an  equivalent  lens.  For  instance,  the  Eye 
may  be  ideally  reduced  for  many  purposes  to  a  single  lens  composed  of 
aqueous  or  vitreous  humour,  having  its  back  coincident  with  the  retina, 
and  its  anterior  aspect  a  spherical  surface  of  5-1248  mm.  radius,  situated  at 
its  most  anterior  point  2-3448  mm.  behind  the  actual  anterior  surface  of  the 
cornea.  Such  a  lens  would  refract  incident  light  and  bring  images  of  dis- 
tant points  to  a  focus  upon  the  retina  in  the  same  way  as  the  actual  Eye 
does. 

Gauss's  Method.  —  Gauss,  followed  by  Listing,  starting  from  this  con- 
sideration as  to  the  equivalence  of  any  system  of  lenses  to  a  single  lens  plus 
a  determinate  shift,  found  that  every  possible  system  of  lenses  could,  if  well 
centred,  be  reduced  to  a  Region  of  Space  to  be  traversed  by  the  incident 
light,  and  presenting  six  characteristic  or  Cardinal  Points,  ranged  along 
the  axis  of  the  system.  These  are  the  Incidental  Focus  F,  the  Incidental 
Principal  Point  P,  the  Incidental  Nodal  Point  N,  the  Refractional  Principal 
Point  P',  the  Refractional  Nodal  Point  N',  and  the  Refractional  Focal  Point 
F'.  The  incidental  points  become  refractional,  and  vice  versa,  when  the 
direction  of  the  rays  is  reversed.  The  rays  are  assumed  to  travel  all  near 
the  axis. 

All  rays  proceeding  from  F  become  after  refraction  parallel  to  each 
other  and  to  the  axis ;  all  parallel  incident  rays,  parallel  to  the  axis  of  the 
system,  pass  through  F'.  An  object  at  P,  or  in  the  same  plane  (at  right 
angles  to  the  axis  of  the  system)  with  it,  forms  an  equal  and  erect  image  at 
P'  or  in  the  same  plane  with  it ;  there  are  only  two  such  points.  Any  ray 
apparently  making  for  N  before  refraction  is,  after  refraction,  parallel  to  its 
former  course,  but  appears  to  be  coming  from  N' ;  there  are  only  two  such 
points.  The  distance  PF  is  the  Incidental,  while  P'F'  is  the  Refractional 
Principal  Focal  Distance. 

These  six  points,  all  in  one  line,  are  closely  related.  The  distance  FN 
is  equal  to  the  distance  FT' ;  and  N'F'  =  FP.'  Therefore  PN  =  P'N'  =  FP 
-  F'P'  ;  and  PP'  =  NN'.  Further,  if  ft  be  the  ratio  between  the  index  of 
refraction  of  the  medium  nearer  the  source  and  that  of  the  medium  beyond 
the  lens,  FP  =  F'P'  •  (1//3)  ;  in  the  case  of  a  lens  in  air,  these  two  principal 
focal  distances  are  equal,  and  further,  P  coincides  with  N  and  P'  with  N'. 

Planes  passing  through  F  and  F'  at  right  angles  to  the  axis  of  the  sys- 
tem are  called  its  Focal  Planes. 

Rays  diverging  from  any  point  in  one  of  these  focal  planes  (of  which 
rays  one  might  be  towards  the  corresponding  nodal  point)  emerge  parallel 
to  one  another ;  and  since  the  ray  from  the  divergence-point  to  the  corre- 
sponding nodal  point  would  have  emerged  parallel  to  its  original  direction, 
all  the  rays  must  necessarily  emerge  parallel  to  the  original  direction  of  that 
ray.  Thus  they  retain  parallelism  with  that  ray,  the  direction  of  which  is, 
on  its  emergence,  determinate. 

Rays  parallel  in  the  firs't  medium  converge  on  some  point  in  the  second 
focal  plane.  That  ray  which  travels  towards  N  emerges  as  if  it  had  come 
from  N'  parallel  to  its  former  course.  Hence  a  line  drawn  from  N'  parallel 
to  the  originally  parallel  rays  will  cut  the  second  focal  plane  in  a  certain 
point ;  towards  that  point  in  the  second  focal  plane  all  the  rays  originally 
parallel  must  converge. 

The  artifice  of  Gauss's  method  (for  which  see  his  Collected  Works,  or, 
for  an  elementary  exposition,  von  Helmholtz's  Physiol.  Optik,  andf  Clerk  Max- 
well, Qu.  J.  Malhem.,  1858,  p.  233;  or  Pendlebury,  Lenses  and  Systems  of 


540 


OF  ETHER-WAVES. 


[CHAP. 


Lenses)  is,  so  to  speak,  the  identification  of  an  incident  set  of  rays  as  they 
cross  the  first  Focal  Plane;  the  rays  are  then  traced  until  they  arrive  at  the 
second  plane ;  from  the  data  thus  obtained  their  subsequent  course  can  be 
ascertained.  Mathematical  difficulties  are  thus  minimised,  for  the  problem 
becomes  mainly  one  of  finding  these  cardinal  points  for  a  lens-system  of 
any  given  form  and  of  any  degree  of  complexity. 

In  the  case  of  a  single  lens  surrounded  by  a  single  medium,  such  as  air, 
let  the  radius  of  the  right-hand  surface  be  r  (  -f  if  the  centre  be  towards  the 
right,  —  if  towards  the  left),  and  let  that  of  the  left-hand  surface  be  r'  (simi- 
larly +  or  —  ) ;  and  if  J3  stand  for  /?,//?„,  the  relative  refractive  index  of 
the  lens  as  compared  with  that  of  the  medium ;  and  if  A  and  B  be  the  left- 
and  right-hand  axial  points  of  the  lens,  so  that  AB  =  t,  the  axial  thickness 
of  the  lens  :  then,  all  measurements  being  reckoned  on  the  footing  that 
+  AX  is  a  distance  from  A  towards  the  right  and  —  AX  the  same  distance 
towards  the  left,  we  have  AF  =  (-  firr'  -((3  -  1)  tr')  +  (j3  -  1)  [0  (r  -  r') 
+  (£  _  1)  f]  ;  BF'  =  (firr1  -  (/?  -  1)  fr)-*-  the  same  divisor ;  AP  =  -  tr1  + 

[/?(r-r')  +  <j3-l)0l  BF=-^[/?(r-r')  +  (/?-l)(j;  PF  =  -  FF' 
=  _  firr'  -s-(0  -  1)  [/?  (r  -  r'}  +  ((3  -  1)  t].  From  these  formulae  the  posi- 
tion of  the  principal  planes  with  respect  to  the  lens,  and  the  distance  PF  or 
P'F'  =  /  between  the  principal  planes  and  the  corresponding  focal  points 
(which  is  the  true  Focal  Distance)  can  be  found  for  a  single  lens  of  any 
form  and  refractive  index,  immersed  in  any  medium.  For  example,  in  a 
convexo-concave  (thin-edged)  lens,  let  r  =  +  12  cm.,  r'  =  -f  9  cm.,  and  Z  =  0-8 
cm. ;  and  (3  =  1-5.  Then,  by  the  above  formulae,  AF  =  -  67-59,  BF  = 
+  64-16,  AP  =  -  147,  BP'=  -  1-96,  PF  =  -  66-12,  and  FF'  =  +  66-12. 
Whence  P  is  outside  the  lens,  1-47  cm.  to  the  left  of  A ;  P'  is  also  outside  the 
lens,  1-16  cm.  to  the  left  of  A ;  and  F  is  67-59  cm.  to  the  left  of  A,  while  F' 
is  64-16  cm.  to  the  right  of  B. 

From  these  formulae  it  will  be  found  that  in  thin-edged  lenses  the  points 
F,  F',  P,  F,  mostly  lie  on  the  axis  in  the  order  FPP'F',  while  in  thick- 
edged  lenses  they  mostly  lie  in  the  order  F'PP'F ;  and  that  unsymmetrical 


(P  OR  N) 


P'  OR  N>) 


Fig.  172  a. 


(FO 


AXIS 


lenses  are  only  truly  reversible  when  the  principal  focal  planes  are  made  to 
exchange  places;  which,  since  in  the  case  of  thin-edged  convexo-concave 
lenses  the  principal  planes  mostly  lie  outside  the  convex  face  and  in  that  of 
thick-edged  convexo-concave  lenses  outside  the  concave  face,  often  involves 
considerable  shifting  of  the  lens  in  its  setting. 

If  the  object  O  be  at  any  distance  d  =  PO  from  the  incidental  principal 


xv.]  LENSES.  541 

plane,  the  distance  of  the  image  I  from  the  other  principal  plane,  d'  =  P'l,  is 
determinable  numerically  by  the  equation  1/PO  +  I/ P'l  —  1/PF,  or  \/d 
+  l/d'  =  I//,  f  being  taken  as  -f  in  divergent,  —  in  convergent  lenses. 

If  we  put  the  thickness  of  the  lens  out  of  view,  making  t  =  0  in  the 
formulae,  we  arrive  at  the  usual  lens-forinulse.  For  example,  in  the  convexo- 
concave  lens  discussed  above,  AF  =  —  firr'  -r-  {(ft  —  1)  •  @(r  —  r')}  =  {(ft  —  1) 
(1/r-  l/r')}-1  =  -72;  BF'=+72;  AP  =  0  ;  BP=0;  AB=0;  that  is, 
the  focal  distances  measured  from  the  imaginarily  coinciding  principal  planes 
or,  in  other  words,  from  the  optical  centre  of  the  lens,  are  each  equal  to  72 
cm.  It  will  be  seen  how  widely  these  values  depart  in  this  case  from  the 
true  values  as  given  by  the  Gauss-formulae  above. 

The  accompanying  diagram  (Fig.  172 a),  in  which  the  obliquity  of  the 
marginal  rays  is  exaggerated,  may  serve  to  illustrate  the  method  as  applied 
to  a  biconvex  lens  in  air :  in  this  case  the  focal  distances  are  equal  and  the 
principal  points,  which  coincide  with  the  nodal,  are  within  the  lens. 

Chromatic  Aberration.  —  When  mixed  coloured-light  is 
passed  through  a  thin-edged  lens,  violet  light  is  most  refracted, 
and  comes  to  a  focus  sooner  than  the  red  rays  do ;  beyond  the 
red  focus  is  the  heat-focus ;  between  the  violet  focus  and  the 
lens  is  the  region  of  the  photographic  focus. 

If  a  beam  of  white  light  be  passed  through  a  single  con- 
vergent-leris,  a  screen  placed  at  the  violet  focus  will  give  an 
image  with  a  red  border  —  the  red  rays  not  having  yet  con- 
verged ;  if  it  be  placed  a  little  farther  off,  at  the  red  focus,  the 
image  is  now  surrounded -by  a  violet  border,  for  the  violet  rays 
are  already  divergent.  Consequently  no  clear  definition  can 
be  obtained  by  the  use  of  such  simple  lenses,  and  it  is  neces- 
sary to  render  them  Achromatic.  A  biconvex  lens  of  flint 
glass,  more  convergent  than  is  necessary,  is  coupled  with  a  bicon- 
cave lens  of  crown  glass  of  proper  curvature  ;  the  latter  destroys 
the  dispersion,  by  bringing  two  colours  to  the  same  focus,  with- 
out wholly  doing  away  with  the  deviation ;  the  couplet  acts  on 
the  whole  as  a  single  lens,  producing  a  somewhat  smaller  refrac- 
tion than  either  of  the  lenses.  This  arrangement  may  be  seen 
in  the  object-glass  of  any  common  telescope.  For  still  further 
accuracy  three,  four,  or  even  a  greater  number  of  lenses  may  be 
combined,  by  which  three,  four,  or  more  colours  are  brought  to 
the  same  focus ;  as  in 'the  achromatic  objectives  of  microscopes. 

Makers  of  photographic  lenses  have  shown  much  skill  in  making  the 
photographic  and  the  visual  focus  coincide ;  for  special  photographic  work, 
such  as  Rutherford's  lunar  photography,  lenses  have  had  to  be  constructed 
whose  curvatures  are  calculated  with  reference  to  the  focus  of  the  highly- 
refrangible  actinic  rays  alone;  and,  while  nothing  can  be  distinctly  seen 
through  such  lenses,  photographs  of  extraordinary  clearness  have  been 
taken  by  their  aid. 


542 


OF  ETHER-WAVES. 


[CHAP. 


Radiant  Heat  may  be  shown  to  be  reflected  and  refracted 
like  Light,  by  concentrating  rays  of  dark  heat  upon  a  Thermo- 
pile by  means  of  a  lens  or  a  mirror,  or  by  refracting  them  by 
means  of  a  prism  into  a  new  path,  in  the  course  of  which  the 
thermopile  must  somewhere  be  placed  before  it  will  indicate 
the  impact  of  Heat-waves :  by  photography  of  the  infra-red 
region  of  the  spectrum ;  by  Langley's  Bolometer  (p.  717) ;  by 
Becquerel's  Phosphorescence-effect,  p.  505. 

INTERFERENCE. 

Ether-waves  are  capable  of  Interference.  Two  systems  of 
equal  waves,  arriving  at  the  same  point  in  opposite  phases,  will 
produce  at  that  point  no  effect,  either  of  light  or  of  heat  or  of 
photographic  action:  at  that  point  the  ether  will  be  at  rest; 
and  thus  light  added  to  light  may  produce  darkness.  In  Fig. 
75  the  two  points  A  and  B  are  centres  of  wave-motion,  and  at 
the  points  &',  d',f,  on  the  screen  MN,  there  is  no  disturbance, 
while  at  intervening  points,  a',  <?',  e\  the  amplitude  of  disturb- 
ance is  doubled. 

Interference  of  waves  thus  affects  the  distribution  of  energy 
in  a  system  of  undulations,  and  such  a  screen  produces  a  system 
of  negative  reflected  waves  from  a',  c>f,  e\  etc. 

Let  us  now  consider  a  monochromatic  beam  of  plane-polar- 
ised light.  Such  a  beam  may  be  divided  into  two  parts  by  reflex- 
ion from  a  silvered  or  platinum  mirror  bent  in  the  middle  at  an 
angle  very  nearly  equal  to  180°,  or  else  by  refraction  through 
a  biprism  whose  angle  is  very  nearly  180°.  The  last  case  is 
shown  in  Fig.  173.  S  is  a  source  of  light;  the  light  from  it 


is  transmitted  through  a  polariser  N :  it  is  now  a  polarised  beam. 
The  rays  are  received  by  a  convergent  lens,  which  makes  them 
converge  upon  S".  In  its  course  it  is  passed  through  a  piece  of 
glass  R,  coloured  red  with  suboxide  of  copper :  it  is  now  to  a 
rough  approximation  monochromatic.  It  is  then  passed  through 
the  biprism  P,  which  refracts  it  in  such  a  way  that  it  seems 
td  come  from  two  equal  and  equally -distant  foci  at  S'"  and  S"". 


xv.]  INTERFERENCE.  543 

The  light  may  then  be  received,  either  on  a  screen,  or  directly 
in  the  observer's  eye  placed  in  the  onward  path  of  the  beam. 
A  series  of  dark  and  bright  fringes  will  be  seen,  corresponding 
to  the  alternate  fringes  of  rest  and  disturbance  of  Fig.  75.  The 
two  beams  apparently  travelling  from  S'"  and  S""  are  polarised 
in  the  same  plane,  and  any  irregularity  of  amplitude  charac- 
terising the  one  is  participated  in  by  the  other.  Hence  they  are 
in  a  position  to  interfere  fully  and  regularly  with  one  another. 
If,  on  the  other  hand,  they  had  been  polarised  in  planes  at  right 
angles  to  one  another,  they  could  not  have  extinguished  one 
another  at  any  point.  When  common  light  is  used,  it  may  be 
at  once  filtered  through  a  piece  of  red  glass  and  then  passed 
through  a  convergent  lens. 

Light  from  two  different  sources  cannot  show  interference-phenomena 
well ;  any  irregularities  in  the  one  vibration  ought  to  be  participated  in  by 
the  other ;  and  hence  even  light  from  the  same  source,  if  one  of  the  beams 
have  been  very  much  delayed,  may  be  rendered  unable  to  show  these  phe- 
nomena, through  the  irregularities  having  had  time  to  produce  a  difference 
between  the  two  beams  of  light,  originating  from  the  same  source  at  dif- 
ferent times. 

To  procure  monochromatic  light  it  is  better  to  project  a 
spectrum  upon  a  screen  in  which  there  is  a  slit,  and  then,  behind 
the  screen,  to  make  use  of  that  part  of  the  spectrum  whose  light 
falls  upon  and  traverses  the  slit. 

It  is  very  easy  to  procure  a  bright  spot  which  may  represent  a  simple 
luminous  point,  by  making  a  small  hole  in  a  metal  screen,  and  in  this  insert- 
ing a  drop  of  glycerine.  This  acts  as  a  powerfully-convergent  lens,  and  if 
sunlight  be  concentrated  upon  it  there  will  appear  on  the  dark  side  of  the 


Fig.174. 


screen  an  intensely  bright  little  spot  of  light  which  may  be  used  as  a  source 
of  light  for  many  experiments;  with  such  a  source  of  light  Fresnel  dis- 
covered the  laws  of  diffraction.  More  elaborately,  the  same  result  may  be 
better  attained  by  means  of  the  electric  light  made  to  converge  by  an  achro- 
matic lens  of  exceedingly  short  focus,  a  high-power  microscopic  objective. 

When  monochromatic  common  light,  proceeding  from  a 
luminous  point,  is  passed  through  a  biprism,  its  vibrations  in 
each  of  two  planes,  at  right  angles  to  one  another,  produce  the 


544  OF  ETHER-WAVES.  [CHAP. 

effects  of  interference  independently  of  one  another,  but  pro- 
duce their  respective  fringes  and  bands  in  coincident  positions 
on  the  screen.  When  mixed  coloured-light  or  white  light  is 
treated  in  this  way,  the  red  fringes  do  not  coincide  with  the 
violet  fringes ;  the  violet  fringes  are  more  numerous  than  the 
red  fringes,  and  are  closer  together.  This  will  be  understood 
from  Fig.  75 ;  if  the  wave-length  be  increased,  the  points  a',  6', 
c',  d',  e\  must  become  farther  distant  from  one  another.  A 
violet  fringe  is  seen  near  the  axial  line  of  the  beam ;  it  is  over- 
lapped by  a  blue,  the  blue  by  a  green,  and  so  on :  each  coloured 
fringe  produced  by  the  interference  of  white  light  presents  a 
complete  spectrum.  The  number  of  such  spectra  is  limited; 
at  a  little  distance  from  the  axial  line  of  the  beam  the  fringes 
overlap  one  another  so  as  to  produce  what  appears  to  the 
eye  to  be  simply  white  light,  but  the  spectrum  of  which  shows 
a  series  of  alternately  dark  and  light  bands:  all  the  colours 
being  equally  encroached  upon  by  dark  bands,  the  result  seems 
white. 

Michelson  has  been  able  to  observe  200,000  fringes. 

A  bent  mirror  used  instead  of  a  biprism  produces,  by  reflexion  of 
white  light  upon  a  screen,  alternate  fringes  of  white  light  and  darkness. 

Measurement  of  Wave-length. — If  S'"S""  be  the  apparent  position 
of  the  two  images  or  apparent  sources  of  light,  which  must  be  monochro- 
matic ;  N  the  position  of  the  central  fringe,  illuminated  by  the  joint  action 
of  S'"  and  S"" ;  the  angle  S"'NS""  =  23;  N'  the  position  of,  say,  the  fourth 
bright  fringe ;  S'"N'  is  shorter  than  S""N'  by  four  wave-lengths ;  this  dif- 
ference is  very  nearly  equal  to  (NN'  x  S"'S""  -*-  AN }  =  NN'  X  2  tan  8. 


Fig.175. 


The  angle  28  can  be  measured  with  a  theodolite ;  the  distance  NN'  can 
be  measured  with  a  micrometer ;  the  value  of  the  four  wave-lengths,  and 
therefore  of  one  wavfi-length,  can  be  determined  from  these  data. 

Fig.  75  shows  that  the  line  of  propagation  of  these  fringes  in  space  is 
hyperbolic ;  the  foci  of  these  hyperbolas  being  the  two  apparent  sources. 

The  bands  vanish  when  one-half  of  the  biprism  or  mirror  is  covered. 

If  the  light  from  one  of  the  sources  be  retarded  by  being 
made  to  pass  through  a  layer  of  a  substance  in  which  light 
travels  more  slowly  than  in  air,  the  whole  of  the  fringes  will  be 
shifted  somewhat  towards  the  side  on  which  the  retardation 


xv.]  INTERFERENCE.  545 

takes  place.  From  the  amount  of  this  shifting  may  be  calcu- 
lated the  amount  of  retardation ;  and  by  means  of  this  the  rela- 
tive velocities  of  light  in  (and  therefore  the  refractive  indices 
of)  such  things  as  hot  air,  cold  air,  hydrogen  gas,  normal  glass, 
compressed  glass,  compressed  liquids,  and  so  forth,  may  be 
estimated. 

Colours  of  thin  films.  —  Thin  films  of  transparent  sub- 
stances, such  as  oil  upon  the  surface  of  water,  iron  oxide  upon 
the  surface  of  tempered  steel,  oxides  deposited  upon  metals  by 
the  galvanic  batteiy,  soap  bubbles,  glass  blown  out  to  an  extreme 
tenuity  or  exfoliating  under  the  influence  of  slow  decomposition, 
present  curious  colours  when  shone  upon  by  a  comparatively 
bright  light. 

Such  films  may  be  rendered  permanent ;  a  solution  of  bitumen  and  a 
little  caoutchouc  in  a  mixture  of  benzene  and  oil  of  naphtha,  dropped  upon 
water,  forms  films  which  solidify  and  may  be  caused  to  adhere  to  a  sheet  of 
paper. 

In  Fig.  176  monochromatic  light  from  S  is  incident  upon  a 
thin  transparent-film  AB  of  uniform  thickness.  A  part  of 
the  light  is  at  once  reflected  to  R  from  Fig.i?6. 

the  first  surface  of  the  film.  Another 
part  is  refracted  to  R'  after  having 
undergone  one  reflexion  at  the  second 
surface.  If  the  path  of  the  beam  in 
the  film  be  an  even  *  number  of  half 
wave-lengths,  the  beam  travelling  to 

R'  is  opposed  in  phase  to  that  travel-  \\T' 

ling  to  R,  and  an  eye  placed  at  RR' 
(these  points  being  supposed  very  close  together)  will  receive 
no  impression  of  light ;  or,  rather,  it  will  receive  but  a  feeble 
impression,  for  tlie  ray  to  R'  cannot  be  quite  equal  in  intensity 
to  that  travelling  to  R.  Again,  an  eye  placed  at  XT'  will  per- 
ceive but  a  feeble  impression  of  light ;  not  absolute  darkness, 
for  the  ray  to  T  is  considerably  more  intense  than  that  to  T', 
and  is  not  completely,  neutralised  by  it. 

There  is,  however,  complete  interference  for  any  one  wave-length  if 
multiple  reflexion  be  taken  into  account. 


*  This  seems  strange  ;  we  might  have  expected  a  retardation  of  an  odd  number 
of  half  wave-lengths  to  produce  a  difference  in  phase  of  half  a  period ;  but  it  will 
be  remembered  that  the  beam  reflected  at  one  of  the  surfaces  of  the>film  —  that  sur- 
face, namely,  which  separates  an  optically  denser  from  a  rarer  medium  —  suffers  a 
loss  of  half  a  wave-length,  which  is  independent  of  the  thickness  of  the  film. 


546  OF   ETHER-WAVES.  [CHAP. 

Let  the  film  be  of  variable  thickness;  a  film  of  air 
between  a  glass  plate  and  a  biprism,  or  between  a  convex  lens 
and  a  plate  of  glass,  varies  in  thickness  with 
the  distance  from  the  centre ;  in  the  former  case 
the  thickness  of  the  film  of  air  varies  as  the  dis- 
tance, in  the  latter  approximately  as  the  square 
of  the  distance  from  the  central  point.  Monochromatic  light 
reflected  from  such  a  system  presents  the  appearance  of  alter- 
nately dark  and  bright  bands  or  circles  —  bright  where  the 
directly-reflected  light  and  the  light  reflected  from  the  second 
surface  of  the  film  are  similar  in  phase  —  dark  where  they  are 
opposed.  In  the  case  of  a  lens  pressed  against  a  plate  they 
are  known  as  Newton's  rings.  The  less  the  curvature  of  the 
lens  the  greater  the  distance  between  two  consecutive  rings. 
The  distance  between  the  consecutive  rings  is,  approximately, 
inversely  proportional  to  the  radius,  so  that  the  external  rings 
are  most  crowded  together.  If  such  a  substance  as  water  be 
used  between  the  lens  and  the  glass,  the  rings  are  closer 
together;  the  width  of  the  rings  varies  inversely  as  /3,  the 
refractive  index  of  the  substance  thus  employed ;  for  a  shorter 
distance  in  an  optically-denser  medium  is  equivalent  to  a  longer 
distance  in  air.  On  inclining  the  incidence  of  the  light,  the 
rings  become  dilated.  By  transmission,  a  second  system  of  rings 
is  produced,  complementary  but  dimmer.  If  mixed  coloured 
or  white  light  be  employed,  the  dark  and  bright  rings  of  the 
several  components  cannot  coincide,  and  the  result  is  a  series  of 
circular  spectra,  in  each  of  which  the  violet  circle  is  the  nar- 
rowest. These  spectra  overlap  one  another  at  a  little  distance 
from  the  centre,  and  blend  into  what  appears  to  the  eye  to  be 
white  light. 

A  series  of  dark  rings  or  fringes  may  be  obtained  by  rubbing  a  film  of 
soap  on  black  glass,  drying  it,  and  breathing  gently  upon  one  point  of  this 
through  a  glass  tube ;  this,  done  in  the  sunshine,  gives  rise  to  bright  colours. 

It  is  not  possible  actually  to  obtain  monochromatic  light ; 
even  that  emitted  by  incandescent  sodium-vapour,  in  which  some 
five  hundred  rings  can  be  seen,  is  not  quite  monochromatic. 

The  centre  of  Newton's  rings  is  dark  if  there  be  approximate  contact ; 
perfect  contact  there  never  can  be,  for  a  dustless  surface.it  is  impossible  to 
obtain ;  even  when  there  is  no  appreciable  thickness  of  film  traversed,  the 
fact  that  one  ray  is  reflected  from  the  upper,  and  the  other  from  the  lower 
surface,  the  one  at  the  bounding  surface  of  an  optically-denser,  the  other  at 
the  surface  of  an  optically-rarer  medium,  causes  the  one  to  lose,  while  the 
other  does  not  lose,  half  a  wave-length  on  reflexion;  they  thus  become 


XV.] 


INTERFERENCE. 


547 


opposed  in  phase,  and  the  centre  is  dark.  If,  however,  both  reflexions  be 
made  to  take  place  from  the  surface  of  an  optically-denser  medium,  as  in 
Young's  experiment,  —  in  which  light  travelling  through  a  lens  of  crown 
glass  was  reflected  first  from  the  upper  surface  of  a  film  of  oil  of  sassafras, 
lying  between  that  lens  and  a  plate  of  flint  glass,  sassafras  being  inter- 
mediate in  its  refractive  power  between  crown  and  flint  glass,  —  there 
is  no  such  relative  retardation,  and  the  centre  of  the  system  of  rings  is 
bright. 

The  Iridescence  of  mother  of  pearl  and  of  objects  with 
a  finely-grooved  or  striated  surface,  such  as  butterfly's  scales,  is 
an  effect  of  interference.  Sunlight  falls  upon  their  surface; 
some  of  this  is  reflected  from  the  ridges,  some  from  the  grooves, 
and  in  this  way  a  difference  of  path  is  set  up  among  the  reflected 
rays,  which  causes  differences  of  phase  among  them,  and,  in 
the  case  of  some  of  them,  opposition  of  phase  and  extinction. 
When  the  incidence  of  the  reflected  light  is  very  oblique,  the 
ridges  alone  may  reflect,  the  differences  of  phase  and  of  path 
produced  will  be  very  small ;  there  will  be  little  iridescence  and 
very  considerable  reflexion. 

The  propagation  of  light  "in  straight  lines"  within  the 
same  isotropic  medium  is  itself  a  result  of  interference.  From 
it  is  derived  the  power  of  making 
a  geometrical  Shadow.  In  Fig. 
178  a  real  focus  at  F  acts  as 
a  source  of  light.  It  casts  a 
sharply-defined  shadow  of  an 
object  O  upon  a  screen.  If  the 
source  of  light  be  an  extended 
one,  not  a  mere  point,  the  shadow 
consists  of  two  regions,  a  central 
umbra  and  a  marginal  penumbra.  In  Fig.  179  the  sun,  S, 
shines  upon  the  earth,  E :  the  earth  being  smaller  than  the  sun, 
there  is  formed  a  cone  of  darkness  behind  the  earth ;  if  the  moon 
travel  wholly  or  partially  into 
this  cone  of  shadow,  it  will  be 
wholly  or  partly  unillumined, 
and  we  have  a  total  or  a  partial 
eclipse  of  the  moon. 

But  outside  this  shadow 
there  is  a  penumbral  region,  in 
which  a  body,  or  any  point  of  a  body,  will  be  in  "  half -shadow," 
not  fully  illuminated,  because  able  only  to  see  a  portion  of  the 
illuminating  body. 


Fig.178. 


Fig.179. 


548 


OF   ETHER-WAVES. 


[CHAP. 


Fig.180. 


When  light  radiating  from  an  extended  object  passes  through 

a  small  aperture,  the  waves 
arriving  at  the  aperture  from 
,  the  object  traverse  the  aper- 
ture, and  there  cross  each 
other;  they  then  diverge, 
and  a  screen  placed  on  the 
opposite  side  of  the  aperture 
receives  an  inverted  image 
of  the  object,  whose  size 
varies  with  the  distance  of 
the  screen,  as  in  the  well-known  Camera  Obscura. 

An  aperture  of  no  appreciable  breadth  would,  at  whatever  distance  the 
screen  might  be  placed,  give  a  perfect  image  in  the  natural  colours,  an 
image  of  which  no  part  would  be  out  of  focus ;  one  of  ^V^110^  diameter  will 
give  on  a  screen  at  40  inches  distance  an  image  which,  though  wanting  in 
brightness,  is  as  perfectly  defined  an  image  as  any  possible  lens  placed  at 
the  aperture  can  produce  :  one  of  y^-inch  will  produce  the  same  definition 
at  a  distance  of  250  inches  ;  or,  in  general  (Lord  Rayleigh),  if  A  be  the 
wave-length,  r  the  semi-diameter  of  the  aperture,  d  the  least  distance  of  good 
definition,  d  —  (2r2/A).  When  the  screen  is  nearer  than  this,  each  point  of 
the  object  makes  on  the  screen  an  image  which  has  the  same  shape  as  the 
aperture,  and  the  superposition  of  these  makes  a  blurred  image.  When  the 
diameter  is  ^-inch  (No.  10  steel  sewing-needle),  d  —  8-57  inches  (pin-hole 
photography)  ;  but  with  smaller  apertures  than  this,  Diffraction  begins  to 
confuse  the  result. 

Light  thus  travels  in  straight  lines,  and  is  incapable  of 
passing  round  corners  under  ordinary  circumstances,  and  as 
examined  by  our  ordinary  senses. 

A  closer  examination  of  the  subject  shows,  however,  that 
light  does  to  a  certain  degree  pass  even  round  corners.  The  phe- 
nomena of  Diffraction,  in  which  this  is  observed,  are  explicable 
on  the  ordinary  principles  of  interference.  Let  S  (Fig.  77)  be 
the  source  of  light ;  waves  diverge  from  this  as  a  centre.  These 
waves  impinge  upon  a  screen  AB.  Fig.  77  shows  that  beyond 
the  screen  AB  there  is  a  series  of  fringes  within  the  geometrical 
shadow ;  that  even  in  the  part  directly  in  view  of  the  source  of 
light  there  are  bands  of  relative  darkness ;  that  the  central 
point  of  the  shadow  may  be  nearly  as  brightly  illuminated  as  if 
there  had  been  no  screen  AB  ;  that  the  broader  the  object  AB, 
the  narrower  will  be  the  fringes ;  that  the  forms  in  space  of  the 
regions  of  approximate  darkness  are  hyperboloids ;  while  if  the 
source  of  light  be  removed  to  an  infinite  distance,  the  hyperbolic 
lines  of  relative  rest  in  the  illuminated  region  are  practically 


xv.]  DIFFRACTION.  549 

reduced  to  straight  lines,  but  sweep  past  the  obstacle  without 
touching  it. 

When  the  obstacle  is  circular — a  minute  circle  of  tinfoil 
pasted  on  a  piece  of  clear  glass  —  the  shadow  cast  upon  a 
screen,  or  received  in  the  eye  directly  or  by  the  aid  of  a  lens  or 
telescope  focussed  on  the  obstacle,  is  seen  to  be  surrounded  by 
a  series  of  dark  and  bright  rings  ;  or,  if  the  light  from  S  be 
mixed-coloured  or  white  light,  by  a  series  of  spectra ;  while  the 
shadow  is  also  modified  by  a  series  of  such  bands  or  spectra, 
and  its  centre  is  bright.  A  similar  construction  for  a  little  cir- 
cular aperture  in  an  opaque  screen  at  AB  will  show  that  the 
bright  spot  produced  on  a  screen  beyond  AB  will  have  fringes 
blurring  the  sharpness  of  its  edges,  and  that  at  certain  distances 
of  the  second  screen  from  AB  the  centre  of  the  bright  spot  will 
be  dark. 

When  the  obstacle  or  chink  is  linear  and  parallel-sided,  the 
fringes  or  spectra  are  parallel  to  one  another ;  when  it  is  not  so 
they  assume  a  curved  form ;  when  it  is  angular  the  fringes  may 
assume  a  great  variety  of  remarkable  and  beautiful  forms. 

The  phenomenon  of  diffraction  can  be  roughly  observed  by 
looking  at  a  distant  gas-flame,  edge  on,  with  the  half-closed 
eyes  ;  the  sun  shining  on  the  eye-lashes  will  also  produce  a  simi- 
lar effect ;  the  morning  sun,  shining  on  twigs  of  trees  situated 
between  the  sun  and  the  eye,  causes  the  shadows  of  some  of 
them  to  become  bright  in  the  centre,  and  a  curious  silvery 
appearance  results. 

The  image  of  any  point  seen  through  a  telescope  or  micro- 
scope has  its  clearness  of  definition  interfered  with  by  the  dif- 
fraction of  rays  of  light  round  the  edges  of  the  diaphragm,  or 
round  the  edges  of  the  lens.  This  effect  is  generally  insignifi- 
cant in  terrestrial  telescopes  ;  it  is  very  noticeable  in  astronom- 
ical telescopes,  where  the  source  of  light,  a  distant  star,  ought 
to  appear  reduced  to  a  point,  but  is  apparently  enlarged  into  a 
perceptible  disc  surrounded  by  rings ;  and  in  the  microscope  it 
sets  a  limit  to  the  powers  attainable,  for  high  powers  involve 
small  lenses  and  small  apertures,  and  these  bring  diffraction  in 
their  train.  The  limit  of  microscopic  definition  is  about  3^QO 
mm.  with  white,  and  about  -g-oVo  mm-  with  blue  light. 

If  a  very  large  number  of  parallel  equidistant  lines  be 
ruled  upon  glass  or  metal,  plane-fronted  light  issuing  from  a 
slit  or  from  the  image  of  a  slit  will,  if  transmitted'  through  or 
reflected  from  this  so-called  Diffraction-grating,  and  focussed 


550  OF  ETHER-WAVES.  [CHAP. 

upon  a  screen  or  in  the  eye,  be  found  to  be  resolved  into  a  cen- 
tral bright  image  of  the  slit,  on  each  side  of  which  is  a  dark 
space,  and  then  a  series  of  successive  spectra,  overlapping  or 
separated  by  dark  spaces,  according  to  the  fineness  of  the 
grating:  these  spectra  have  their  violet  ends  turned  towards 
the  central  bright  image  (see  Fig.  77a).  By  multiplying  the 
number  of  lines  in  the  Diffraction-grating,  as  in  Prof.  Rowland's 
gratings,  which  have  43,000  equidistant  lines  to  the  inch,  the 
spectra  may  be  rendered  almost  perfectly  pure,  so  that  Fraun- 
hofer's  lines  may  be  easily  seen  in  them. 

A  microscopical  preparation  of  muscular  tissue  will  often  be  found  to 
act  as  a  more  or  less  efficient  diffraction-grating ;  the  striations  of  the  mus- 
cular fibres  take  the  place  of  the  grooves  engraved  on  the  glass. 

The  value  of  diffraction-spectra  is  that  the  deviation  in  the 
successive  spectra  depends  directly  upon  the  wave-length  ;  their 
disadvantage  the  mechanical  difficulties  of  uniform  grooving  of 
the  grating,  and  of  making  clean-cut  grooves. 

If  any  kind  of  light  have,  in  air,  the  wave-length  A  centimetres,  and  if 
N  be  the  average  number  of  lines  per  centimetre  engraved  on  the  grating ; 
and  if  8  be  the  angular  deviation  of  any  particular  coloured  light  (or, 
better,  of  any  particular  Fraunhofer  line),  —  then  sin  8  is  equal  to  NX  for 
the  first  spectrum,  to  2xA  for  the  second  spectrum,  and  so  forth ;  and  since 
N  and  8  can  be  measured,  A  can  be  accurately  found.  At  the  spot  where 
light  of  wave-length  A.  appears  in  the  third  spectrum,  that  of  wave-length 
3A./2  in  the  second  and  that  of  wave-length  3A.  in  the  first  spectrum  coincide. 

The  definition  in  the  diffraction-spectrum  is  best  in  the  position  of 
minimum  deviation  (p.  141);  and  the  normal  spectrum  is  pro- 
duced when  the  angle  of  incidence  is  so  regulated  that  the  angle  of  diffraction 
is  zero  (see  p.  141). 

The  Twinkling  of  Stars  is  another  effect  of  interfer- 
ence :  light,  coming  to  the  eye  from  a  star  so  distant  as  to  be 
practically  a  single  luminous  point,  arrives  in  rays  which  have 
traversed  slightly  unequal  distances  in  an  irregularly-refracting 
atmosphere  and  thus  enter  the  eye  in  irregularly-unequal  phases. 
Now  one  colour  is  extinguished,  now  another  ;  the  eye  perceives 
coloured  light  complementary  to  that  momentarily  lost.  No 
two  persons  can,  as  a  rule,  see  any  star  twinkling  in  precisely 
the  same  manner.  The  planets  twinkle  only  at  their  edges: 
their  discs  present  many  points  or  sources  of  light,  whose  scin- 
tillations, on  the  whole,  mask  one  another. 

If  a  planet  and  a  twinkling  star  —  say  Jupiter  and  Sirius  —  be  severally 
looked  at  through  an  opera-glass  which  is  rapidly  whirled  across  the  field  of 
view,  the  image  of  the  planet  will  appear  to  be  drawn  out  into  a  continuous 


xv.  INTERFERENCE.  551 

streak,  while  that  of  the  star  will  be  broken  up  into  a  chain  of  unequally- 
bright  and  differently-coloured  spots  of  light. 

The  colours  of  light  from  a  bright  point  twinkling  through  a  dusty  haze, 
or  shining  through  a  piece  of  glass  covered  with  lycopodium  ;  the  Corona 
(red  externally)  which  surrounds  the  moon  as  it  shines  through  an  atmos- 
phere charged  with  particles  of  condensed  aqueous  vapour;  the  coloured 
rings  seen  when  particles  float  in  the  vitreous  humour  of  the  eye, — these  are 
all  different  diffractive  effects  of  interference  ;  and  the  smaller  the  size  of 
the  particles  which  produce  them,  the  greater  the  breadth  of  the  coloured 
rings.  Each  particle  acts  as  a  partially  opaque  small  screen. 

The  interference  of  Actinic  Rays  may  be  shown  by  pho- 
tography; of  Dark  Heat,  by  passing  a  delicate  Thermopile, 
a  Tasimeter  (p.  636),  or  a  Bolometer  (Langley's  Thermic 
Balance,  p.  717)  through  an  invisible  diffraction-fringe  system  of 
dark  heat-waves,  obtained  by  treating  rays  of  dark  heat  with  a 
bent  mirror  or  a  biprism  ;  under  these  circumstances  the  instru- 
ment employed  will  alternately  indicate  and  cease  to  indicate 
the  impact  of  heat-waves. 

DOUBLE  REFRACTION. 

If  a  transparent  medium  have  the  same  properties  in  all 
directions  it  is  homogeneous,  or,  optically,  isotropic.  A  wave  of 
mechanical  disturbance  starting  from  a  single  point  of  disturb- 
ance in  it  will  be  spherical.  The  properties  of  the  ether-waves 
within  transparent  substances  are,  in  some  fashion,  correlated 
with  the  molecular  structure  ot  the  substance,  and  thus  any 
ether-waves  propagated  from  centres  within  homogeneous  or 
isotropic  substances  are  themselves  also  spherical. 

Substances  in  which  the  propagation  of  light  is  in  spherical  waves  are 
either  amorphous,  or  else  belong  to  the  cubical  system  of  crystals,  the  system 
in  which  the  three  crystallographic  axes  of  the  crystal  are  equal. 

In  some  crystalline  substances  one  of  the  crystallographic 
axes  differs  from  the  other  two ;  the  crystal  is  then  symmetrical 
in  reference  to  this  axis  only,  and  is  said  to  be  uniaxial.  A 
mechanical  disturbance  is  propagated  in  such  a  crystal  in  the 
form  of  an  ellipsoid. 

A  slice  cut  out  of  such  a  crystal  in  such  a  way  that  its  faces 
are  parallel  to  this  principal  axis,  is  said  to  have  been  cut 
parallel  to  the  Principal  Section  of  the  crystal. 

The  propagation  of  an  ether-wave  in  a  uniaxial  crystal  is 
peculiar.  Fig.  181  shows  an  equal-sided  rhombohedron  cut  out 
of  a  crystal  of  Iceland  spar  by  splitting  it  along  its  natural  cleay- 


552 


OF  ETHEE- WAVES. 


[CHAP. 


age-planes ;  its  axis  AB  joins  the  opposite  obtuse-angles.     Let  a 
point  C  on  this  axis  be  a  centre  of  optical  disturbance.     Then 

two  concentric  sets  of 
waves  are  produced; 
the  one  spherical  just 
as  in  glass,  the  other 
ellipsoidal ;  one  of  the 
axes  of  the  ellipsoid 
coincides  with  the  axis 
of  the  crystal,  and  is 
equal  for  a  given  in- 
terval of  time  to  the  diameter  of  the  sphere  developed  in  an 
equal  time ;  the  other  two  axes,  which,  to  avoid  circumlocu- 
tion, we  shall  here  call  the  extraordinary  axes,  are  equal  to 
one  another,  and  are  either  longer  or  shorter  than  the  former, 
according  to  the  nature  of  the  crystal.  The  next  question  is,  — 
Which  part  of  a  general  disturbance  at  C  is  propagated  in  the 
spherical,  and  which  in  the  ellipsoidal  wave  ? 

It  may  roughly  be  stated  that  just  as  we  have  seen  beams  of 
polarised  light  differently  affected  by  simple  reflexion  and  refrac- 
tion according  to  the  plane  of  their  polarisation,  so  in  double 
refraction  the  behaviour  of  a  beam  of  light  depends  upon  its 
state  of  polarisation. 

On  referring  to  Fig.  59  we  find  that  the  construction  there 
given  for  the  course  of  a  refracted  plane-fronted  wave  may  be 
.reduced  to  the  following  construction  (due  to  Huyghens)  for  a 
single  ray  refracted  at  the  surface  of  an  ordinary  isotropic 
medium.  AB  is  an  incident  ray  travelling  through  the  medium 

M ;  CD  a  circular  arc, 
drawn  from  centre  B,  with 
radius  proportionate  to  the 
velocity  of  light  in  the 
medium  M.  Continue  the 
arc  CD  into  the  second 
medium  M';  produce  AB 
until  it  cuts  that  arc  in  E ; 
from  E  draw  a  tangent  line 
(or  plane)  cutting  the  re- 
fracting surface  in  T.  From  B  as  centre  draw  a  semicircular 
arc  in  the  medium  M',  with  a  radius  proportionate  to  the  velocity 
of  light  in  M'.  From  T  draw  a  tangent  to  this  arc  ;  the  tangent 
touches  the  arc  at  Bf ;  join  BB'.  BBf  is  the  refracted  ray. 


XV.] 


DOUBLE   REFRACTION. 


553 


Fig.183. 


A  series  of  somewhat  similar  constructions  will  enable  us  to 
study  a  certain  number  of  cases  of  double  refraction. 

Suppose  a  block  to  be  cut  out  of  a  crystal  of  Iceland  spar 
in  such  a  way  that  one  of  its  cut  surfaces  is  parallel  to  the  axis ; 
and  suppose  an  incident  beam  to  fall  upon  that  surface  in  a 
direction  at  right  angles  to  the  axis.  Fig.  183  shows  that  if  GH 
represent  such  a  block,  and  if 
the  incident  beam  be  in  the 
plane  of  the  paper,  the  axis  is 
in  such  a  case  looked  at  end- 
on  ;  and  then  we  find  that  the 
incident  ray  is  divided  into  two 
parts,  which  travel  at  different 
rates,  the  slower  one,  BO,  in 
the  central  sphere,  the  more 
rapid  one,  BE,  in  the  outer 
ellipsoid,  which,  looked  at  in 
this  aspect,  has  a  circular  section  ;  the  former,  BO,  the  Ordinary 
Ray  (which  obeys  the  ordinary  law  sin  L  =  /3  sin  g),  being  more 
refracted  than  the  latter,  BE,  the  Extraordinary  Ray.  Both 
these  rays  are  in  this  case  in  the  plane  of  the  paper,  like  the 
original  incident-ray.  The  relative  radii  of  the  two  circles  may 
be  found  from  the  respective  amounts  of  refraction  of  the  two 
rays  at  this  kind  of  incidence. 

For  the  light  emitted  by  sodium-vapour,  the  ordinary  index  and  the 
extraordinary  index  of  Iceland  spar  are  respectively  1-65850  and  1-48635; 
the  reciprocals  of  these  numbers  represent  the  relative  velocities  of  the 
ordinary  and  the  extraordinary  rays  in  Iceland  spar  as  compared  with  that 
of  light  in  air,  this  being  reckoned  as  unity.  In  such  crystals  as  those  of 
Iceland  spar  the  ordinary  ray  is  more  retarded  than  the  extraordinary. 

Let  us  now  turn  the  block 
of  spar  round  so  that  its  axis 
is  brought  into  the  plane  of  the 
paper  —  that  is,  into  the  same 
plane  with  the  incident  light ; 
the  incident  light  now  travels 
in  a  principal  section  of  the 
crystal.  One  of  the  extraor- 
dinary axes  of  the  ellipsoid, 
being  at  right  angles  to  the 
axis  of  the  crystal,  is  at  right  angles  to  the  refracting  surface ; 
its  semi-axis,  BF  in  the  sectional  figure  (Fig.  184),  bears  to  the 


Fig.184. 


554 


OF   ETHER-WAVES. 


[CHAP. 


Fig.185. 


radius  of  the  circle  the  ratio  of  1-65850  to  1-48635  if  the  light 
used  be  that  emitted  by  sodium-vapour. 

If  the  block  of  spar  be  cut  by  a  plane  at  right  angles  to  the 
principal  sections,  but  not  parallel  to  the  axis,  we  obtain  the 

result  shown  in  Fig.  185. 

A 


The  incident  light  is  in 
the  plane  of  the  paper; 
the  axis  of  the  crystal  is 
also  in  the  plane  of  the 
paper. 

When  the  surface  which 
receives  the  incident  beam 
has  been  cut  at  right  angles 
to  the  axis,  and  the  light 
falls  upon  it  normally  (that 
is,  at  right  angles  to  the 
surface,  or  parallel  to  the  axis),  there  is  no  double  refraction ; 
the  ordinary  and  the  extraordinary  rays  coincide. 

A  parallel-sided  slice  of  Iceland  spar  cut  in  any  other 
direction  than  at  right  angles  to  the  axis  will  divide  an  incident 
ray  into  an  ordinary  and  an  extraordinary  ray,  except  in  the 

case  in  which  one  of  the  rays  is  so 
refracted  as  to  become  parallel  with 
the  axis,  in  which  case  the  other  ray 
coincides  with  it.  In  Fig.  186  the 
incidence  is  normal,  and  an  observer 
at  A  will  see  two  images  of  a  spot  at  B,  of  which  one,  the 
ordinary,  is  produced  as  it  would  have  been  by  ordinary  glass : 
while  if  he  turn  the  slice  round,  the  extraordinary  image  will 
rotate  round  the  ordinary  one.  This  can  be  readily  observed 
with  an  ordinary  crystal  of  Iceland  spar. 

Light  striking  on  a  plate  or  a  common  crystal  of  Iceland 
spar  is  thus  split  into  two  rays,  and  a  single  point  or  a  page  of 
,A  print  looked  at  through  such  a  crystal  gives 
a  double  image.  Conversely,  a  pair  of  points, 
C,  D,  if  looked  at  by  an  observer  at  A,  will 
have  their  images  blended,  and  by  finding  for 
various  distances  between  C  and  D  the  angle 
ABE,  at  which  these  points  appear  to  blend,  the  two  refractive 
indices  may  be  found:  the  rays  CB,  DB,  and  BA  being  caused 
to  lie  all  in  a  principal  section  of  the  crystal.  BD  represents 
the  ordinary  and  BC  the  extraordinary  ray. 


Fig.  186. 


Fig.187. 


C      D 


xv.]  DOUBLE   REFRACTION.  555 

When  the  incident  ray  is  oblique  to  the  principal  section,  the  extraor- 
dinary ray  is  no  longer  in  the  same  plane  with  the  incident  and  the 
ordinary  refracted  ray,  but  is  deflected  to  one  or  the  other  side :  the  tangent 
plane  to  the  ellipsoid  does  not  touch  it  in  the  plane  of  incidence. 

The  above  figures  are  drawn  for  crystals  such  as  Iceland 
spar,  beryl,  emerald,  mica,  ruby,  sapphire,  tourmaline,  the  ordi- 
nary index  of  refraction  of  which  is  greater  than  the  extraor- 
dinary, and  in  which  the  ordinary  ray  travels  more  slowly  than 
the  extraordinary,  and  lies  between  the  extraordinary  ray  and 
the  axis ;  such  crystals  are  called  Negative  Crystals.  In  others, 
such  as  ice,  quartz,  boracite,  the  extraordinary  ray  lies  between 
the  ordinary  ray  and  the  axis ;  such  crystals  are  called  Positive 
Crystals.  In  the  latter,  the  extraordinary  axes  of  the  ellipsoid 
are  shorter  than  the  diameter  of  the  sphere,  which  thus  encloses 
the  ellipsoid :  the  extraordinary  index  of  refraction  is  in  them 
greater  than  the  ordinary  index. 

The  two  rays,  the  ordinary  and  the  extraordinary,  are  found 
to  be  polarised  in  planes  almost  exactly  at  right  angles  to  one 
another.  The  ordinary  ray  is  polarised  in  a  plane  contain- 
ing both  the  incident  ray  and  the  crystalline  axis. 

If  the  incidence  be  that  of  Fig.  184,  the  incident  ray,  the 
reflected  ray,  and  both  refracted  rays  are  in  the  same  plane, 
the  plane  of  the  paper,  and  the  axis  is  parallel  to  that  plane ; 
the  ordinary  ray  is  said  to  be  polarised  in  that  plane ;  light 
polarised  in  such  a  plane  of  incidence  passes  through  the  spar 
as  an  ordinary  ray.  The  extraordinary  ray,  when  the  whole 
three  rays  thus  travel  in  a  principal  section  of  the  crystal,  is 
found  to  be  polarised  in  a  plane  exactly  at  right  angles  with 
the  plane  of  polarisation  of  the  ordinary  ray. 

The  second  face  of  the  block  of  crystal  may  be  so  cut  that 
it  receives  the  ordinary  and  the  extraordinary  rays  at  such  an 
angle  as  to  transmit  the  one,  but  totally  to  reflect  the  other.  In 
Nicol's  prism  a  long  rhomb  of  Iceland  spar  is  cut  in  this  way, 
and  the  portions  are  so  cemented  by  Canada  balsam  that  when 
common  light  enters  the  Nicol  it  is  divided  into  two  rays,  of 
which  one,  the  Ordinary,  is  totally  reflected  when  it  meets  the 
cemented  surface,  while  the  Extraordinary  ray  is  trans- 
mitted and  emerges  (the  faces  of  the  prism  having  been,  in 
order  to  permit  this,  cut  down  to  the  proper  angle)  in  a  direc- 
tion parallel  to  that  of  the  incident  ray.  The  whole  arrange- 
ment is  thus  capable  of  acting  as  a  polariser ;  and  if  polarised 
light  be  sent  through  it  in  one  rotational  position,  the  Nicol  will 


556  OF   ETHER- WAVES.  [CHAP. 

transmit  it  freely ;  while  if  the  Nicol  be  rotated  through  90°  in 
either  direction,  on  either  side  of  the  most  favourable  position, 
it  will  transmit  none  of  it.  It  can  thus  serve  as  a  means  not 
only  of  producing  polarised  light,  but  also  of  detecting  polarised 
light,  and  of  finding  in  what  plane  it  is 
polarised ;  and  when  it  does  this  duty  it 
is  called  an  analyser. 

Foucault's  prism  is  shorter  than  Mcol's,  and 
•O   an  air-film   replaces   the    balsam.      In   Rochon's 
prism  two   similar  pieces  of  quartz  make  up  a 
parallel-faced   block    (Fig.   187 a):    the    ordinary 
ray  emerges  without    deviation :    the    extraordi- 
Axis  nary  is  sent  away  to  one  side. 

In  tourmaline  there  is  double  refraction ;  but  one  of  the 
rays,  the  ordinary,  is  absorbed,  and  the  extraordinary  alone 
passes  through.  Thus  a  thin  plate  of  tourmaline  acts  as  a 
polariser  of  common  light  incident  upon  it,  and  another  plate 
rotating  in  front  of  it  may  act  as  an  analyser,  —  a  convenient 
arrangement,  were  it  not  that  tourmaline  is  always  dark  in 
colour,  and  absorbs  much  of  the  light  incident  upon  it.  For 
this  reason  Nicol's  prisms  are  commonly  used  as  sources  of 
polarised  light. 

Crystals  of  sulphate  of  iodo-quinine  act  like  tourmaline,  but 
are  useless  because  they  are  dark,  small,  and  brittle. 

We  may  here  recall  the  different  modes  of  obtaining  a  beam  of  plane- 
polarised  light. 

1.  Reflexion  of  ordinary  light  from  glass  at  the  angle  of  complete 

polarisation. 

2.  Transmission  through  a  pile  of  glass  plates  with  parallel  sides ;  the 

angle  of  incidence  being  the  angle  of  complete  polarisation,  or  an 
angle  approximating  to  it. 

3.  Separation  of  the  ordinary  from  the  extraordinary  ray  produced  by 

double  refraction ;  this  being  done 

(a)  by  tourmaline,  which  extinguishes  the  ordinary  ray ; 
(6)  by  a  Mcol  or  a  Foucault  prism,  which  turns  aside  the  ordinary 

ray; 
(c)  by  a  Rochon  prism,  which  turns  aside  the  extraordinary  ray. 

Some  crystals,  such  as  topaz  and  arragonite,  have  two  axes, 
and  are  called  Binaxial  Crystals  :  in  these  the  wave-surface  is 
very  complex,  and  they  have  three  indices  of  refraction. 

In  general,  in  these  crystals,  the  wave-front  is  oblique  to  the  rays,  and 
there  is  no  ray  which  obeys  the  ordinary  law  of  refraction  that  sin  i  =  /3  sin  g; 
but  that  ray  which  does  so  most  nearly  in  general,  and  which  does  so  per- 
fectly when  the  incidence  is  in  one  of  the  principal  sections,  is  called  the 
ordinary  ray ;  while  the  other  of  the  two  rays,  into  which  a  ray  of  incident 


XV.] 


DOUBLE   REFRACTION. 


557 


light  is  divided  on  non-axial  incidence,  is  called  the  extraordinary  ray.  In 
such  crystals  the  positions  of  the  optic  axes,  which  have  no  invariable  rela- 
tion to  the  crystallographic  axes,  are  variable ;  they  vary  with  the  tempera- 
ture of  the  crystal,  and  with  the  kind  of  light  employed ;  and  in  some  cases 
a  crystal  is  found  to  be  binaxial  for  one,  uniaxial  for  another  kind  of  light ; 

Fig.  1876. 


Fig.  187  c. 


187  d. 


Glauberite  (native  sulphate  of  soda  and  lime),  for  example,  being  binaxial 
to  red,  uniaxial  to  violet  light. 

If  we  take  a  crystal  of  arragonite,  which  happens  to  have  its  three  indices 
of  refraction  in  directions  at  right  angles  to  one  another,  we  find  that  in  the 
three  principal  sections  the  circles  and  ellipses,  which  represent  the  simulta- 
neous propagations  of  the  different  parts  of  the  wave  from  a  central  point 
O,  are  differently  related  to  one  another.  Let  vt  be  the  greatest  velocity 
(the  velocity  in  air  being  taken  as  equal  to  unity),  vul  the  least,  and  vn  the 
intermediate,  these  three  velocities  determining  the  three  indices  of  refrac- 
tion for  any  particular  colour ;  then  the  propagation  of  a  disturbance  from 
a  central  point  O  results  in  the  formation  of  a  complex  wave-surface  which 
may  be  understood  by  looking  at  Figs.  187  b,  c,  and  d,  which  represent  the 
three  principal  sections :  or  better,  by  studying  the  models  of  this  kind  of 
surface  which  are  now  procurable.  One  of  these  principal  sections  (Fig. 
187  d),  that  of  greatest  and  least  elasticity,  presents  the  peculiarity  that  the 
ellipse  is  partly  inside,  partly  outside  the  circle.  If  a  line  NM  be  drawn, 
touching  both  ellipse  and  circle  at  N  and  M,  it  will  be  seen  that  the  disturb- 
ance from  O  reaches  M  and  N"  at  the  same  time ;  and  after  successive  intervals 
of  time,  as  ellipse  and  circle  expand,  the  successive  tangent-lines  M'N',  M"N", 
etc.,  remain  parallel  to  MN.  The  portions  of  the  wave-front  at  M  and  N 

Fig.  187  e. 


therefore  move  forward,  with  respect  to  the  direction  OM,  with  equal 
velocity;  and  this  direction  is  one  of  the  Optic  Axes  of  the  crystal,  for 
the  particular  colour  employed;  there  being  another,  OM',  in  the  same 
principal  section. 

If  a  single  ray  of  natural  monochromatic  light,  SO,  Fig.  187  e,  fall  upon 
a  plate  of  arragonite  at  such  an  angle  that  the  ordinary  ray  travels  along  the 
optic  axis  OM,  the  common  tangent-plane  NM  advances  parallel  to  itself ; 


558  OF   ETHER- WAVES.  [CHAP. 

and  a  separate  ordinary  and  extraordinary  ray,  O  and  E,  might  be  expected 
to  emerge,  producing  two  images  of  S. 

Internal  Conical  Refraction.  —  So  far  as  the  diagram  187  d  can  show, 
we  would  not  expect  more  than  these  two  images  :  but  if  we  refer  to  a  model 
of  the  wave-surface,  and  if,  instead  of  applying  a  mere  tangent-line  MN,  we 
apply  a  tangent-plane  such  as  a  piece  of  flat  glass,  we  shall  find  that  the 
tangent-plane  is  in  contact  with  the  wave-surface  along  the  whole  periphery 
of  a  circle  of  contact.  Each  and  every  point  of  this  circle  satisfies  the  same 
conditions  as  the  points  M  and  N" ;  and  if  non-polarised  monochromatic  light 
reach  the  point  O  (Fig.  187  e),  there  to  be  refracted  so  that  the  ordinary  ray 
tends  to  go  along  the  optic  axis  OM,  and  the  extraordinary  along  ON,  the 
result  is  that  the  two  images  of  S,  visible  at  other  angles  of  incidence  of 
SO,  open  out  into  a  complete  circle  of  light,  which  has  been  produced  by  the 
splitting  of  the  incident  ray  of  natural  light,  SO,  into  a  hollow  cone  of  rays, 
ONM,  within  the  crystal ;  the  rays  composing  this  cone  all  pass  through  the 
above-mentioned  circle  of  contact.  The  light  at  every  point  of  this  circular 
image  is  polarised,  and  at  opposite  points  it  is  polarised  in  planes  at  right 
angles  to  one  another. 

External  Conical  Refraction.  —  In  Fig.  187  d  we  see  a  point  P  —  there 
being  four  such  points  —  at  which  rays  from  O  in  both  waves  arrive  at  the 
same  time.  The  direction  of  vibration  in  the  two  simultaneous  rays  is  not, 
however,  the  same.  If  the  light  from  O  emerged  at  P  into  air,  the  rays  OP 
would  be  refracted,  so  far  as  the  diagram  can  show  us,  in  two  directions. 
But  on  referring  to  a  model  of  the  wave-surface  we  would  find  that  at  P 
there  was  a  conical  dimple,  into  the  bottom  of  which  a  hollow  tangent-cone 
might  be  fitted,  with  its  apex  at  P.  Every  radial  line  in  this  tangent-cone 
would  determine  a  different  refraction  of  a  pencil  of  light  travelling  along 
OP  and  emerging  at  P  into  air.  Light  travelling  in  the  direction  OP  there- 
fore opens  out,  when  it  emerges  into  the  air,  into  a  hollow  cone  of  light ; 
and  conversely,  a  solid  cone  of  rays  concentrated  by  a  lens  may  in  part  be 
collected  and  made  to  run  in  the  common  direction  PO. 

When  light  has  passed  through  a  crystal  of  Iceland  spar 
and  been  divided  into  an  ordinary  and  an  extraordinary  ray,  if 
it  be  caused  to  fall  upon  a  second  crystal  whose  faces  are  par- 
allel to  those  of  the  first,  the  two  rays  pass  through,  suffering 
no  further  division;  the  ordinary  ray  emerging  from  the  first 
crystal  is  still  the  ordinary  ray  in  the  second  crystal,  which  acts 
like  a  mere  prolongation  of  the  first.  If  the  second  crystal  be 
turned  90°  round  a  longitudinal  axis  parallel  to  the  line  AB  in 
Fig.  188,  there  is  still  no  Fig.iss. 

division    of    the   rays  ;    but  ^u^^m^a  A 

J  Kffih  ffiffii 

the  ordinary  ray  on  emer- 
gence from  the  first  crystal 
is  an  extraordinary  ray  relative  to  the  second  crystal,  and  is 
refracted  as  such  in  that  crystal ;  and  the  converse  applies  to  the 
extraordinary  ray  emerging  from  the  first  crystal.  If  the  second 
crystal  occupy  any  rotational  position  intermediate  between 


XV.] 


DOUBLE   REFRACTION. 


559 


these,  each  ray  incident  on  it  is  decomposed  into  an  ordinary 
and  an  extraordinary  ray.  There  are  thus,  in  the  ordinary  case, 
four  images  of  a  bright  point  seen  through  a  pair  of  crystals 
arranged  end  to  end,  at  a  distance  from  one  another,  and  these 
images  blend  into  two  when  the  crystals  are,  by  rotation,  placed 
parallel  or  at  right  angles  to  one  another. 

Interposed  Lamina.  —  When  a  polariser  and  an  analyser 
of  any  kind  are  arranged  at  right  angles,  so  that  a  plane-fronted 
beam  incident  on  the  system  is  wholly  cut  off  or  deflected  by  it, 
an  eye  placed  beyond  the  analyser  can  perceive  no  light;  but 
if  a  thin  film  of  mica,  or  other  double-refracting  substance, 
uniaxial  or  bin  axial,  of  uniform  thickness,  be  caused  to  intervene 
between  the  polariser  and  the  analyser,  the  field  may  become 
filled  with  light,  coloured  or  white,  according  to  the  position  of 
the  interposed  film. 

In  Fig.  189  the  line  AB  represents  a  plane  vertical  to  the  paper,  and 
cutting  the  paper  in  AB :  we  call  this  the  vertical  plane,  or  the  plane  AB. 
Then  let  us  by  any  convenient  means  produce  a  beam  of  plane-polarised 
monochromatic  light,  polarised  in  the  plane  AB,  and  let  us  suppose  this 
beam  to  be  seen  end-on,  travelling  away  from  the  observer's  eye.  Interpose 
a  thin  plate  of  some  birefringerit  substance  in  the  path  of  the  beam :  let  the 
axis  of  this  lie  in  the  plane  CD.  The  beam  AB  is  broken  up  by  the  inter- 
posed plate  into  two :  one  in  which  the  plane  of  polarisation  is  parallel  to 
CD,  one  in  which  it  is  at  right  angles  to  that  plane.  The  former  is  trans- 
mitted through  the  interposed  plate  as  an  ordinary  ray,  the  latter  as  an 
extraordinary.  The  lines  Oa,  Of,  Oc,  indicate  the  relative  amplitudes  of 
vibration  in  the  incident  polarised  beam,  in  the  extraordinary,  and  in  the 
ordinary  transmitted  beams  respectively.  The  interposed  plate  may  be  so 
thin  that  although  the  incident  beam  is  divided  into  two  transmitted  beams, 
these  have  not  perceptibly  separated  from  one  another,  and  on  emergence 
are  not  only  parallel,  but  are 
also  practically  coincident.  In 
a  wide-fronted  wave-system  this 
coincidence  may  be  held  to  be 
absolute  except  at  the  edges  of 
the  beam.  Though  the  two 
beams  coincide  in  direction, 
their  undulations  do  nox  coin- 
cide in  phase ;  in  positive  crys- 
tals the  extraordinary,  in  nega- 
tive crystals  the  ordinary  ray  is 
more  retarded  than  its  compan- 
ion. Let  us  suppose  that  the 
more  retarded  ray  has  lost  one 
Avave-length  :  then  the  result  of 
superposition  of  the  two  emer- 
gent rays  will  be  a  plaiie-polar- 
ised  beam  similar  to  that  which  had  originally  fallen  upon  the  interposed 


Fig.189. 


560 


OF  ETHER-WAVES. 


[CHAP. 


Fig.190, 


plate,  and  in  the  same  plane  AB  ;  if  half  a  wave-length  (=  |-\)  be  lost,  the 
result  will  be  an  equal  plane-polarised  beam,  polarised  in  the  plane  EE'. 

If,  again,  CD  coincide  with  AB  —  that  is,  if  the  principal  plane  of  the 
interposed  crystalline  plate  be  -parallel  to  the  plane  of  polarisation  of  the 
incident  light  —  there  is  no  extraordinary  beam  O/;  and  the  light,  hav- 
ing been  transmitted  through  the  interposed  film  as  an  ordinary  ray,  emerges 
as  it  entered,  plane-polarised  in  the  original  plane  AB.  If,  again,  CD  be 
at  right  angles  to  AB,  the  incident  beam  is  wholly  transmitted  as  an 
extraordinary  ray,  and  emerges  polarised  in  the  original  plane. 

Let  us  now  suppose  that  CD  is  inclined  to  AB  at  an  angle  of  45°  :  if 
one  of  the  rays  be  retarded  by  some  even  multiple  of  |X,  the  result  is  plane- 
polarised  light,  either  polarised  in  the  original  plane  (when  the  retardation 
may  be  measured  in  whole  wave-lengths),  or  in  one  at  right  angles  to  it 

(when  the  retardation  is  some  odd 
number  of  half  wave-lengths),  for 
EE'  is  at  right  angles  to  AB  when 
CD  makes  45°  with  it.  Again,  if 
the  retardation  be  some  odd  mul- 
tiple of  |A,  the  extraordinary  and 
E  ordinary  rays  are  compounded  into 
a  circularly  -  polarised  ray  of 
light;  and  if  the  retardation  be 
of  any  value  other  than  some  mul- 
tiple of  a  quarter  wave-length,  the 
result  is  an  elliptically-polarised 
beam,  the  ellipse  being,  according 
to  the  amount  of  retardation,  some 
one  of  those  indefinitely  numerous 
ellipses  which  may  be  described 
within  the  rectangle  EaE6'. 
In  the  general  case,  AB  (Fig.  190)  being  the  plane  of  the  incident  beam, 
CD  the  principal  section  of  the  interposed  plate,  the  angle  AOC  having  any 
value,  and  Oa,  Oc,  Of  being  respectively  the  relative  amplitudes  of  the 
incident  ray  polarised  in  the  plane  of  AB,  and  of  the  ordinary  and  extraor- 
dinary rays  emergent  from  the  interposed  plate;  compounded,  their  result 
is  an  Elliptically-Polarised  beam,  of  which  the  limits  are  :  — 

(a)  A  plane-polarised  beam,  whose  plane  of  polarisation  is  AB  and 
whose  amplitude  is  represented  by  Oa. 

(1)  When  CD  coincides  with  AB. 

(2)  When  CD  is  at  right  angles  to  AB. 

(3)  When  the  relative  retardation  of  cd  and  fg  is  0,  or  an  even 

multiple  of  ^A. 

(&)  An  equal  plane-polarised  beam  whose  plane  of  polarisation  is  EE' ; 
the  angle  AOE'  being  equal  to  twice  AOC  :  this  is  the  result  when 
the  relative  retardation  is  an  odd  number  of  half  wave-lengths. 

(c)  A  circularly-polarised  beam  when  the  angle  AOC  is  equal  to  45°, 
and  the  relative  retardation  is  some  odd  multiple  of  ^A. 

(1)  Right-handed  (rotation  contrary  to  hands  of  a  watch)  when 

the  component  polarised  in  fg  loses  (together  with  any 
number  of  whole  wave-lengths)  one  quarter  wave-length  or 
gains  three  quarters  relatively  to  that  in  cd. 

(2)  Left-handed  when  it  relatively  gains  one  quarter  or  loses  three. 


xv.] 


DOUBLE   REFRAC' 


,7 

561 


Fig.lDl. 


Elliptically-polarised  light. is  produced  in  every  other  relative  position 
of  CD.  This  is  right-handed  if  the  relative  retardation  of  the  extraordinary 
T&jfg  transmitted  through  CD  lie  between  0  and  |A  or  between  n\  and 
n\  +  |A,  where  n  is  any  whole  number ;  left-handed  if  its  relative  retar- 
dation lie  between  \\  and  A,  or  between  n\  +  \\  and  (n  +  1)A.  If  the 
plane  CD  lie  so  that  the  angle  AOC  lies  to  the  left  of  AB  (the  observer 
being,  as  hitherto,  supposed  to  be  stationed  near  the  source  of  light),  these 
conditions  of  left-  and  right-handedness  respectively  are  reversed. 

A  plate  of  birefringent  substance  of  such  a  thickness,  that  when  it  is 
interposed  in  the  path  of  a  beam  of  plane-polarised  light  of  a  particular 
colour,  with  its  principal  section  at  an  angle  of  ±  45°  to  the  plane  of  polari- 
sation, it  converts  that  plane-polarised  light  into  circularly-polarised  light, 
is  called  a  quarter-undulation  plate. 

Quarter-undulation  plates  are  of  two  kinds:  (a)  Where  the 
thickness  is  just  such  as  to  cause  a  relative  retardation  equal  to  £A,  or  to 
(«A  +  5 A)  ;  (6)  Where  the  plates  are  thicker,  but  are  opposed  in  their  action. 
In  Fig.  191  two  plates  cut  out  of  a  doubly-refracting  crystal  are  shown 
fitted  together;  the  one  is  cut  so  that  its  axis  is 
parallel  to  the  plane  of  the  paper ;  the  other  has  its 
axis  at  right  angles  to  the  paper.  Incident  light 
arrives  already  polarised ;  it  is  divided  by  double 
refraction  into  two  rays,  an  ordinary  and  an  extraor- 
dinary; then,  since  the  second  plate  has  its  axis  at  right  angles  to  the 
axis  of  the  first,  the  ordinary  ray  of  the  first  plate  is  refracted  in  this  as 
an  extraordinary  ray,  while  the  extraordinary  ray  of  the  former  passes 
through  as  an  ordinary  ray.  On  emergence  both  rays  are  parallel  and 
practically  coincident ;  and  the  amount  of  relative  retardation  is  equal  to 
that  produced  by  a  thin  plate  equal  in  thickness  to  the  difference  between 
the  thicknesses  of  the  two  plates. 

When  light,  plane-polarised,  is  totally  reflected  from  glass,  it  is  found 
to  be  elliptically-polarised,  unless  it  had  been  originally  polarised  in  the 
plane  of  incidence,  or  in  a  plane  at  right  angles  to  this.  Reflexion  from 
metals  presents  this  peculiarity  at  all  angles  of  incidence.  The  vibratory 
movement  actually  extends  beyond  the  surface  of  the  glass  into  the  rarer 
medium  beyond,  as  may  be  proved  on  bringing  a  second  piece  of  glass  close  to 
the  totally  reflecting  surface,  when  interference- 
colours  will  be  seen.  As  a  result  of  this,  a  differ- 
ence of  phase  is  set  up  between  the  two  components 
(polarised  in  and  at  right  angles  to  the  plane  of 
incidence)  into  which  the  incident  light  may  be 
resolved.  A  similar  result  occurs  in  metallic 
reflexion,  for  some  of  the  light  penetrates  to 
a  slight  depth  below  the  reflecting  surface.  A 
wave  cannot  have  its  direction  abruptly  changed ; 
and  during  the  gradual  change  of  its  direction, 
its  phase  becomes  altered  to  a  slight  extent : 
and  this  effect  differs  in  amount  according  to 
the  direction  of  vibration  of  the  incident  waves. 
When  the  angle  of  incidence  is  such  that  the 
difference  of  phase  set  up  corresponds  to  a  relative  retardation  |A,  two  such 
total  reflexions  would  convert  a  plane-polarised  ray  into  a  circufarly-polarised 
one.  If  a  rhomb  of  glass  be  cut  in  such  a  form  that  a  ray  of  light  may  pass 

2o 


Fig.192. 


562 


OF  ETHER-WAVES. 


[CHAP. 


normally  through  one  surface,  strike  a  second  surface  at  the  appropriate 
angle  of  incidence  and  be  there  totally  reflected,  strike  the  third  surface  at 
an  equal  angle,  and  pass  out  normally  through  a  fourth  surface,  a  ray  so 
travelling  through  it  will,  on  emergence,  be  found  to  be  circularly-polarised. 
Such  a  rhomb  is  known  as  a  Fresnel's  Rhomb,  and  acts  as  a  quarter-undu- 
lation plate  for  every  kind  of  light,  while  a  film  of  mica,  real  or  virtual,  can 
only  act  as  such  towards  light  of  one  kind. 

If  plane-polarised  light  pass  successively  through  two  similar  quarter- 
undulation  plates,  similarly  placed,  the  emergent  light  is  plane-polarised  in 
a  plane  at  right  angles  to  the  original  plane  of  polarisation  ;  whereas,  if 
the  two  quarter-undulation  plates  be  opposed  in  their  action,  the  light  is 
restored  by  the  second  to  its  original  state,  plane-polarised  in  the  original 
plane.  A  second  quarter-undulation  plate  of  known  action  affords  us  a 
means  of  distinguishing  right-  from  left-handed  elliptically-  or  circularly- 
polarised  light. 

In  metallic  reflexion  there  is  always  a  particular  angle  of  incidence,  at 
which  circularly-polarised  light  is  converted  by  reflexion  into  plane-polarised 
light. 

The  two  vibrations  which  make  up  the  circular  or  elliptic  vibration  of 
the  ether  in  a  circularly  or  elliptically-polarised  beam  of  light  are  not  in  a 
condition  to  interfere  with  one  another  on  account  of  their  difference  of 
phase,  because  they  are  executed  in  planes  at  right  angles  to  one  another. 
If  a  beam  circularly  or  elliptically  polarised  by  an  interposed  lamina  be 
received  upon  a  birefringent  analyser,  it  is  split  into  two  parts,  one  an  ordi- 
nary ray,  the  other  an  extraordinary  ray,  and  each  of  these  is  plane-pola- 
rised. In  Fig  193  AB  is  the  plane  of  original  polarisation,  CD  a  principal 
section  of  the  interposed  lamina,  EE'  a  principal  section  of  the  analysing 
crystal.  Then  a  plane-polarised  ray  whose  amplitude  is  represented  in 
magnitude  by  the  line  Oa,  and  whose  plane  of  polarisation  is  AB,  is  resolved 
by  the  interposed  lamina  into  two,  Oc  and  Of,  which  are  upon  emergence 
compounded  into  a  plane-  or  an  elliptically-  or  circularly-polarised  beam, 

according  to  their  relative  retarda- 
tions. When  this  strikes  the  analyser, 
its  components  Oc  and  Of  are  them- 
selves resolved  each  into  a  pair  of  com- 
ponents parallel  and  at  right  angles 
to  EE' ;  these  are  respectively  Oef  and 
Oh  from  Oc,  and  —  Oe  and  Og  from 
Of.  In  the  plane  EE'  we  have  there- 
fore two  vibrations,  Oe'  and  —  Oe ;  in 
the  plane  at  right  angles  to  EE'  we 
have  the  vibrations  Og  and  OA.  But 
Oe'  and  —  Oe  differ  in  phase ;  so  do 
Og  and  Oh.  These  are  therefore  in 
a  condition  for  interference.  The 
ordinary  ray,  passing  through  the 
analyser,  is  made  up  of  the  mutually- 
interfering  components,  Oe'  and  —  Oe, 
and  the  extraordinary  of  Og  and  Oh ; 
the  effect  of  interference  is  to  cause 
a  distribution  of  energy  such  that  the  ordinary  ray  gains  or  loses  as  much 
energy  as  the  extraordinary  loses  or  gains,  and  thus  the  energies  of  the 


Fig.193. 


xv.]  DOUBLE   REFRACTION.  563 

ordinary  and  the  extraordinary  rays  are,  taken  together,  equal  to  the  energy 
of  the  incident  plane-polarised  ray.  The  amount  of  relative  retardation 
caused  by  the  interposition  of  the  doubly-refracting  lamina,  when  meas- 
ured in  wave-lengths,  depends  upon  the  particular  kind  of  light  employed. 
Hence  when  the  original  plane-polarised  light  is  a  white  light,  each  colour 
obeys  its  own  law ;  each  colour,  if  strong  in  the  ordinary,  is  weak  in  the 
extraordinary  ray,  and  vice  versa ;  thus  the  extraordinary  ray  and  the  ordi- 
nary are  coloured,  and  their  colours  are  complementary. 
The  following  are  the  limiting  cases  :  — 

1.  There  is  no  extraordinary  ray  when  — 

(a)  AB,  CD,  and  EE'  (Fig.  193)  coincide. 

(6)    AB  and  EE'  coincide,  and  CD  is  at  right  angles  to  them. 

2.  There  is  no  ordinary  image  when  — 

(a)  AB  and  CD  coincide,  and  EE'  is  at  right  angles  to  them. 

(b)  CD  and  EE;  coincide,  and  AB  is  at  right  angles  to  them. 

3.  The  two  images  are  equal  for  every  colour,  and  are  therefore  white  — 

(a)  When  AB  and  CD  coincide,  and  the  angle  AOE'  =  ±  45°. 
(&)  When  AB  and  CD  are  at  right  angles,  and  the  angle  AOE'  = 
±45°. 

(c)  When  CD  and  EE'  coincide,  both  being  at  an  angle  of  ±  45° 

with  AB. 
In  every  other  position  the  two  images  are  complementarity  coloured. 

Determination  of  the  character  of  a  Beam  of  Light.  —  A  crystal 
of  Iceland  spar  capable  of  rotation  round  a  longitudinal  axis  may  be  used  as 
an  analyser,  and  will  enable  one,  with  the  intervention  of  a  doubly-refracting 
lamina,  to  determine  the  character  of  a  beam  of  light  falling  upon  it. 

Plane-polarised  light :  as  the  prism  is  rotated,  the  ordinary  and  the 
extraordinary  images  appear  and  alternately  wax  and  wane,  disappearing 
and  reappearing.  In  this  instance  the  doubly-refracting  lamina  is  dispensed 
with. 

Elliptically-polarised  light  and  partially-polarised  common 
light :  the  two  images  never  entirely  disappear,  though  they  become  alter- 
nately brighter  and  dimmer. 

Circularly-polarised  light,  and  natural  light:  the  two  images 
do  not  vary  in  their  relative  intensity  with  the  rotation  of  the  prism ;  they 
continue  nearly  equal. 

Elliptically  and  circularly-polarised  light  on  the  one  hand,  and  common 
light  unpolarised  or  partially  polarised  on  the  other,  are  distinguished  by 
the  respective  actions  upon  them  of  a  quarter-undulation  plate,  interposed 
between  the  source  and  the  analyser ;  the  former  are  converted  by  this  plate 
into  plane-polarised  light,  the  latter  are  not ;  and  the  former  then  produce 
only  one  image  in  some  positions  of  the  analyser,  while  the  latter  always 
produce  two. 

Colours  produced  by  interposed  film.  —  When  a  polariser 
and  an  analyser  are  so  placed  that  the  latter  quenches  the  light 
which  the  former  transmits,  the  interposition  between  them  of  a 
plate  of  mica  or  selenite,  or  any  other  doubly-refracting  sub- 
stance, will  cause  light  again  to  reach  the  eye,  provided  that 
the  principal  section  of  the  interposed  substance  be  neither  par- 


564  OF   ETHER-WAVES.  [CHAP. 

allel  nor  at  right  angles  to  the  principal  sections  either  of  the 
polariser  or  analyser. 

In  Fig.  193  above,  let  the  angle  AOE'  be  made  a  right  angle ;  Og  and 
O^  come  to  coincide  in  direction  with  AB ;  Oe  and  Oe'  with  GH,  at  right 
angles  to  AB.  The  polariser  allows  ab  to  pass  :  the  analyser  cuts  off  all 
components  polarised  in  the  plane  AB ;  whence  crossed  prisms  produce  per- 
fect darkness. 

But  the  intervention  of  the  doubly-refracting  substance  resolves  the 
light  which  cannot  traverse  the  analyser  into  two  rays,  of  each  of  which 
there  is  some  part  that  can  traverse  that  obstruction.  If  the  doubly- 
refracting  substance  interposed  be  uniform  in  thickness,  the  whole  field 
under  crossed  prisms  becomes  filled  with  uniform  coloured  light ;  if  the 
polariser,  or  the  analyser,  or  the  interposed  film,  be  turned  round,  the  light 
first  becomes  white,  and  then  passes  into  the  complementary  colour. 

The  colours  produced  by  a  given  film  depend  upon  the 
amount  of  relative  retardation  produced  by  it  in  light  of  each 
kind.  This  depends  upon  (a)  the  substance  of  the  film  and 
its  refractive  indices ;  (6)  its  thickness ;  (c)  the  inclination  at 
which  the  ray  traversing  it  strikes  it;  (c£)  the  relation  of  its 
optic  axis  or  axes  to  the  plane  of  its  surface. 

When  an  irregular  film  of  mica  or  selenite,  flaked  off  with 
a  penknife  from  a  large  mass,  is  interposed  between  crossed 
prisms,  the  eye,  looking  through  the  analyser,  sees  the  darkness 
of  crossed  prisms  transformed  by  the  interposition  into  a  series 
of  gorgeously  brilliant  colours ;  and  as  the  analyser  is  turned 
round,  these  fade  away  into  white  light,  and  reappear  in  comple- 
mentary hues.  If  the  film  be  a  very  thin  wedge,  each  thickness 
of  it  produces  its  own  colour,  and  a  kind  of  spectrum  is  thus 
produced.  A  double  wedge  of  quartz,  known  under  the  name 
of  Babinet's  compensator,  and  shown  in  Fig.  194,  acts  as  a 
virtual  film  of  graded  thickness,  and  gives  a  series  of  fringes 
pi  1M  or  spectra.  This  chromatic  property  of  a 

Axis  doubly-refracting  film  and  an  analyser  may 

be  made  use  of  to  detect  polarised  light: 
if  the  light  looked  at  through  such  a  system 
be  wholly  or  even  partially  polarised,  the 
phenomena  of  polarisation-colours  come  into 
view ;  and  while,  for  example,  natural  light  in  such  a  case  gives 
two  nearly  equal  white  images  when  a  crystal  of  Iceland  spar  is 
used  as  the  analyser,  circularly-polarised  light,  on  the  other 
hand,  gives  two  complementary  coloured-images  of  almost 
exactly  equal  intensity  —  equal,  that  is,  from  the  physical  point 
of  view,  though  to  the  eye  these  coloured  images  may  not  seem 
equally  bright. 


xv.]  DOUBLE   REFRACTION.  565 

When  a  divergent  or  a  convergent  beam  of  white  light 
passes  normally  through  an  interposed  film  cut  at  right  angles 
to  its  axis,  the  centre  of  the  ordinary  image  is,  when  the 
analyser  is  parallel  to  the  polariser,  found  to  be  bright  and 
colourless,  while  round  this  there  is  a  series  of  annular 
fringes  or  spectra,  the  local  colours  of  which  depend  upon  the 
local  relative  retardations ;  the  whole  being  traversed  by  a 
colourless  cross,  whose  branches  are  parallel  and  at  right 
angles  to  the  plane  of  polarisation.  At  the  same  time,  the 
extraordinary  image  presents  the  complementary  appear- 
ances —  a  black  centre,  a  black  cross,  and  complementary  col- 
ours. When  the  analyser  is  turned  round  through  90°,  so  that 
the  ordinary  image  becomes  an  extraordinary  one,  it  reverses 
its  appearance. 

This  cross  is  really  a  coincidence  of  two  crosses,  one  parallel  and  at 
right  angles  to  the  primitive  plane  of  polarisation,  and  the  other  parallel 
and  at  right  angles  to  the  principal  section  of  the  analyser. 

When  a  lamina  is  interposed  whose  axis  is  not  at  right  angles  to  its 
surface,  the  coloured  (or  isochromatic)  lines  are  modified  into  hyperbolic 
curves,  or  even  into  lines  nearly  straight. 

When  the  lamina  used  has  been  cut  from  a  binaxial  crystal,  the  isochro- 
matic lines  are  converted  into  a  series  of  curves  known  as  lemniscates,  and 
the  dark  or  colourless  crosses  are  represented  by  a  pair  of  hyperbolic  curves. 

The  doubly-refracting  power  of  a  body  may  be  de- 
tected when  it  is  placed  between  crossed  prisms,  and  by  this 
means  it  is  found  that  substances  which  are  ordinarily  iso- 
tropic  become  doubly  refracting  when  they  are  exposed  to 
compression,  or  to  dilatation,  or  flexure,  or  torsion,  or  vibra- 
tion (especially  at  the  nodes),  or  to  molecular  stress,  as  where 
they  are  heated  and  then  suddenly  cooled,  or  to  electrical 
stress ;  and  crystals  ordinarily  isotropic  become  double-refract- 
ing when  exposed  to  mechanical  stress,  or  when  they  crystallise 
irregularly  or  are  not  homogeneous.  Organic  tissues  are  by 
this  means  for  the  most  part  found  to  be  double-refracting,  and 
they  seem,  when  placed  between  crossed  prisms,  to  shine  by 
their  own  light  again.st  a  dark  background  —  a  circumstance 
favourable  to  definition,  for  there  is  no  diffraction  of  light  round 
the  fibres,  but  practically  of  little  utility,  for  it  is  difficult  to 
get  prisms  of  Iceland  spar  sufficiently  clear  to  be  interposed  in 
the  path  of  the  rays  coining  from  a  high-power  objective. 

It  has  been  proposed  to  make  use  of  a  dynamometer  which  measures 
forces  by  the  compressions  exerted  on  glass  which  is  interposed  between 
crossed  prisms,  these  compressions  being  estimated  by  the  colours  produced : 


566  OF  ETHER-WAVES.  [CHAP. 

the  greater  the  compression,  the  greater  the  difference  of  phase  set  up 
between  the  ordinary  and  the  extraordinary  rays,  and  the  greater  the  wave- 
length corresponding  to  that  colour  which  is  cut  out  of  the  emergent  light. 

It  has  also  been  found  that  slices  of  different  minerals  placed  between 
crossed  prisms  act  in  very  characteristic  manners,  and  are  thus,  in  many 
cases,  easily  identified. 

Andrews  proposed  as  a  test  for  sodium  to  make  sodium-platinum  chlo- 
ride, which  produces,  when  placed  between  crossed  prisms,  colours  so  vivid 
and  characteristic  that  the  millionth  part  of  a  grain  of  sodium  can  be 
detected  by  this  means. 

ROTATORY  POLARISATION. 

When  natural  white-light  is  passed  through  a  polariser,  then  through  a 
film  of  mica  or  selenite  cut  parallel  to  the  axis,  and  lastly,  through  an 
analysing  prism  of  Iceland  spar,  it  gives,  as  we  have  seen,  two  colourless 
images  of  the  source  of  light.  If  now  we  replace  the  mica  or  selenite  by  a 
slice  of  quartz  cut  parallel  to  the  axis,  the  two  images  produced  are  com- 
plementarily  coloured. 

If  their  light  be  examined  with  a  prism,  it  is  found  that  the  spectrum 
of  the  light  of  the  extraordinary  image  is  lacking  in  a  particular  region, 
which  presents  a  dark  band,  while  that  particular  region  is  bright  in  the 
spectrum  of  the  ordinary  image.  Further,  as  the  analyser  is  turned  round, 
the  dark  band  in  the  spectrum  of  the  extraordinary  ray  seems  to  travel  up 
or  down  the  spectrum ;  and  if  the  piece  of  qiiavtz  used  be  very  thin,  this 
dark  band  may  traverse  the  whole  spectrum  while  the  analyser  is  rotated 
through  an  angle  of  less  than  180°.  That  particular  kind  of  light  which  is 
absent  in  the  extraordinary  ray  leaves  the  quartz  plate  in  a  condition  of 
polarisation  in  a  plane  parallel  to  the  principal  section  of  the  analyser. 

Each  position  of  the  analyser  cuts  off  a  distinct  kind  of  light  in  the 
extraordinary  ray  :  hence  light  of  each  colour  must  have  become  polarised 
in  a  special  plane,  and  the  plane  of  polarisation  of  the  light  incident  upon 
the  quartz  has  been  rotated,  that  of  each  component  colour  to  a  specific 
extent. 

Rotation  is  easier  to  detect  with  polarised  than  with  common  light; 
but  common  light  is  similarly  rotated,  as  interference-experiments  may  be 
made  to  show. 

Biot  found  that  a,  the  amount  of  angular  rotation  of  the  plane  of  polarisa- 
tion of  each  colour,  was,  very  roughly,  proportional  to  the  square  of  its  wave- 
frequency,  or  inversely  proportional  to  the  square  of  A.,  the  wave-length.  Boltz- 
mann  showed  that  the  true  law  is  that  a  =(A -r- A2)  +  (B -4- A4)  :  in  quartz, 
for  example  (Stefan),  a  =  [(7-07018/106)  f  A2]  +  [(0-14983/1012)  -*-  A4], 
where  A.  is  the  wave-length  in  mm.,  and  a  the  rotation  produced  by  a  slice 
1  mm.  thick,  this  rotation  being  measured  in  degrees  of  angle. 

We  have  seen  that  a  plane-polarised  beam  is  equivalent  to  two  equal 
and  opposite  circularly-polarised  beams ;  but  quartz  allows  a  right-handed 
circularly-polarised  beam  to  travel  faster  througn  it  than  a  left-handed  one  ; 
at  any  given  point  the  right-handed  component  is  therefore  not  so  advanced 
in  its  phase  as  its  left-handed  companion  :  this  is  equivalent  to  a  relative 
gain  of  phase  by  the  so-called  left-handed  component  (see  definition,  p.  515); 
this  causes  the  plane  of  the  plane-polarised  ray  gradually  to  turn  to  the  right, 
in  the  same  direction  as  the  hands  of  a  watch  when  the  ray  is  looked  at  from 


xv.]  ROTATORY  POLARISATION.  567 

behind,  from  polariser  towards  analyser.  Ward  thinks  the  periods  are 
altered,  not  the  velocities. 

A  piece  of  quartz  1  mm.  thick  thus  turns  the  plane  of  polarisation  of 
yellow  rays  about  22° ;  a  piece  about  16-36  mm.  thick  will  turn  it  through 
360°,  for  the  amount  of  rotation  is  proportional  to  the  thickness  of  the 
rotating  medium.  For  the  Fraunhofer  line  B  the  specific  rotatory 
power  of  quartz  (1  mm.  thick)  is  15°-55;  for  line  D,  21°-67 ;  for  line  IIr 
50°-98. 

A  substance  which  acts  in  the  same  sense  as  quartz  is  said  to  be 
dextro-rotatory  or  positive;  one  which,  causing  a  relatively- slow  pro- 
pagation of  right-handed  circularly-polarised  light,  rotates  the  plane  of 
polarisation  to  the  left,  is  laevo-rotatory  or  negative.  This  property  is  not 
confined  to  crystals.  The  following  list  comprises  a  few  examples  of  bodies 
of  each  kind :  — 

Dextro-rotatory.  — =  Some  samples  of  quartz  ;  cane  sugar,  grape  sugar, 
camphor  ;  many  essential  oils,  such  as  oil  of  orange,  oil  of  caraway ;  cincho- 
nine,  quinidine ;  castor-oil. 

Lsevo-rotatory.  —  Some  samples  of  quartz  ;  oil  of  anise,  oil  of  mint,  oil  of 
turpentine ;  quinine ;  sugar  of  fruits,  starch  ;  albumin. 

The  rotatory  powers  of  different  substances  are  compared  by  means  of 
two  constants,  (a)  The  real  specific  rotatory  power;  the  rotation, 
for  a  given  colour  or  Fraunhofer  line,  produced  by  a  layer  1  mm.  thick  of  the 
substance  itself.  The  symbol  aD  denotes  the  real  rotation  for  the  Fraunhofer 
line  D. 

(6)  The  apparent  specific  rotatory  power,  [a],  for  a  given  line 
or  colour  ([a]D,  that  for  the  line  D) ;  the  rotation  produced  by  a  substance 
in  a  state  of  dilution.  It  is  equal  to  a/dp,  where  a  is  the  observed  rotation 
(measured  in  degrees),  e  the  quantity  of  active  substance  per  gramme  of 
solution,  I  the  length  of  the  column  employed,  and  p  the  density  of  the 
solution. 

The  apparent  specific  rotatory  power  is  slightly  increased  by  rise  of 
temperature  and  modified  by  the  nature  and  proportion  of  the  diluent 
substance. 

With  these  variations,  for  cane  sugar  [a]D  is  about  67°;  for  milk-sugar 
—  a-lactose  80°,  /^-lactose  54°-5;  for  crystallised  grape-sugar  in  7-68%  solu- 
tion [a]D  =  52°-89,  in  82-6%  solution  [a]D  =  57°-8. 

A  column  20  cm.  in  length  of  solution  of  cane  sugar,  containing  in  each 
100  cubic  cm.  16-350  grms.  of  cane  sugar,  is  equivalent  in  rotatory  power 
to  a  plate  of  right-handed  quartz  1  mm.  thick.  This  fact,  coupled  with  the 
fortunate  circumstance  that  the  rotatory  dispersion  for  quartz  is  the 
same  as  that  for  cane  sugar  and  glucose,  enables  the  strength  of  solutions  of 
sugars  to  be  approximately  determined  by  means  of  a  Saccharimeter. 

Essential  oils  are  found  to  retain  their  rotatory  power  unimpaired  (due 
allowance  being  made  for  proportionate  dilution)  when  in  dilute  solution,  or 
even  when  in  the  state  of  vapour,  provided  that  they  undergo  no  chemical 
change.  When  substances  Isevo-  or  dextro-rotatory  are  mixed  with  each 
other  or  with  indifferent  substances,  and  if  there  be  no  chemical  change,  the 
rotatory  effect  of  the  whole  is  found  by  multiplying  the  rotatory  index  of 
each  substance  by  the  proportion  in  which  it  is  present,  and  finding  the  joint 
effect  of  the  components  of  the  mixture  by  a  process  of  simple  addition.  If 
a  rotatory  substance  assume  the  crystalline  form,  its  rotatory  action  is  very 
often  masked  by  double  refraction :  whence  solids,  such  as  camphor,  are 


568  OF  ETHEE-WAVES.  [CHAP. 

generally  best  examined  in  solution ;  exceptions  to  this  being  found  in  some 
cases,  such  as  those  of  benzile  and  chlorate  of  soda,  where  the  rotatory  power 
depends  upon  the  crystalline  structure,  and  in  which  the  crystals  are  gen- 
erally hemihedric,  or,  as  it  were,  distorted  towards  one  side. 

Rotatory  polarisation  is  thus  due  either  to  crystalline  arrangement  of 
molecules  or  to  the  structure  of  the  molecules  themselves ;  and  it  has  been 
shown  (van  't  Hoff)  that  bodies  gifted  with  the  molecular  power  of  rotation 
have,  in  their  chemical  graphic  formulae,  a  marked  want  of  symmetry.  It 
is  thought  that  wherever  there  is  such  asymmetry  of  the  molecule  there 
ought  to  be  rotatory  power,  but  that  this  is  masked  by  different  molecules 
possessing  opposite  asymmetries :  this  is  illustrated  by  dextro-rotatory  and 
Isevo-rotatory  tartaric  acid,  whose  crystals  possess  opposite  asymmetries,  but 
which,  when  mixed  in  solution,  form  non-rotatory  racemic  acid. 

We  are  now  in  a  position  to  understand  the  pieces  which  make  up  a 
Soleil's  saccharimeter.  1.  A  Nicol's  prism,  achromatised  by  a  prop- 
erly shaped  prism  of  glass  through  which  the  transmitted  extraordinary 
ray  passes  :  the  achromatic  Nicol,  thus  acting  as  a  polariser,  is  so  placed  that 
the  light  transmitted  by  it  is  polarised  in  a  vertical  plane. 

2.  A  double-quartz  plate,  or  Biquartz;    two   semicircular  plates  of 
quartz  joined  by  a  vertical  cement-line,  and  thus  forming  a  circular  disc 
of  uniform  thickness  :  the  two  halves  have  opposite  rotatory  power,  and  their 
thickness  is  so  adjusted  that  they  respectively  deviate  through  90°  in  oppo- 
site directions  the  plane  of  polarisation  of  incident  plane-polarised  greenish- 
yellow  light ;   they  therefore   both   deviate  greenish-yellow   light,  incident 
upon  them  and  polarised  in  a  vertical  plane,  into  the  same  horizontal  plane. 

3.  A  Liquid-holder;  a  tube  fitted  with  clear  glass  at  each  end,  in 
which  is  placed  a  layer  of  the  liquid  to  be  examined,  10  centimetres  in 
length,  such  being  the  distance  between  the  terminal  glass-plates. 

4.  A  Compensator.     This  is  in  its  effect  a  quartz  plate  of  variable 
thickness.     It  consists  of  two  pieces  of  quartz  of  a  wedge  shape.     One  of 

these  can  be  made  to  slip  over  the  other ;  the  central 
thickness  is  thus  variable  at  will.     The  amount  of 

P """"%?'%•-  ^•^Wljjijj^     movement  can  be  measured  by  means  of  a  vernier 
connected  with  the  one  wedge,  and  a  scale  connected 
with  the  other.     When  the  zero  of  the  vernier  coin- 
cides with  the  zero  of  the  scale,  the  thickness  of  the  compensator  is  such  that 
it  exactly  neutralises  the  rotatory  effect  of  one  of  the  halves  of  the  biquartz, 
while  it  doubles  that  of  the  other,  the  effect  being,  in  both  cases,  to  bring 
the  light  back  to  the  original  vertical  plane  of  polarisation. 

5.  An  Analyser:  this  is  generally  a  Nicol's  prism. 

6.  A  Lens  to  be  focussed  on  the  biquartz. 

To  use  the  instrument:  —  Fill  the  liquid-holder  with  water,  and  put 
it  in  position ;  focus  the  lens  6  so  as  to  obtain  a  clear  image  of  the  biquartz  ; 
make  the  vernier  and  the  scale  of  the  compensator  to  coincide ;  turn  the 
analyser  round  until  there  is  observed  to  fill  the  field  a  particular  hue,  lying 
between  the  red  and  the  blue,  and  called  the  teinte  de  passage ;  this  hue  being 
chosen  out  of  the  many  which  will  successively  come  into  view,  on  the  ground 
that  as  the  instrument  is  constructed,  the  appearance  of  this  colour  denotes 
that  the  principal  section  of  the  analyser  is  parallel  to  the  plane  of  polarisa- 
tion of  the  yellow  ;  if  the  yellow  could  have  passed  through  the  analyser  it 
would  have  been  transmitted  as  an  ordinary  ray :  but  a  Nicol  prism  cuts  off 
the  ordinary  ray ;  the  yellow  is  therefore  cut  off ;  the  extraordinary  ray, 


xv.]  EOTATOKY  POLAEISATION.  569 

which  alone  passes  through  the  analyser,  is  thus  represented  in  colour  by 
white  daylight,  minus  its  bright  yellow  :  the  remainder  produces  in  the  eye 
the  effect  of  a  dim  lavender  gray,  which,  with  great  sensitiveness,  merges 
into  red  on  the  one  hand,  or  into  blue  on  the  other,  when  the  analyser  is 
slightly  rotated.  Both  halves  of  the  quartz  plate  appear  of  the  same  colour, 
because,  from  them  both,  the  yellow  light  issues  polarised  in  the  same 
horizontal  plane. 

If  now  the  water  in  the  liquid-holder  be  replaced  by  the  liquid  to  be 
tested,  and  if  that  liquid  have  rotatory  power,  the  two  halves  of  the  quartz 
will  cease  to  appear  of  the  same  colour  :  the  liquid  aids  the  rotatory  effect  of 
the  one,  and  is  opposed  to  that  of  the  other.  The  effective  thickness  of  the 
compensator  is  now  varied  until  the  rotatory  effect  of  the  liquid  is  neutral- 
ised :  the  vernier  shows  by  its  displacement,  how  much  the  thickness  has  been 
increased  or  diminished :  the  graduation  of  the  vernier  is  arbitrary,  but  a 
displacement  of  one  step  on  the  scale  amounts  generally  to  a  difference  of 
one-tenth  of  a  millimetre  in  the  thickness  of  the  quartz ;  and  as  the  vernier 
reads  to  tenths,  the  positive  or  negative  alteration  of  thickness  of  the  quartz, 
found  necessary  to  restore  the  uniform  coloration  of  the  field,  may  be  meas- 
ured to  the  hundredth  of  a  millimetre.  If  the  thickness  of  the  compensator 
have  to  be  diminished,  the  liquid  has  rotatory  power  similar  to  that  of  the 
quartz  used  in  making  the  compensator ;  if  it  have  to  be  increased,  its  action 
is  contrary  to  that  of  the  quartz.  It  is  necessary  to  know  of  what  kind  this 
quartz  is ;  this  being  known,  it  can  be  stated  that  100  mm.  of  the  liquid  are 
equal,  positively  or  negatively,  to  so  many  millimetres  of  dextro-rotatory  or 
Isevo-rotatory  quartz,  as  the  case  may  be ;  and  thus  the  rotatory  power  of 
the  liquid  can  be  specified  with  precision. 

Thus  a  layer  of  water  10  cm.  thick,  containing  diabetic  sugar  in  solution 
in  the  proportion  of  10  grammes  per  litre,  is  equivalent  to  a  thickness  of 
342  mm.  of  right-handed  quartz  :  the  thickness  of  a  dextro-rotatory  com- 
pensator of  quartz  would  have  to  be  diminished  by  an  amount  corresponding 
to  34-2  divisions  of  the  scale  on  the  interposition  of  a  solution  of  that  sub- 
stance of  the  given  thickness  and  the  given  strength;  while  if  the  solution 
were  weaker  or  stronger,  the  amount  of  change  of  thickness  of  the  quartz, 
as  shown  by  the  amount  of  displacement  of  the  vernier,  would  be  approxi- 
mately proportional  to  the  strength.  If  the  liquid  to  be  examined  be 
coloured,  a  Nicol  and  a  quartz  in  front  of  the  saccharimeter  will  frequently 
enable  this  to  be  corrected,  by  filling  the  field  of  view  with  the  complemen- 
tary colour. 

For  other  Saccharimeters  in  use,  see  Watt's  Dictionary  of  Chemistry, 
Suppt.  iii.  p.  1198. 

TRANSFORMATIONS  OF  THE  ENERGY  OF  ETHER-WAVES. 

We  have  already  seen  this  energy  transformed  into  molec- 
ular work  in  the  processes  of  photography,  and  it  is  now  merely 
necessary  to  remark  that  whatever  increases  the  absorption  of 
light  by  a  set  of  molecules,  increases  the  chemical  work  done  by 
the  incident  ether-waves ;  if,  for  example,  a  spectrum  be  cast 
upon  a  photographic  plate  prepared  with  collodion 'in  which 
chlorophyll  has  been  dissolved,  the  local  of  the  chlorophyll 


570  OF   ETHER- WAVES.  [CHAP. 

absorption-bands  comes  out  most  strongly  in  the  resultant  pho- 
tograph of  the  spectrum. 

The  impact  of  ether-waves  upon  some  substances  gives  their 
molecules  a  new  arrangement :  selenium  is  thus  so  acted  upon 
by  light  that  it  becomes  a  better  conductor  of  electricity  than  it 
is  in  the  dark ;  and  hard  rubber  is  superficially  acted  upon  by 
light,  so  that  when  the  incident  beam  is  intermittent  or  har- 
monically variable  in  intensity,  the  rubber  emits  a  sound  which 
reproduces  in  its  pitch  or  its  complexity  the  peculiarities  of  the 
incident  light. 

It  has  been  proposed  to  call  the  last-mentioned  property  of 
hard  rubber  the  sonorescence  of  that  substance. 

As  to  the  mechanical  or  molar  work,  the  pressure  exerted 
by  the  impinging  ether- waves,  though  small,  is  definite.  The 
energy  in  one  cub.  cm.  of  sunlight  at  the  earth's  surface  is  about 
6,oooooo  erg,  and  the  pressure  of  direct  sunlight  per  square  cm. 
is  about  67yiroo~oo  dyne,  or,  roughly  speaking,  about  the  weight 
of  4  Ibs.  on  each  square  mile  of  ground. 

This  would,  if  the  earth  were  a  rigid  obstacle,  press  upon  its  whole  sur- 
face with  an  aggregate  force  of  about  884  x  10n  dynes.  This  force,  acting 
upon  the  earth's  mass  (614  x  10-5  grammes)  would  produce  an  outward 
acceleration  of  1-44  x  10~14  cm.-per-sec.  per  second,  or  a  yearly  acceleration 
of  14-33  cm.-per-annum.  The  effect  due  to  this,  say  10,000  miles  in  15,000 
years,  would  be  small  in  comparison  with  the  uncertainties  which  have 
existed  as  regards  the  earth's  true  distance  from  the  sun  ;  but  it  would  pro- 
duce other  retardations  which  do  not  appear  to  have  occurred.  The  earth 
cannot,  however,  be  regarded  as  a  rigid  obstacle ;  it  is  to  a  great  extent 
pervious  to  the  ether  in  which  it  travels. 

The  mechanical  effect  of  ether-waves  is  rather  to  be  looked 
for  in  their  heating  effect  than  in  direct  pressure.  They  may 
heat  absorbent  gases,  such  as  ammonia,  and  cause  them  to  do 
mechanical  work,  or  to  produce  sound  if  the  incident  beam  be 
intermittent  or  harmonically  variable. 

OPTICAL  INSTRUMENTS. 

The  Eye,  considered  as  a  simple  lens,  brings  parallel  rays  incident  upon 
the  cornea  to  a  focus  upon  the  retina.  Hence,  when  it  is  at  rest,  as  when 
one  meditatively  contemplates  space,  it  is  adapted  for  vision  of  infinitely- 
distant  objects,  or,  as  the  phrase  goes,  it  is  accommodated  for  infinity.  To 
look  at  nearer  objects  requires  an  effort  for  each  — -an  effort  of  accommo- 
dation. This  is  effected  by  increasing  the  convexity  of  that  part  of  the  eye 
called  the  crystalline  lens,  which  is  normally  flattened. 

The  range  of  accommodation  provided  by  our  power  of  varying  the 
form  of  the  crystalline  lens  is  the  same  as  if  we  were  provided  with  a  set 
of  lenses  of  all  focal  lengths  between  infinity  and  about  ten  centimetres. 


xv.]  OPTICAL   INSTRUMENTS.  571 

The  front  of  the  eye  has  the  form  of  a  prolate  spheroid,  and  parallel 
rays  tend,  after  refraction  thereat,  to  converge  upon  one  point,  without 
spherical  aberration.  The  external  rays  are  cut  off  by  the  iris,  which  acts 
as  a  diaphragm  whose  aperture  can  be  automatically  adjusted  according  to 
the  brightness  of  the  incident  light,  and  thus  again  tends  to  counterbalance 
spherical  aberration.  The  crystalline  lens  of  the  eye  varies  in  density  from 
surface  to  centre,  and  thus  again  spherical  aberration  is  reduced.  By  these 
means  the  image  formed  on  the  retina  is  made  as  clear  as  usually  need  be : 
but  spherical  aberration  is  never  completely  absent. 

The  eye  presents  several  other  faults,  as  we  find  when  we  expose  it  to 
severe  tests.  Its  several  parts  are  not  truly  centred.  Its  surfaces  are  never 
truly  symmetrical  round  an  axis.  It  is  often  too  long  in  the  bulb,  so  that 
rays  are  brought  to  a  focus  before  arriving  at  the  retina,  and  produce,  instead 
of  clear  images  of  the  several  points  of  an  object,  a  number  of  overlapping 
diffusion-circles,  and  there  is  consequently  produced  a  blurred  image  of  the 
whole ;  this  condition  requires  the  use,  in  front  of  the  eye,  of  thick-edged 
lenses,  in  order  somewhat  to  diverge  the  incident  beam.  The  bulb  may, 
on  the  other  hand,  be  too  short,  so  that  converging  lenses  are  necessary. 
The  images  of  equally-distant  coloured  objects  can  never  appear  equally 
distinct,  for  they  are  not  in  focus  at  the  same  time.  The  front  of  the 
cornea  has  frequently  a  somewhat  cylindrical  form,  in  consequence  of  which 
horizontal  and  vertical  objects  do  not  come  to  the  same  focus.  The  field 
of  vision  is  extremely  limited,  and  the  most  sensitive  part  of  the  retina  is 
excentric.  Yet  for  all  this  we  are  for  the  most  part  insensible  to  these 
defects ;  we  have  the  power  of  adjusting  the  eye  with  extreme  rapidity  for 
all  the  parts  of  an  extended  object,  and  we  have  been  educated  by  experi- 
ence to  use  both  our  eyes,  and  thus,  by  blending  the  separate  pictures  pro- 
vided by  the  two  eyes,  to  form  judgments  as  to  the  solid  form  and  distance 
of  remote  objects,  —  a  power  which  we  discover  to  have  depended  greatly 
upon  binocular  vision  when  we  try,  shutting  one  eye,  suddenly  to  touch 
any  given  object  at  arm's  length,  though  it  can  be  cultivated  even  with  one 
eye ;  just  as  microscopists  who  have  long  used  a  monocular,  and  cultivated  a 
habit  of  keeping  the  fine  adjustment  in  action,  find  no  perspective  advan- 
tage in  the  use  of  a  binocular  microscope.  .  With  age  the  power  of  accom- 
modation wanes ;  for  near  objects  the  image  cannot  be  brought  to  a  focus 
on  the  retina ;  and  then,  in  order  clearly  to  see  near  objects,  the  aid  of  con- 
vergent lenses  must  be  sought. 

The  Microscope.  —  An  ordinary  thin-edged  lens  is  called  a  simple 
microscope  or  magnifying  glass.  The  simplest  compound  microscope  is 
formed  of  an  objective  —  a  combination  of  lenses  which  converges  the  rays, 
divergent  from  the  object,  into  an  inverted  real  image,  achromatic  and 
aplanatic  (i.e.  devoid  of  the  effects  of  spherical  aberration),  in  a  plane  in 
space  between  itself  and  the  eyepiece  —  and  of  a  convergent  eyepiece,  also 
compound,  and  corrected  for  spherical  and  chromatic  aberration,  which  mag- 
nifies this  inverted  real  image,  and  produces  an  inverted  virtual  image  at  an 
apparent  distance  from  the  eye,  not  less  than  that  of  the  nearest  distinct 
vision.  The  real  image  formed  by  the  objective  must  be  at  the  focus  of  the 
eyepiece ;  hence,  when  a  more  highly-magnifying  eyepiece  is  used,  in  order 
to  throw  the  real  image  up  to  the  focus  of  the  eyepiece  the  objective  must 
more  closely  approach  the  object  examined.  In  practice  the  eyepiece  con- 
sists essentially  of  two  parts;  that  nearer  the  object  is  the  field-glass, 
which  catches  the  rays  that  were  on  their  way  to  form  a  real  image  far  up 


572  OF    ETHER-WAVES.  [CHAP. 

the  tube,  and  brings  them  to  another  focus,  so  that  they  form  a  real  image 
at  a  point  which  coincides  with  the  focus  of  the  upper  part  of  the  eyepiece 
—  that  is,  at  the  place  where  the  diaphragm  within  the  eyepiece  is  situated. 
To  make  this  real  image  visible,  remove  the  upper  lens  of  the  eyepiece,  and 
drop  a  disc  of  oiled  tissue-paper  down  upon  the  diaphragm  within  the  eye- 
piece. The  field-glass  brings  the  whole  image  of  the  object  within  the  field 
of  view  of  the  eye-glass. 

The  rays  from  the  objective,  instead  of  being  received  by  an  eyepiece, 
may,  the  eyepiece  being  removed,  be  allowed  to  fall  upon  a  screen ;  they 
will  there  form  a  real  image  of  any  size,  which  may  be  traced  by  hand,  if 
the  illumination  be  sufficient ;  if  the  screen  be  a  sensitised  photographic 
plate,  a  photograph  may  be  produced.  In  this  case,  the  image  being  more 
remote,  the  object  must  be  nearer  the  lens  than  it  is  in  the  ordinary  use  of 
the  instrument. 

The  minute  virtual  image  of  surrounding  objects  produced  by  reflexion 
from  a  globule  of  mercury  is  one  of  the  most  trying  tests  for  an  ordinary 
microscope. 

In  the  astronomical  telescope  parallel  rays  from  a  distant  star  are 
made  to  converge  and  form  a  small  real  image;  this  is  examined  by  a 
simple  achromatic  eyepiece.  The  image  is  inverted  like  that  in  the  micro- 
scope. 

In  the  terrestrial  telescope  rays  nearly  parallel  are  made  to  converge 
and  form  a  small  inverted  real  image ;  this  small  image  is  magnified  and 
reinverted  by  an  arrangement  of  lenses  equivalent  to  a  compound  microscope. 

In  the  opera  glass  a  convergent  lens  directs  incident  rays  towards  an 
inverted  real  image,  but  before  this  is  formed  the  rays  meet  a  divergent  lens, 


Fig.196. 


which  causes  them,  instead  of  converging  towards  a  real  inverted  image, 
to  diverge  as  if  from  a  virtual  erect  image,  as  is  shown  in  Fig.  196.  This 
combination  of  lenses  —  Galileo's  doublet  —  is  one  of  the  simplest  and  most 
useful. 

In  the  ophthalmoscope,  as  used  for  the  observation  of  an  erect  image 
of  the  fundus  of  the  eye,  the  principle  of  Galileo's  doublet  is  sometimes  util- 
ised. In  the  first  place,  light  is  made  to  fall  upon  the  fundus  of  the  eye  by 
means  of  a  concave  mirror  held  in  the  hand  or  fixed  upon  the  forehead  of 
the  observer.  The  fundus  is  thus  illuminated,  and  becomes  a  source  of  light. 
Rays  from  it  pass  towards  the  eye  of  the  observer  through  a  central  aper- 
ture in  the  mirror,  placed  opposite  the  eye  of  the  observer.  These  rays  from 
the  fundus  are,  if  the  eye  observed  be  myopic  (too  long  in  the  bulb),  ren- 
dered convergent  by  the  media  of  the  observed  eye  itself,  and  a  thick-edged 
lens,  placed  near  the  eye  observed,  causes  them  to  enter  the  eye  of  the  observer 
as  if  they  had  proceeded  from  an  enlarged  erect  virtual  image.  The  con- 
vergent lens  of  Galileo's  doublet  is  thus  represented  by  the  observed  eye 
itself,  while  the  biconcave  lens  employed  makes  up  the  pair  of  lenses.  If 
the  eye  observed  be  normal,  and  accommodated  for  an  infinite  distance,  rays 


xv,J  OPTICAL   INSTRUMENTS.  573 

proceeding  from  any  point  of  its  retina  emerge  parallel,  and  a  second  lens  is 
not  absolutely  necessary  if  the  observing  eye  be  normal,  for  the  rays  come 
to  a  focus  on  the  observing  retina,  if  the  observing  eye  be  also  accommo- 
dated for  infinity ;  if  the  observed  eye  be  too  short  in  the  bulb  the  rays  are, 
on  emergence  from  it,  still  divergent,  and  in  this  case  a  convex  lens  is 
necessary. 

The  ophthalmoscope  may  also  be  used  in  such  a  way  as  to  give  an 
inverted  image,  not  so  much  magnified  as  in  the  preceding  case,  but  more 
extensive  in  its  field,  brighter,  and  more  easy  of  attainment.  A  beam  of 
light  reflected  from  the  mirror  converges  upon  and  passes  through  a  focus ; 
it  then  diverges  on  its  way  towards  the  eye,  but  encounters  a  thin-edged  lens 
which  causes  it  rapidly  to  converge  into  and  then  to  pass  through  a  focus 
within  the  eye,  and,  after  traversing  this  focus,  to  illuminate  a  wide  area  of 
the  fundus  of  the  eye.  Light  from  the  illuminated  f undus  is  collected  by 
the  biconvex  lens  before  mentioned,  which  forms  a  real  image;  the  rays 
from  this  image  pass  on,  through  the  aperture  in  the  mirror,  into  the  eye  of 
the  observer,  who  then  perceives  an  inverted  and  magnified  image  of  the 
fundus  of  the  eye,  —  an  image  which  may  be  still  further  enlarged  by  means 
of  a  second  convergent-lens  placed  behind  the  aperture  of  the  mirror. 

VISUAL  PERCEPTION. 

The  retina  is  not  a  uniform  surface,  but  is  made  up  of 
elements  whose  average  distance  from  one  another,  in  the 
yellow  spot,  is  about  -005  mm.  Distant  points,  whose  angu- 
lar distance  is  such  that  their  images  on  the  retina  are  less  than 
•005  mm.  from  one  another,  seem  to  blend  into  one,  and  thus 
two  stars,  whose  angular  distance  is  less  than  70",  appear  to  the 
eye  as  a  single  star. 

The  stimulation  of  nerves  is  associated  with  chemical  work 
in  the  nerve-ends  ;  and  this  with  absorption.  In  this  respect  it 
is  interesting  to  find  that  the  retina,  which  is  particularly  sensi- 
tive to  yellow  and  green  light,  absorbs  green  and  yellow  light, 
and  in  white  light  appears  purple.  It  has  been  pointed  out  that 
the  blindness  of  the  eye  to  heat-waves  and  actinic  waves  is  of 
advantage :  for  the  energy  of  heat-radiation  is  relatively  so 
great  that  everything  would  appear  intensely  bright,  and  our 
ordinary  vision  of  objects  would  be  impossible  if  the  rays  of 
dark  heat  were  visible ;  while  if  the  ultra-violet  rays  were  visi- 
ble, the  image  of  every  point  would  be  shrouded  in  a  haze  due 
to  chromatic  aberration. 

Prof.  S.  P.  Langley  estimates  that  the  amount  of  energy  which  is  neces- 
sary to  produce  vision  ranges  from  j^  erg  for  the  extreme  red  to  100.oooooo 
erg  for  the  green  of  the  spectrum. 

When  two  colours  affect  any  one  element  of  the  retina  at 
the  same  time,  the  resultant  sensation  is  that  of  a  single  colour, 


574  OF   ETHER-WAVES.  [CHAP. 

not  the  same  as  either  of  the  components.  Thus,  when  the  stim- 
ulations which  would  separately  give  rise  to  red  and  yellow 
are  superposed,  the  resultant  impression  is  one  of  orange  ;  red 
light  and  yellow  light  together  make  orange  light.  In  the  same 
way  yellow  and  green  make  yellowish-green  or  greenish-yellow ; 
and  in  general,  colours  near  one  another  in  the  spectrum  give 
rise,  when  compounded,  to  an  average  or  intermediate  sensation. 
But  in  the  same  way  green  and  purple-red  in  different  propor- 
tions, that  is,  of  different  intensities,  will  produce  all  hues  of 
purplish-red,  red,  orange,  yellow,  greenish-yellow,  and  yellowish- 
green  ;  green  and  indigo-blue,  all  hues  of  greenish-blue  and  blue ; 
indigo-blue  and  purple-red,  all  hues  of  violet-blue,  violet,  and 
purple,  up  to  the  purple-red  employed. 

Methods  of  Mixture  of  Colours.  —  1.  A  source  of  light :  a  prism  :  a 
screen  upon  which  a  spectrum  is  formed :  two  slits  in  the  screen,  so  placed 
as  to  admit  the  passage  of  two  selected  colours  of  the  spectrum :  achromatic 
lenses  behind  the  screen  converge  the  two  coloured  beams  towards  a  com- 
mon crossing  point :  a  screen  there  placed  indicates  the  mixed  colour. 

2.  A  V-shaped  slit  in  a  screen  (von  Helmholtz) ;  a  prism  behind  this :  the 
two  spectra  produced  overlap  each  other  and  produce  a  very  extensive  series 
of  combination-colours. 

3.  Maxwell's  discs:  e.g.,  a  disc  of  red-  and  one  of  green-painted  card- 
board :  each  disc  is  slit  down  to  the  centre,  and  cut  out  at  the  centre  so  as 
to  be  fitted  upon  a  rotating  top :  the  one  disc  being  slipped  through  the 
other,  the  relative  proportions  of  red  and  green  in  view  can  be  modified  at 
will :  the  whole  is  rotated  at  such  a  rate  that  the  successive  impressions  of 
red  and  green  enter  the  eye  at  least  from  twenty-five  to  fifty  times  per 
second :  each  local  impression  of  red  in  the  retina  is  still  vivid  while  that  of 
green  has  already  commenced,  and  vice  versa ;  the  colours  blend  in  the  eye, 
and  various   shades  of   orange-red,  orange,  yellow,  or  yellowish-green  are 
produced,  according  to  the  relative  proportion  of  the  colours  blended. 

4.  Parallel  rays  are  caused,  by  a  lens,  to  converge  upon  a  focal  point ; 
the  light  traversing  different  portions  of  the  lens  is,  by  the  interposition  of 
transparent  coloured   screens,  diversely  coloured;    all   comes   to   the  same 
focus ;  the  eye,  placed  axially  at  the  focus,  receives  mixed  rays ;  the  colours 
blend  in  the  eye  (Aitken). 

But  a  colour  on  one  side  of  the  green,  when  blended  with 
one  on  the  other  side  of  it,  produces  always  a  certain  amount  of 
white  light,  which  dilutes  the  resultant  colour :  and  if  we  try 
to  blend  yellow  with  blue,  we  obtain  nothing  but  white  light. 
Yellow  and  blue  are  said,  then,  to  be  complementary  to  one 
another  :  and  to  every  colour  there  is  some  complementary  col- 
our. The  complement  to  green  is  purple-red,  which  does  not 
happen  to  be  in  the  spectrum,  but  is  intermediate  between  the 
spectral  red  and  violet. 


xv.]  VISUAL   PERCEPTION.  575 

That  yellow  light  and  blue  light  make  white  light  is  contrary  to  the 
general  impression,  which  is  that  yellow  and  blue  make  green :  when  yellow 
and  blue  pigments  are  mixed,  the  yellow  and  the  blue  lights  reflected 
from  the  mixture  destroy  one  another,  forming  white  light ;  and  the  resid- 
ual green,  never  absent  even  from  the  purest  pigment-reflected  blue  or 
yellow  light,  is  perceived,  somewhat  wanting  in  brightness,  and  diluted  by 
the  white  light  produced  by  the  complementary  colours. 

The  phenomena  of  Double  Refraction  enable  us  to  produce  an  indefi- 
nite number  of  complementary  colours. 

Any  pure  colour  or  hue  may,  by  means  of  Maxwell's  discs,  be  diluted 
with  varying  proportions  of  white  or  of  black.  Pure  red  thus  treated  passes 
through  pink  tints  to  white,  or  through  brown  shades  to  black.  Every  one 
of  these  tints  or  shades  will  be  complementary  to  the  colour  (greenish-blue) 
which  was  complementary  to  the  original  pure  red ;  but  the  result  of  the 
mixture  is  not,  in  the  case  of  the  darkened  shades,  a  pure  white,  but  a 
lighter  or  a  darker  gray ;  and  in  all  cases  the  proportion  of  the  pure  red  to 
the  pure  greenish-blue  (to  keep  by  our  example)  remains  fixed  and  inde- 
pendent of  the  percentage  of  white  or  of  black  added  to  the  original  pure 
red.  Any  colour  in  nature  can  be  matched  by  finding  out  a  proper  angular 
proportion  (including  the  case  of  complete  omission  of  one  or  more)  for  a 
set  of  five  Maxwell's  discs,  viz.,  white,  black,  vermilion-red,  emerald-green, 
and  ultramarine-blue. 

From  this  point  of  view,  green  would  appear  to  affect  the 
eye  as  a  primary  colour,  the  others  being  a  purple-red  arid  an 
indigo-blue. 

The  primary  colours  are,  according  to  von  Helmholtz,  a  slightly  pur- 
plish red,  a  vegetation-green,  slightly  yellowish  (wave-length  about  5600 
tenth-metres),  and  an  ultramarine-blue  (about  4820).  Young's  original 
statement  was  red,  green,  and  violet ;  Clerk  Maxwell  concluded  they  were 
vermilion,  emerald-green,  and  ultramarine-blue ;  and  Fick,  red,  green,  and 
blue.  Ilering,  on  the  other  hand,  contends  that  there  are  four  primary 
colours;  Helmholtz's  purplish-red,  and  a  green,  complementary  to  it,  be- 
tween the  Fraunhofer  lines  b  and  F ;  a  yellow  (about  5750  tenth-metres), 
and  a  blue,  complementary  thereto  (about  4830). 

The  explanation  of  this  peculiar  blending  power  of  the  eye  has  been, 
that  every  element  of  the  eye  which  is  broad  enough  to  perceive  white  light 
consists  of  three  ultimate  elements,  each  of  which  is  capable  of  perceiving 
one  of  the  physiologically  primary  colours ;  and  that  relatively  varying 
degrees  of  stimulation  of  the  respective  nerve-ends  give  rise  to  blended 
sensations  of  intermediate  character.  Then  it  was  supposed  that  those 
who  were  "  colour-blind  "  had  lost  their  sensitiveness  to  one  (or  more)  of 
these  primary  colours.  But  it  now  appears,  as  Pole  and  others  have  shown, 
that  whenever  sensitiveness  to  green  is  lost,  that  to  red  is  lost  also ;  while 
that  to  yellow  and  that  to  blue  generally  remain,  but  may  also  be  lost 
together,  so  as  to  leave  no  colour-sense  at  all.  Hence  Hering's  view  seems 
preferable  to  the  trisensatioiial  theories. 

A  spectrum  formed  by  light  travelling  from  a  waning  source  is  found 
to  modify  its  tints  as  the  light  fades ;  the  orange-red  seems  to  become  more 
purely  red,  the  yellow-green  more  purely  green,  and  so  on ;  at  length  the 
faint  spectrum  is  approximately  restricted  to  red,  green,  and  violet  or  violet- 
blue  bands ;  of  each  group  of  nerve-ends,  one  is  feebly  stimulated  by  a  given 


576  OF   ETHER- WAVES.  [CHAP.  xv. 

colour,  the  others  are  inappreciably  so,  though  if  one  be  stimulated  the 
others  can  never  remain  wholly  unaffected.  On  the  other  hand,  if  a  col- 
oured light  be  rendered  exceedingly  bright,  the  other  nerve-ends  participate 
in  the  excitement:  very  bright  red  seems  somewhat  orange;  violet  very 
easily  passes  over  into  whiteness  when  its  brilliancy  is  excessive. 

A  black  colour  is  due  to  the  absence  of  stimulation  of  any  of  the  nerve- 
ends;  and  between  bright  white  and  black  there  is  a  gradation  of  weak 
whites  which  are  called  grays.  The  purest  black  (Chevreul's  black)  is 
obtained  on  looking  at  a  comparatively  small  hole  in  the  lid  of  a  deep  black- 
velvet-lined  box. 

Fatigue  of  the  retina  causes  it  to  become  insensible  to  a  colour  long 
looked  at:  when  white  light  is  then  looked  at,  it  appears  of  a  hue  comple- 
mentary to  that  colour,  the  sense  for  which  has  been  temporarily  exhausted. 

When  some  of  the  nerve-ends  of  the  retina  are  stimulated,  the  stimula- 
tion spreads  to  some  degree  :  a  very  narrow  white-hot  wire  appears,  especially 
from  a  little  distance,  to  be  much  wider  than  it  really  is ;  this  phenomenon 
being  named  Irradiation.  In  consequence  of  this,  the  crescent  moon  appears 
larger  than  that  part  of  the  moon  which  is  illuminated  by  light  reflected 
from  the  earth ;  a  candle-  or  gas-flame  appears  continuous,  though  its  incan- 
descent particles  are  by  no  means  in  contact  with  one  another ;  and  the 
glowing  filament  of  an  electric  incandescent  lamp  appears  much  thicker 
than  it  really  is. 

Perception  of  Form.  —  The  two  eyes  receive  images  of 
different  form ;  these  are  blended  by  a  mental  operation  into  a 
compound  image,  which  experience  has  taught  us  to  associate 
with  the  distance  of  the  several  parts  of  the  object.  This  is 
applied  in  the  Stereoscope ;  two  pictures  of  images  taken  from 
different  photographic  standpoints  are  formed,  one  in  each  eye, 
and  the  effect  is  that  of  outstanding  relief.  This  may  be  exag- 
gerated with  singular  effect  where  the  photographs  are  taken 
from  standpoints  situated  at  a  mutual  distance  of  several  feet : 
mountain  scenery  is  thus  brought  into  perspective.  The  same 
exaggerated  effect  may  be  observed  when  a  landscape  is  looked 
at  through  a  pair  of  telescopes,  parallel  but  at  several  inches' 
distance  from  one  another,  the  light  traversing  each  being 
brought  into  the  corresponding  eye  by  an  arrangement  of 
reflecting  prisms. 

The  images  in  the  two  eyes  may  often  differ  in  brightness : 
when  this  is  the  case,  there  is  a  struggle  between  the  two  fields 
of  view,  which  causes  the  impression  known  to  us  as  that  of 
Lustre ;  this  effect  being  specially  well  marked  in  the  case  of 
metals. 

One  of  the  most  curious  things  in  the  action  of  the  eyes  is  that  a  single 
mental  image  is  not  formed  in  binocular  vision  unless  the  images  be  formed 
on  Corresponding  points  of  the  two  retinae :  if  we  displace  one  eye  we  see 
two  images;  and  the  relative  positions  of  the  eyes  are  adjusted  by  a  system 
of  muscles,  so  as  to  secure  this  correspondence. 


CHAPTER  XVI. 

ELECTRICITY   AND   MAGNETISM. 

ELECTRICITY  and  MAGNETISM  are  not  in  themselves  forms  of 
Energy ;  neither  are  they  forms  of  Matter. 

They  may  perhaps  be  provisionally  defined  as  properties  or 
Conditions  of  Matter;  but  whether  this  Matter  be  the  ordi- 
nary matter,  or  whether  it  be,  on  the  other  hand,  that  all-pervad- 
ing Ether  by  which  ordinary  matter  is  everywhere  surrounded 
and  permeated,  is  a  question  which  has  been  Tinder  discussion, 
and  which  is  now  held  to  be  settled  in  favour  of  the  latter  view. 

At  first  sight  it  would  appear  that  the  electricity  of  an  elec- 
trified body  is  a  condition  of  that  body  itself.  When  a  small 
piece  of  resin  and  a  small  piece  of  glass  are  rubbed  together,  it 
is  found  that  after  they  are  pulled  asunder,  the  resin  and  the 
glass  attract  one  another  with  a  definite  and  measurable 
force ;  and  that  this  force  varies  (beyond  a  certain  small  dis- 
tance) inversely  as  the  square  of  the  distance  between 
them.  This  attraction  across  an  intervening  space  has  been  by 
some  held  to  be  due  to  a  so-called  Mutual  Action  at  a  Distance ; 
but  when  the  bodies  are  pulled  away  from  one  another,  work  is 
done  upon  them,  which  will  be  restored  when  they  are  allowed 
to  approach  one  another  ;  and  it  seems  probable  that  this  work 
has  been  done,  not  upon  two  isolated  bodies  mutually  acting  at 
a  distance,  but  upon  a  system,  which  consists  of  the  two  bodies 
together  with  the  Ether  between  them.  This  Ether  has  been 
stressed  by  their  separation ;  the  tendency  of  the  two  bodies  to 
approach  one  another  is  the  elastic  tendency  of  the  Ether  to 
recover  its  original  condition ;  and  these  phenomena  of  electric 
attraction  and  repulsion  may  be  explained  as  phenomena  of 
Ether-stress. 

Two  masses  of  resin  rubbed  on  glass  are  found -to  repel  one 
another ;  two  masses  of  glass  which  have  been  rubbed  with  resin 

2p  577 


578  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

also  repel  one  another ;  in  other  words,  two  masses  in  a  s  i  m  i  la  r 
electric  condition  generally  repel  one  another. 

According  to  the  nature,  the  size,  the  dryness,  of  the  pieces 
of  material  exposed  to  mutual  friction,  and  according  to  some 
other  circumstances,  it  is  found  that  after  friction  and  separa- 
tion the  force  of  mutual  repulsion  or  attraction  of  two  electrified 
bodies  varies.  One  body  may  thus  be  more  or  less  highly  elec- 
trified than  another ;  it  is  said  to  possess  or  be  charged  with  a 
greater  or  a  less  quantity  of  electricity. 

Two  bodies  are  said  to  be  equally  charged  or  to  be  charged 
with  equal  Quantities  of  Electricity  when  (being  of  the  same 
size  and  form)  they  can  precisely  replace  one  another  in  their 
action  upon  other  electrified  bodies. 

When  two  equally  electrified  bodies,  at  a  mutual  distance  of 
one  centimetre  in  air,*  repel  or  attract  one  another  with  a  force 
which  balances  one  dyne,  they  are  each  said  to  be  charged  with 
a  quantity  equal  to  one  C.G.S.  Electrostatic  Unit  of  Electricity. 
If  one  of  these  bodies,  thus  said  to  be  charged  with  a  unit  of 
electricity,  be  brought  to  an  exact  centimetre's  distance,  in  air, 
from  a  body  charged  with  an  unknown  quantity  of  electricity, 
the  force  between  the  two  electrified  bodies  may  be  measured 
directly;  and  if  it  be  equal  to  Q  dynes,  the  body  tested  is 
shown  to  bear  a  charge  of  Q  units  of  electricity.  Further,  if  a 
body  bearing  Q  units  be  brought  to  the  same  distance  from  a 
body  charged  with  Q'  units,  the  force  between  them,  in  air,  will 
be  equal  to  Q  x  Qf  =  QQ'  dynes. 

Quantity  of  Electricity  is  thus  treated  of  as  if  it  were  analo- 
gous to  Mass,  or  Quantity  of  Matter ;  but  only  as  a  means  of 
expression.  The  facts  observed  can,  to  a  great  extent,  be  stated 
and  systematized  by  means  of  the  device  of  attributing  the  phe- 
nomena to  the  existence  of  Electric  Matter,  which  may  be 
variously  distributed ;  but  it  will  soon  be  seen  that  this  is  purely 
a  device,  and  that  the  electric  matter,  with  whose  quantities  and 
actions  we  deal,  is  imaginary  merely. 

A  piece  of  glass,  after  being  rubbed  with  resin,  is  said  to 
bear  a  charge  of  vitreous  electricity ;  the  resin,  on  the  other 
hand,  is  said  to  be  charged  with  resinous  electricity.  If  any 
body  become  electrified  in  any  way,  it  must  become  either  vitre- 
ously  or  resinously  electrified. 

Similarly-electrified  bodies  repel   one   another;    dissimi- 

*  We  shall,  in  the  meantime,  assume  that  the  medium  surrounding  the  electrified 
bodies  is  air  in  all  cases. 


xvi.]  "ELECTRIC   MATTER."  579 

larly-electrified  bodies  attract  one  another;  these  statements 
being,  when  the  bodies  are  very  near  one  another,  subject  to  an 
exception  hereafter  to  appear  (p.  601). 

When  a  jet  of  water  issues  from  a  metallic  nozzle  connected  with  an 
electric  machine,  the  particles  of  the  issuing  stream,  being  similarly  electri- 
fied, repel  one  another,  and  the  jet  is  broken  up  into  spray.  When  the 
nozzle  has  a  capillary  orifice,  the  surface-tension  at  the  aperture  is  overcome 
by  the  electric  self-repulsion,  and  the  liquid  rapidly  issues  as  if  its  viscosity 
were  greatly  diminished. 

If  a  body  charged  with  resinous  electricity  and  one  equally 
charged  with  vitreous  be  brought  into  contact,  the  charges  of 
both  apparently  disappear  and  the  bodies  resume  a  neutral  state. 
Vitreous  and  resinous  electricities  are  thus  found  to  bear  to  one 
another  the  same  relation  as  positive  and  negative  quantities  in 
algebra,  and  by  a  purely  arbitrary  convention  charges  of  vitreous 
electricity  are  said  to  be  positive,  and  resinous  negative. 

The  above  statements  are  comprised  within  the  statement  that  if  F  be 
the  mechanical  force  of  repulsion  between  two  charges  of  electricity, 
these  charges  being  Q  units  in  one  body  and  Q'  units  in  another,  and  d  the 
distance  between  them,  F  =  k  •  QQ'/^2  '•>  and  if  our  units  of  quantity  be 
so  chosen  that  k  =  1  when  the  medium  between  the  charges  is  air,  then 
F  =  QQ'  /d'2.  If  Q  and  Q'  be  both  positive  or  both  negative,  the  product 
QQ'  is  positive,  and  the  stress  is  expansive  or  repulsive ;  while  if  one  of  the 
charges  be  resinous  and  the  other  vitreous,  QQ,'  is  negative,  and  the  stress  is 
such  that  the  bodies  appear  to  attract  one  another. 

When  an  electrified  body  presents  a  charge  of  Q  units  uni- 
formly distributed  over  a  superficial  area  of  A  sq.  cm.,  its  charge 
per  sq.  cm.  is  Q/A  =<r,  the  so-called  Superficial  Density  of  the 
Electric  Charge. 

If  the  distribution  be  not  uniform,  the  density  over  any  minute  area  may 
be  expressed  as  the  ratio  of  the  charge  borne  by  that  area  to  the  area  itself. 

The  superficial  density  of  a  given  charge  may  be  increased 
by  diminishing  the  free  surface  of  the  charged  conductor. 

A  piece  of  tinfoil  charged  and  connected  with  a  gold-leaf  electroscope 
(p.  604)  will  cause  a  divergence  in  the  leaves  of  that  electroscope,  which 
increases  when  the  tinfoil  is  partly  rolled  up.  When  charged  aqueous  par- 
ticles coalesce  to  form  raindrops,  their  free  surface  diminishes,  and  the 
density  of  their  charge  increases. 

The  superficial  density  of  a  charge  borne  by  a  conductor 
varies  from  point  to  point,  according  to  the  form,  but  inde- 
pendently of  the  material,  of  the  conductor.  Only  upon  a 
sphere,  in  free  space,  is  it  uniform. 


580  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

A  charge  borne  by  an  ellipsoid  assumes  at  the  points  a,  b,  c,  d  (Fig.  197) 
densities  proportional  to  Oa,  Ob',  Oc',  Od ;  bbr,  cc'  being  tangents  at  b,  c,  and 

Ob',  Oc'  at  right  angles  to  these.  The  density 
at  the  extremities  is  thus  greater  than  it  is  else- 
where. A  needle-point  resembles  the  extrem- 
ity of  a  very  elongated  ellipsoid,  and  the  den- 
sity of  a  charge  borne  by  a  needle  tends  to  be 
^  extremely  great  at  the  apex. 

The  density  varies  over  a  surface  of  any 
form  in  the  same  way  as  the  thickness  of  a 
hollow  shell  of  the  same  form  would  vary  if 
that  thickness  were  so  adjusted  as  to  produce, 
under  the  law  of  inverse  squares,  no  interior  effect. 

A  body  cannot  bear  an  indefinite  charge  of  electricity ;  if  the 
density  be  very  great  over  the  surface  or  at  any  part  of  the  sur- 
face of  a  conductor,  sparks  will  fly,  generally  from  the  point  at 
which  the  density  is  greatest,  either  to  surrounding  objects  or 
into  the  surrounding  air. 

If  two  bodies  equally  and  strongly  charged  with  opposite 
electricities  be  brought  sufficiently  near  one  another,  a  spark 
will  pass  between  them,  their  electricities  will  combine,  and 
they  will  be  discharged  and  return  to  the  neutral  state. 
Sparks  will  pass,  as  a  rule,  when  two  bodies  differing  greatly 
in  their  electrical  condition  are  brought  sufficiently  near  one 
another. 

The  thickness  of  air  across  which  a  spark  can  leap  is  known  as  the 
Striking  Distance  in  air;  and  it  depends  in  general  upon  the  nature  of 
the  substance  through  which  the  spark  passes,  and  the  nature,  condition, 
and  shape  of  the  surfaces,  as  well  as,  in  particular  cases,  upon  the  density 
of  the  charges  at  the  points  from  which  the  sparks  leap.  The  hotter  the 
charged  bodies,  the  greater  the  striking  distance. 

An  electric  spark  is  disruptive  in  its  effect,  and,  in  air,  it  tears  its  way 
from  one  dust-particle  to  another ;  it  heats  air ;  it  produces  sound  and  light ; 
in  water  it  may  jar  the  liquid  and  shatter  the  containing  vessel ;  it  can  pierce 
glass,  and  will  scatter  but  not  ignite  gunpowder. 

When  a  body  is  very  highly  charged,  the  air  in  its  imme- 
diate neighbourhood  becomes  similarly  charged,  is  repelled, 
and  masses  of  it  are  torn  off  and  repelled  in  constant  succession, 
an  electrical  wind  or  stream  of  electrified  air  being  produced. 

When  a  body  is  charged  with  electricity,  there  is  always  an 
equal  charge  of  the  opposite  kind  of  electricity  somewhere : 
every  distribution  of  electricity  has  a  corresponding  comple- 
mentary distribution  of  an  equal  amount  of  electricity  of  the 
Opposite  kind.  In  the  case  of  the  mutually-rubbed  pieces  of 
glass  and  resin  previously  adverted  to,  the  charges  borne  by  the 


• 


xvi.]  COMPLEMENTARY   DISTRIBUTIONS.  581 

two  masses  are  equal  and  opposite  :  when  a  single  object,  elec- 
trified by  friction,  stands  within  a  room,  the  walls  of  the  room 
are,  over  their  whole  inner  surface,  oppositely  electrified,  and 
bear  a  charge  numerically  equal  to  that  of  the  electrified  body. 
When  an  object  is  electrified  in  the  open  air,  the  earth  itself 
(together  with  the  heavenly  bodies)  takes  up  an  equal  and 
opposite  charge ;  and  thus  the  algebraical  sum  of  the  posi- 
tive and  negative  electricities  in  the  Universe  is  con- 
stantly equal  to  zero. 

This  doctrine  is  (Lippmann,  Silvanus  Thompson)  called  the  Law  of  the 
Conservation  of  Electricity :  whatever  charge  one  body  gains,  others  lose. 

To  feign,  as  we  have  already  done,  that  there  are,  in  cases 
of  electric  phenomena,  distributions  of  positive  or  negative 
imaginary  electric  matter,  is  convenient  for  many  purposes  of 
calculation.  For  example :  a  metallic  sphere  or  hollow  globe, 
when  electrified,  presents  no  electric  phenomena  within  its  sub- 
stance or  its  cavity ;  its  electrified  condition  is  manifest  only 
externally,  and  is  uniform  all  round  its  surface ;  whence  it  may 
be  imagined  that  a  uniform  film  of  electric  matter  covers  or 
coincides  with  the  surface  of  the  metallic  body,  and  this  imag- 
inary film  repels  or  attracts  equally  imaginary  films  of  matter 
distributed  over  the  surfaces  of  neighbouring  electrified  bodies, 
and  does  so  with  forces  whose  amount  may  be  calculated  in 
accordance  with  the  propositions  of  the  section  on  Attraction, 
p.  188. 

The  law  of  the  resultant  force  resembles  that  of  Gravitation. 
Every  particle  of  this  imaginary  electric  matter  in  the  Uni- 
verse repels  every  other,  existent  at  any  given  moment,  with 
a  Force  F  proportional  to  the  product  of  the  Quantities,  and 
varying  inversely  as  the  Square  of  the  Distance  between  them. 
[In  air,  F  =  QQ'/d2.] 

From  this  ensue* the  following  propositions:  —  (1.)  An  electrified 
metallic  sphere  acts  upon  all  external  electrified  particles  as  if  its  charge 
were  concentrated  at  its  centre;  and  therefore  its  repulsion  of  (or  its 
attraction  for)  a  Unit  of  Electricity,  external  to  the  sphere  and  situ- 
ated at  a  point  at  a  distance  d  from  the  centre  —  i.e.,  the  resultant 
"  Electric  Force  "  there  —  is,  in  air,  Q/ d2  =  <|>,  where  Q  is  the  charge  on 
the  sphere. 

(2.)  The  density  of  the  surface-distribution  on  the  sphere  =  charge/area 
=Q/4irris=<r;  and  at  a  point  immediately,  but  distinctly,  outside  the  sphere, 
so  near  to  it  that  rf,  the  distance  from  its  centre,  can  be  taken  as  equal  to  r, 
the  local  Electric  Force,  <|>,  =  Q/</2  =  4irrV/d2  =  ±TT<T.  This  result  is  inde- 
pendent of  the  shape  of  the  conductor,  provided  that  the  distribution  of  its 
charge  be  such  (see  Fig.  197)  as  to  produce  no  interior  effect. 


582  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

(3.)  A  superficial  distribution  of  electricity  presents  from  point  to  point 
such  variations  of  density  that  it  has  no  action  upon  particles  within  it. 
A  metallic  chamber  of  any  form  may  be  electrified  until  sparks  fly  from  its 
outer  surface ;  yet  no  electrical  effect  will  be  perceived  internally.  This  is 
the  strongest  proof  of  the  law  of  the  inverse  square  that  can  be  imagined : 
no  other  law  of  repulsion  or  attraction  could  result  in  a  force  imperceptible 
internally,  the  distributions  being  such  as  those  actually  observed. 

(4.)  Every  element  of  the  imaginary  superficial  shell  or  film  itself  is 
subject  to  repulsion  from  its  fellow-elements.  This  repulsion  of  the  film 
itself  amounts,  per  unit  quantity,  not  to  <|>,  but  to  i<|>*;  hence  it  is,  per  sq. 
cm.,  $$  •  a-  =  27TO-2  dynes.  A  soap-bubble  when  electrified  expands ;  the 
atmospheric  pressure  is  resisted  by  a  self-repulsion  or  so-called  Electric 
Tension  over  the  surface,  whose  outward  resultant,  f  =  pQ,  is  equal  to 
!<J>.<r  —  27TO-2,  all  in  dynes  per  sq.  cm.  A  soap-bubble  may  be  electrified 
by  blowing  it  on  a  metallic  pipe,  and  connecting  the  pipe  with  an  electric 
machine. 

From  this  we  must  conclude  that  the  surface  of  every  electrified  body 
is  in  a  state  of  expansive  tension,  whose  resultant  is  at  right  angles  to 
the  surface,  and  that  the  film  of  air  in  contact  with  it  is  subject  to  a  disrup- 
tive tendency,  which  varies  as  the  square  of  the  superficial  density.  Sparks 
fly  from  the  surface  of  a  charged  conductor  into  the  surrounding  air,  when 
pQ=  *><!>•  <r  =  66,708  dynes  per  sq.  cm. 

Since  the  Self -repulsion  or  outward  Electric  Tension  pn  =  %  <|>  •  a-  per  sq.  cm., 
and  the  Electric  Force  <|>  =  47TO-,  it  follows  that  ;;0  =  <|>'2/ STT  ;  and  this  is  equal 
to  the  Mechanical  Force  /•=  t,  in  dynes  per  sq.  cm.,  across  the  field,  in  the 
direction  of  the  lines  of  force.  That  is  to  say,  the  space  surrounding  the 
conductor  is  under  tension,  and  pulls  upon  the  conductor  with  a  force  or 
traction  t  =f=p0  —  ^<j>-  a  =  27ro-2  =  <J>'2/S7r,  all  in  dynes  per  sq.  cm.  of  the 
surface  of  the  conductor. 

Let  it  be  borne  in  mind,  however,  that  what  is  actually 
observed  in  electrostatic  phenomena  is  the  measurable  repul- 
sion or  attraction,  which  is  found  to  vary  as  if  it  were  due  to 
positive  or  negative  charges  under  the  law  of  inverse  squares. 
The  actual  movements  or  tendencies  to  movement  are  equally 
explainable  under  the  assumption  that  they  are  due  to  Ether- 
stress,  which  implies  that  there  shall  be  two  localities  between 
which  the  Ether  may  be  stressed.  Accordingly,  if  an  electrified 
body  be  "  insulated  "  by  being  placed  on  a  dry  glass  stand  within 
a  room,  the  walls  of  the  room  are  oppositely  electrified,  and  bear 
a  complementary  charge,  numerically  equal  in  the  aggregate  to 
the  charge  of  the  insulated  body.  The-space  comprised  between 
the  electrified  body  anxLihe  oppositely-electrified  walls  of  the 
room  is  a  Field  of  Force,  permeated  by  Lines  of  Force  and 
Equipotential  Surfaces.  The  lines  of  force  traversing  such  a 
field  quit  the  free  surface  of  the  insulated  body  at  right  angles, 
and  strike  the  walls  of  the  room,  again  at  right  angles.  They 

*  For  a  gravitational  analogy,  see  problem  10,  p.  189. 


xvi.]  FIELD   OF   ELECTRIC   FORCE.  583 

are,  in  general,  of  a  curved  form.  A  certain  number  of  lines  of 
force  may  be  grouped  within  a  bundle  or  Tube  of  Force,  whose 
cross-sectional  area  increases  as  the  lines  of  force  diverge  from 
one  another,  or  diminishes  as  they  converge  ;  and  <|>,  the  resultant 
local  Electric  Force,  or  the  mechanical  force  on  a  unit  of  elec- 
tric quantity  placed  at  any  point  within  any  such  tube,  must 
vary  inversely  as  the  local  cross-sectional  area  of  the  tube.  The 
Intensity  of  the  Field,  or  the  Number  of  Lines  of  Electric 
Force  per  sq.  cm.  of  cross-sectional  area  of  any  Tube  of  Force, 
at  any  point,  is  also  equal  to  the  local  value  of  <|>. 

The  force  <j>  upon  a  unit-charge  brought  within  distance  d  of  a  charge 
Q  would  be  <f>  =  Q  x'unity/rf2.  The  Intensity  of  the  Field,  or  the  local 
value  of  <j>,  is  therefore  Q/t/2.  This,  taken  numerically,  will  be  the  Num- 
ber of  Lines  of  Force  per  sq.  cm.  cross-section  of  a  Tube  of  Force  cut  across 
at  distance  d  cm.  from  the  charge  Q.  But  the  area  of  a  sphere  of  radius 
d  cm.  is  4ird'2  sq.  cm. :  hence  the  total  number  of  lines  of  force  cut  by  such 
a  sphere,  i.e.  the  Total  Number  of  Lines  of  Force  radiating  from  a  charge  Q, 
=  c|>  x  area  =  4?rQ. 

If  the  tubes  of  force  be  constant  in  cross-sectional  area,  the 
lines  of  force  are  parallel  to  one  another,  and  the  equipotential 
surfaces  are  equidistant  and  plane ;  the  field  is  then  a  Uniform 
Field  of  Force,  the  Ether  in  which  is  exposed  to  a  uniform 
intensity  of  stress. 

Such  a  field  we  find  in  the  central  part  of  the  space  between  two  parallel 
plates  insulated  from  one  another  and  brought  to  different  potentials. 

The  imaginary  electric  matter  or  imaginary  superficial  film 
appears  only  at  the  free  ends  of  the  lines  of  force ;  and  on  a 
conductor  its  imaginary  local  "  superficial  density "  is  always 
o-  —  <)>/47r  =  V//27r,  where/  is  the  actual  mechanical  force  or 
traction  t  per  sq.  cm.  in  the  field,  and  <j>  the  actual  force  upon 
a  unit  electric  quantity  put  in  the  field,  close  to  the  charged 
conductor. 

The  lines  of  force  tend  to  shorten  themselves  and,  if  they 
run  in  the  same  general  direction,  to  repel  one  another.  The 
Lines  of  Force  are,  it  will  be  borne  in  mind,  merely  convenient 
means  of  representing  to  ourselves  the  actual  forces  or  stresses 
within  the  field  of  electric  force. 

Dust  floating  in  the  air  between  two  charged  surfaces  finds  its  way  along 
the  lines  of  force  to  one  surface  or  the  other,  and  if  sticky,  it  agglomerates 
(Lodge).  "  Thunder  clears  the  air." 

The  conception  of  Potential  (Chapter  VII.)  is  'one  of  the 
highest  importance  in  the  theory  of  Electricity :  but  it  must  be 


584  ELECTKICITY  AND   MAGNETISM.  [CHAP. 

remembered  that  "Potential"  is  not  in  itself  a  physical 
state,  nor  is  it  an  explanation  of  electric  any  more  than  it  is 
of  gravitational  phenomena;  it  is  a  scientific  concept;  it  is  an  aid 
to  calculation,  and  it  enables  us  to  see  and  to  make  use  of  gravi- 
tational analogies ;  it  is  based  upon  the  law  of  the  repulsion  and 
attraction  of  the  imaginary  electric  matter,  which  law  is  itself 
merely  a  mode  of  statement  of  the  observed  forces  in  the  field 
of  force. 

The  Absolute  Electrical  Potential  at  a  point  is  a  mathe- 
matical expression,  possessing  a  numerical  value:  it  measures 
the  tendency  which  the  existing  electric  forces  would  have  to 
drive  an  electrified  particle  away  from  or  to  prevent  its  approach 
to  the  point  in  question,  if  such  a  particle,  bearing  a  charge 
equal  to  one  unit  in  quantity,  were  situated  at  that  point  or 
were  brought  up  to  that  point;  and  it  is  numerically  equal  to 
the  number  of  ergs  of  work  that  must  be  done  in  order 
to  bring  a  positive  unit  of  electricity  from  a  region  where 
there  is  absolutely  no  electric  force  —  e.g.,  from  a  region  at  an 
infinite  distance  from  all  electrified  bodies  —  by  any  path 
up  to  the  point  in  question;  provided  always  that  the  trans- 
fer of  the  positive  unit  of  electricity  be  supposed  to  have  no 
effect  whatsoever  upon  the  distribution  of  the  electricity  of 
other  bodies  in  the  neighbourhood  of  that  point. 

Difference  of  potential  between  two  points.  —  If  VQ 
ergs  of  work  must  be  done  in  order  to  remove  a  quantity  Q  of 
electricity  from  the  point  A  to  the  point  B  against  electric 
repulsion,  then  the  two  points  A  and  B  are  at  potentials 
which,  considered  absolutely,  may  be  unknown,  but  which  differ 
numerically  by  V:  and  B  is  said  to  be  at  a  higher  potential  than 
A,  by  V  units  of  potential. 

When  there  is  a  difference  of  potential  between  any  two 
points  in  space,  a  body  bearing  a  charge  of  positive  electricity, 
and  placed  at  the  point  at  which  the  potential  is  greater,  is 
driven  towards  the  point  of  less  potential,  just  as  in  the 
corresponding  gravitation-problem,  a  mass  tends  to  fall  towards 
a  lower  level;  and  if  free  to  move  it  will  follow  the  track  of  the 
lines  of  force,  travelling  thus  from  each  equipotential  surface  to 
the  next  one,  indefinitely  near  it,  by  the  shortest  path.  The  path 
between  the  two  points  is  not  necessarily  the  shortest,  for  the 
lines  of  force  are  often  curved  (see  Fig.  234). 

A  positively-charged  particle,  placed  in  a  region  of  posi- 
tive potential,  will  be  repelled  along  the  lines  of  force  into  a 


xvi.]  ELECTROSTATIC  POTENTIAL.  585 

region  of  less  or  of  zero  or  of  negative  potential :  a  negatively- 
charged  body  under  the  same  circumstances  travels  in  the  oppo- 
site direction.  In  other  words,  the  Lines  of  Force  correspond 
at  each  point  to  the  Direction  in  which  the  Potential  most 
rapidly  decreases. 

The  Mean  Electric  Force  <|>  acting  upon  a  Unit-Charge  of  electricity 
within  an  electrical  field  is  equal  to  the  difference  between  the  potentials  of 
two  points  within  that  field  and  situated  at  a  mutual  distance  of  one  centi- 
metre, that  distance  being  measured  along  the  lines  of  force :  for  if  Vy  and 
Vy/  be  the  potentials  of  two  points  whose  mutual  distance  is  d,  the  Work 
done  in  moving  a  unit  of  electricity  from  the  point  of  lower  to  the  point  of 
higher  potential  is  Vy  —  Vy/ ;  but  this  Work  done  is  also  equal  to  <|K/,  where 
4>  is  the  mean  force  resisting  the  transfer ;  whence  $d  =  V,  -  V/y,  and 
4>  =  (V,  -  V/y)  -  d.  When  d  =  l  cm.,  <|>  =  V,  -  V//;  and  when  (V,  -  V,,) 

=  l,4»  =  i/d. 

But  this  Electric  Force  on  Unit  Quantity,  <|>  =  (V,  -  V/y)  •*•  d,  is  the 
Potential-Slope  or  Potential-Gradient;  and  in  a  uniform  field  this  is 
uniform,  for  the  potential  diminishes  equably  throughout  the  field.  Hence 
in  a  uniform  field  the  Electric  Force,  which  is  47rcr  near  one  of  the  charged 
conductors,  remains  equal  throughout  the  field  up  to  the  opposed  conductor ; 
and  the  superficial  density  a'  of  the  opposed  charge  must  be  equal,  for 
<J>  =  47TO-  =  47TO-'.  In  a  non-uniform  field,  the  gradient  varies  as  the  equi- 
potential  surfaces  lie  nearer  or  farther  apart ;  and  as  Tubes  of  Force  widen 
out  or  become  narrower,  the  Electric  Force  <|>  —  i.e.  the  number  of  lines  of 
force  per  sq.  cm.  of  cross-area  of  the  tubes  of  force  —  is  inversely  propor- 
tional to  their  area ;  and  when  they  reach  the  opposed  conductor,  the  tubes 
of  force  which  have  left  an  area  A  engage  an  opposing  area  A' ;  hence 
4>'  =  47T0--A/A';  but  this  is  also  equal  to  47ro-';  whence  AV  =  A<r,  and 
the  charges  on  the  two  conductors  which  determine  a  field  of  force  must  be 
equal,  as  well  as  opposite. 

Since  under  the  law  of  inverse  squares  the  potential  due  to  repelling 
mass  Q  at  distance  d  is  Q/d,  and  at  distance  d'  is  Q/d',  the  difference  of 
potential  at  points  situated  at  distances  d  and  d'  respectively  from  the  repel- 
ling mass  Q  is  Q/d  —  Q/d'  =  Q(d'  —  d)/dd':  and  since  the  difference  of  poten- 
tials of  any  two  equipotential  surfaces  is  numerically  equal  to  the  Work 
done  in  transferring  a  unit-charge  from  the  surface  of  lower  to  the  surface 
of  higher  potential,  it  follows  that  the  Work  done  by  or  on  a  unit-charge, 
on  its  moving  or  being  moved  from  distance  d  to  distance  d'  from  a  charge 
Q,  is  Q(//'  —  d)/ddf,  positive  or  negative  as  the  case  may  be. 

If  a  body  charged  with  electricity  be  not  free  itself  to  move 
along  the  lines  of  force,  we  find  this  most  remarkable  phenome- 
non —  that  in  a  field  of  force,  the  points  of  which  correspond- 
ing to  the  extremities  of  the  body  are  at  different  potentials, 
the  electrical  condition  of  the  body  tends  to  travel :  one  aspect 
of  the  charged  body  —  the  aspect,  namely,  which  looks  towards 
that  direction  in  which  the  charged  body  would  itself  travel  if 
it  were  free  to  do  so  —  tends  to  become  more  strongly  charged 
or  to  acquire  a  greater  superficial  density  of  charge  ;  the  oppo- 


586  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

site  aspect  tends  to  become  less  strongly,  or,  it  may  be,  even 
oppositely  charged.  This  redistribution,  if  it  take  place,  has  the 
effect  of  equalising  the  potential  throughout  the  body 
placed  within  the  field  of  force  ;  and  it  reminds  us  of  the  read- 
justment of  level  and  accumulation  of  water  towards  the  lower 
end  of  a  tank  laid  on  a  sloping  surface  or  tilted  up  at  one  end, 
during  which  readjustment  a  difference  of  level  produces  a  flow 
of  water.  And  further,  it  does  not  matter  whether  a  body  in 
the  field  were  originally  charged  or  not :  if  an  electric  field  be 
set  up  round  an  uncharged  body,  that  body  tends  to  become 
positively  charged  in  the  region  of  lower  potential,  and  nega- 
tively charged  in  that  of  higher  potential.  The  new  distribu- 
tion, once  assumed,  is  permanent  so  long  as  the  field  of  force 
which  immediately  surrounds  the  body,  and  which  tends  to  de- 
termine a  difference  of  potential  between  its  opposite  aspects, 
remains  unchanged ;  but  while  that  distribution  is  being  assumed 
we  have  a  brief  Current  of  electricity. 

A  Difference  of  Potential,  in  whatever  way  it  may  be 
set  up  within  a  body,  produces  a  tendency  to  prompt  equalisa- 
tion of  potential  throughout  that  body,  and  thus  to  the  establish- 
ment of  a  momentary  Current  of  Electricity;  a  permanent 
difference  of  potential,  in  whatever  way  kept  up,  tends  to  pro- 
duce a  continuous  current. 

A  lightning  conductor  is  the  seat  of  a  continuous  current  so  long  as  the 
earth  at  its  base  and  the  air  at  its  apex  are  at  different  potentials. 

If  we  employ  the  analogy  of  air  drawn  up  by  a  fan  through 
a  mine,  down  one  shaft  and  up  the  other,  while  air  flows  by  a 
return  path  from  the  upcast  to  the  downcast  shaft  to  equalise 
any  differences  of  pressure  set  up  in  the  atmosphere,  we  may 
see  that  all  displacements  and  currents  of  electricity  must  take 
place  in  closed  circuits.  For  the  most  part  we  attend  only  to 
potential-differences  and  flows  of  electricity  between  two  points 
or  places.  But  it  is  important  to  remember  that  every  current 
in  a  conductor  is  accompanied  by  a  simultaneous  Displacement- 
Current  in  the  surrounding  Ether.  This  displacement-current, 
along  with  the  Conduction-Current  along  the  conductor,  makes  a 
closed  circuit ;  and  it  comes  to  an  end  when  the  Ether  has  been 
put  into  a  condition  of  Electric  Stress  (or  want  of  stress),  such 
as  shall  correspond  to  the  final  difference  of  potential  between 
the  ends  of  the  conductor.  When  this  has  been  done,  a  condi- 
tion of  equilibrium  between  the  conductor  and  the  surrounding 
medium  is  reached,  and  Energy  is  stored  in  the  medium,  up 


xvi.]  DIFFERENCE   OF   POTENTIAL.  587 

to  a  certain  definite  amount.  When  the  conductor  returns  to  a 
neutral  state,  this  energy  is  restored  by  the  Ether,  which  regains 
its  normal  condition. 

Even  within  a  conductor  there  may  be  this  storing  up  of  energy,  due 
to  the  imperfect  conductor  itself  acting  more  or  less  as  a  dielectric.  The 
true  current  will  accordingly  consist  of  the  Conduction-Current  plus  this 
Displacement-Current,  which  only  ceases  when  a  uniform  state,  whether  of 
quietude  or  of  steady  conduction,  has  been  attained. 

Difference  of  potential  is  analogous  to  difference  of  level 
or  Head  of  Water  in  hydraulics;  and  when  it  determines  a 
flow  of  electricity,  it  is  often  called  electromotive  force 
or  E.M.F.  — a  term  which,  in  this  sense,  might  with  advantage 
be  abandoned,  and  instead  of  which  we  shall  use  the  phrase 
electromotive  difference  of  potential  or  E.M.D.P.  In 
place  of  this  phrase  the  reader  who  has  any  reason  for  doing  so 
will  easily  read  the  words  Electromotive  Force.  The  objection 
to  employing  this  expression,  when  that  which  is  meant  to  be 
spoken  of  is  in  fact  a  potential-difference,  is  simply  that  a  dif- 
ference of  potential,  like  a  difference  of  water-level,  is  not  itself 
a  Force,  and  does  not  even  completely  specify  the  Electric 
Force  <|>  which  determines  an  electric  flow  :  to  determine  this  the 
form  and  the  dimensions  of  the  electric  conductor  must  be 
known,  as  well  as  the  difference  of  electric  potential  between 
its  extremities,  just  as  the  dimensions  of  a  water-pipe  must  be 
known,  as  well  as  the  available  head  of  water,  before  we  can 
calculate  the  local  falls  or  slopes  of  pressure  and  the  forces  pro- 
ducing flow. 

We  have  already  seen  that  <j>  =  (V,  —  Vy/)  -s-  d;  and  this  is  Clerk  Max- 
well's "  Electromotive  Intensity  "  or  true  Electromotive  Force,  the  Electric 
Force  <J>  acting  upon  a  unit-charge  of  electricity  at  any  point  referred  to. 
This  is,  in  other  words,  the  Potential-Slope  across  the  point  in  question,  in 
the  direction  of  the  Lines  of  Force. 

The  Cause  of  D.P.,  whatever  that  may  be,  is  also  called  E.M.F. : 
and  this  operates  in  a  direction  opposed  to  that  in  which  the  resulting 
E.M.D.P.  tends  to  act;  just  as  the  upward  pressure  of  a  force-pump  driving 
water  into  a  cistern  is  opposed  to  the  downward  pressure  and  flow  obtain- 
able from  the  cistern  when  filled. 

This  may  sometimes  operate  from  point  to  point,  so  as  to  heap  up  the 
resultant  D.P. :  and  when  we  sum  up  the  effects  of  all  the  local  E.M.F.'s,  we 
may  arrive  at  the  aggregate  resultant  D.P.  produced. 

Difference  of  Potential  is  often  spoken  of  by  electrical  engineers  as 
Electric  Pressure :  thus  we  hear  of  high-pressure  and  low-pressure  currents. 
It  is  also  known  as  Voltage,  when  measured  in  the  particular  units  known 
as  Volts. 

If  two  bodies  be  at  different  potentials,  when  they  are  con- 


588  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

nected  by  a  metallic  wire  the  charge  over  them  will  be  read- 
justed by  a  momentary  current  along  the  wire,  and  they  will 
come  to  the  same  potential. 

Two  bodies  are  said  to  be  at  the  same  potential  when  elec- 
tricity has  no  tendency  to  travel  from  one  to  the  other,  even 
though  they  be  brought  into  communication  by  a  metallic  wire. 

Difference  of  potential  is  thus  also  analogous  to  difference  of  tempera- 
ture, and  "  electromotive  force  "  <f>  to  temperature-gradient. 

The  earth  itself  is  arbitrarily  assumed  to  be  at  zero 
potential :  and  bodies  in  such  a  condition  that  when  they  are 
placed  in  contact  or  in  metallic  communication  with  the  earth 
their  electric  condition  is  unaltered,  have  a  potential  whose 
value  is  equal  to  this  arbitrary  zero. 

The  arbitrary  or  conventional  potential  —  or,  briefly,  The 
Potential  —  of  a  point  in  an  electric  field  of  force  is,  numeri- 
cally, the  number  of  ergs  of  work  necessary  to  bring  a  unit  of 
electric  charge  up  to  the  point  in  question  from  a  region  of 
nominal  zero-potential  —  e.g.,  from  the  surface  of  the  earth. 

Between  a  positively-charged  body  within  a  room  and  the 
negatively-charged  walls  of  the  room  there  must  lie,  in  the  inter- 
vening field  of  force,  one  equipotential  surface  which  has  a  Zero 
Potential,  its  potential  being  the  same  as  that  of  the  earth  out- 
side the  room.  Within  this  equipotential  closed  surface  there 
is  a  region  of  Positive  Potential ;  exterior  to  it  there  is  a  region 
of  Negative  Potential.  The  potential  of  the  inner  region  is 
greatest  at  the  surface  of  the  electrified  body ;  the  potential  in 
the  negative  region  is  most  negative  on  the  surface  of  the  walls. 

If  the  walls  of  the  room  be  at  the  same  potential  as  the  earth,  then  the 
whole  field  of  force  is  at  positive  potentials,  relatively  to  the  arbitrary  or 
conventional  zero  of  potential.  The  stress  in  the  field,  the  forces  and  poten- 
tial-difference across  it,  are  not  affected  by  this.  The  walls  and  the  earth 
have  still  been  negatively  charged ;  but  a  change  has  been  effected  in  the 
absolute  value  of  the  nominal  zero  of  potential. 

The  potential  cannot  be  a  maximum  or  a  minimum  at  any  point  within 
a  field  of  force,  if  that  point  be  not  upon  the  surface  of  one  of  the  con- 
ductors whose  surfaces  bound  the  field. 

Conductors  and  Non-conductors.  —  In  the  familiar  case  of 
a  lightning-conductor  we  see  a  marked  distinction  between  the 
conductive  copper  along  which  a  continuous  current  of  electricity 
can  flow,  and  the  air  or  an  unprotected  building  which  can  only 
be  traversed  by  a  disruptive  discharge.  A  conductor  is,  when 
a  charge  is  borne  by  it  and  retained  by  it  in  equilibrium,  a  sub- 
stance throughout  the  whole  volume  and  over  the  whole  surface 


xvi.]  CONDUCTORS   AND   NON-CONDUCTORS.  589 

of  which  the  potential  is  uniform ;  while  if  inequalities  of  poten- 
tial were  set  up  within  it,  the  conducting  material  of  a  perfect 
conductor  would  offer  no  resistance  to  the  readjustment  of  poten- 
tial by  means  of  a  current.  A  perfect  non-conductor  or 
dielectric  would,  on  the  other  hand,  be  a  substance  the  different 
parts  of  which  may,  after  an  electric  disturbance,  remain,  without 
any  process  of  readjustment  and  for  an  indefinite  period  of  time, 
at  potentials  differing  to  any  extent.  There  are  no  bodies  which 
are  absolute  non-conductors ;  all  conduct  electricity  more  or  less 
slowly.  There  are  no  bodies  which  are  perfect  conductors ;  all 
offer  more  or  less  resistance  to  the  flow  of  electricity.  Bodies 
which  conduct  extremely  badly  are  called  Non-conductors  or 
insulators  :  bodies  which  offer  comparatively  small  resistance  to 
the  passage  of  electricity  are  in  practice  called  Conductors. 

The  ether  in  an  insulator  can  stand  exposure  to  a  moderate  stress ;  that 
in  a  perfect  conductor,  for  some  reason,  cannot;  that  in  an  ordinary  con- 
ductor yields  continuously,  with  a  greater  or  less  quasi-plasticity. 

When  a  charged  body  is  placed  upon  an  insulator,  such  as 
ebonite,  guttapercha,  indiarubber,  dry  glass,  sealing-wax,  quartz, 
it  is  said  to  be  insulated  ;  its  potential  cannot  become  equal  to 
that  of  the  earth  for  a  long  period  of  time ;  it  is  said  to  retain 
its  charge  for  a  long  period. 

Air  at  a  high  pressure  is  almost  an  absolute  insulator :  cold  air,  damp 
or  dry,  at  the  ordinary  pressure  is  one  of  the  best  insulators :  but  even 
within  cold  air,  bodies  charged  with  electricity  gradually  lose  their  charge  ; 
a  partial  vacuum  is  a  good  conductor ;  a  good  vacuum  is  again  an  extremely 
good  insulator.  Ice  insulates,  water  is  a  bad  conductor;  obsidians  and 
lavas  insulate  when  hot,  and  steatite  even  when  red-hot ;  glass  when  diy  is 
an  insulator,  but  when  very  hot  is  a  conductor.  A  body  charged  and  sup- 
ported upon  a  dry-glass  stem  within  a  vacuum  or  a  very  dry  cold  atmos- 
phere will  retain  its  charge  for  a  very  long  period ;  but  if  the  air  be  damp, 
so  that  the  insulating  glass  stem  condenses  upon  its  surface  a  film  of  mois- 
ture from  the  air,  that  film  will  slowly  conduct  the  charge  to  earth.  The 
insulating  character  of  air  seems  (J.  J.  Thomson)  to  indicate  that  free 
molecules  of  air  cannot  be  charged,  but  that  there  must  be  free  atoms,  or 
that  dust  or  extremely  minute  water-drops  must  be  present,  before  this  can 
be  done.  A  metal  or  a  phosphorescent  substance,  negatively  charged  and 
exposed  to  ultra-violet  light,  readily  loses  its  charge ;  and  it  appears  that 
the  surface  of  the  conductor  is  then  to  some  extent  broken  up  into  dust  and 
repelled.  In  a  high  vacuum  (yrooToFo  atmo.),  silver  negatively  charged 
will  (Crookes)  evaporate  in  this  way  so  vehemently  as  to  assume  a  super- 
ficial glow,  so  intense  is  the  agitation  of  its  particles. 

A  sufficient  difference  of  potential  will  cause  a  spark  to  fly  between  two 
charged  conductors  across  the  intervening  dielectric :  in  the  case  of  turpen- 
tine, paraffin,  and  olive  oil,  the  striking  distance  is  when  the  discharge  is 
continuous  regularly,  when  the  discharge  is  interrupted  irregularly  proper- 


590  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

tional  to  the  difference  of  potential ;  in  air  the  striking  distance  increases 
faster  than  the  difference  of  potential,  and  the  curve  indicating  the  ratios 
of  striking  distances  to  differences  of  potential  is  a  parabola.  A  steeper 
potential-slope,  <j>,  is  required  for  thin  than  for  thick  layers  :  but  a  potential- 
difference  less  than  1  to  1|  electrostatic  unit  will  not,  in  air,  produce  a  spark 
at  all. 

Kinds  of  Conductors.  —  There  are  two  kinds  or  classes  of 
Conductors  :  first,  those  which  act  as  conductors  without  any 
apparent  displacement  of  their  own  substance,  such  as  metals 
and  other  substances  ordinarily  known  as  good  conductors ;  and 
second,  those  in  which  the  transfer  of  electricity  is  accompanied 
by  a  relative  displacement  of  particles  within  the  conductor. 
If  a  current  of  electricity  be  sent  through  a  solution  of  common 
salt  in  water,  it  will  be  found  that  those  particles  of  Na  and  Cl, 
into  which  the  salt  is  likewise  on  other  grounds  believed  to  break 
up  on  its  being  dissolved  in  water,  are  displaced  within  the  solu- 
tion, and  that  they  travel,  the  Na  towards  the  electrically  nega- 
tive and  the  Cl  towards  the  positive  extremity  of  the  solution, 
each  kind  of  particle  with  its  own  specific  velocity.  Each  such 
particle  or  sub-molecule  or  ion  parts  at  its  extremity  of  the 
solution  with  a  certain  definite  quantity  of  electricity,  positive 
or  negative,  and  no  more.  Conductors  of  this  kind  are  called 
Electrolytes;  and  they  consist  mainly  of  solutions  of  salts 
and  of  chemically-strong  acids  or  bases,  in  which  the  whole  or  a 
part  of  the  substance  dissolved  has  been  broken  up  into  these 
sub-molecules  or  ions,  but  in  which  the  water  plays  no  part  as  a 
conductor. 

The  phenomena  of  electricity  present  themselves  within  a 
conductor  only  while  a  current  is  actually  passing  through  it; 
for  then  only  are  there  any  differences  of  potential  within  the 
conductor.  And  even  this  can  only  occur  when  the  conductor 
is  an  imperfect  one ;  for  within  a  perfect  conductor  there  never 
could  be  any  difference  of  potential  set  up.  With  a  perfect  con- 
ductor, or  when  there  is  no  current,  but  a  more  or  less  perma- 
nent condition  of  Statical  Equilibrium  of  the  charge,  electrical 
phenomena  are  restricted  to  the  Field  of  Force' — that  is,  to 
the  non-conductor  or  Dielectric  external  to  the  conductor;  for 
within  non-conductors  alone,  not  in  conductors,  can  any  electrical 
stress  or  difference  of  potential,  permanently  or  for  any  length  of 
time,  be  maintained. 

If  the  air  had  been  as  good  a  conductor  as  copper  we  would  probably 
never  have  known  anything  about  electricity,  for  our  attention  would  never 
have  been  directed  to  any  electrical  phenomena. 


xvi.]  CONDUCTION.  591 

Phenomena  of  electricity  in  a  state  of  equilibrium,  associated 
with  more  or  less  permanent  differences  of  potential  and  evinced 
within  a  dielectric,  are  said  to  be  electrostatic ;  those  evinced 
during  adjustment  of  electric  potential  by  the  passage  of  a  cur- 
rent along  a  conductor  are  said  to  be  electrokinetic. 

If  electrostatic  phenomena  be  due  to  stresses  in  the  Ether, 
electrokinetic  are  due  to  movements  of  the  same ;  and  a  moment- 
ary current  of  electricity  in  a  copper  wire  is  a  throb  due  to  a 
readjustment  of  the  stresses  in  the  Ether  or  dielectric  surround- 
ing that  wire  ;  the  throb  is  accompanied  by  a  readjustment  of  the 
Lines  of  Force  in  the  field  surrounding  the  wire ;  these  lines,  as 
it  were,  slip  along  the  wire,  carrying  Energy  with  them  in  the 
Ether  or  other  dielectric,  not  in  the  wire  itself,  except  in  so  far 
as  the  imperfectly-conducting  character  of  the  wire  may  lead  to 
its  acting  to  some  extent  as  if  it  itself  were  a  dielectric  or  non- 
conductor. The  lines  of  force  in  that  case  traverse,  and  slip 
along  within,  the  substance  of  the  wire  itself. 

"Free"  and  ' '  Bound  "  Charges.  —  A  distinction  is  fre- 
quently made  between  afreeandabound  charge  of  electricity. 
The  former  is  understood  to  be  a  charge  borne  by  an  insulated 
body,  and  independent  of  surrounding  objects,  while  the  latter 
is  such  a  charge  as  is  held  in  position  by  the  presence  and  attrac- 
tion of  a  charge  of  the  opposite  character  upon  a  neighbouring 
body.  In  truth,  however,  all  charges  are  bound  charges ;  the 
complementary  distribution  must  be  somewhere;  the  field  of 
force  may  be  great  or  small,  but  it  must  have  its  limits.  It  may 
be  small,  as  when  a  little  electrified  body  is  suspended  within  a 
metal  flask  which  is  not  insulated;  it  may  be  great,  like  the 
field  of  force  between  a  thundercloud  and  the  earth :  in  the  for- 
mer case  the  complementary  charge  is  distributed  over  the  inner 
surface  of  the  flask ;  in  the  latter  it  travels  about,  and  is  at  its 
densest  upon  the  surface  of  the  earth  beneath  the  travelling 
thundercloud,  or  else  upon  adjacent  clouds.  Even  when  an 
electrified  body  is  placed  at  an  extremely  great  distance  from  all 
surrounding  objects,  it  cannot  be  held  to  have  a  free  charge,  for 
its  charge  is  bound  by  the  complementary  distribution  upon  the 
far-distant  objects;  and  a  particle  isolated  in  otherwise  vacu- 
ous infinite  space,  if  such  a  thing  were  possible,  could  not  become 
charged  with  electricity  at  all,  for  the  complementary  charge 
could,  in  such  a  case,  have  no  locus. 

If  the  Ether  be  stretched  or  compressed,  it  must  be  stretched 
or  compressed  between  at  least  two  points,  which  may  be  near 


592  ELECTKICITY  AND   MAGNETISM.  [CHAP. 

or  far  from  one  another.  Bearing  this  in  mind,  however,  it  is 
undoubtedly  convenient  in  many  respects  to  permit  ourselves 
the  use  of  such  expressions  as  "  a  body  freely  charged  with  Q 
units  of  +  electricity,"  and  in  so  doing  to  omit,  provisionally, 
all  consideration  of  the  complementary  charge,  which  is  sup- 
posed sufficiently  distant. 

This  mode  of  expression  is  also  in  accord  with  the  convention  that  the 
potential  of  the  earth  is  always  zero. 

Division  of  Charge.  —  When  a  conductor  charged  with 
electricity  is  brought  into  contact  or  into  metallic  communica- 
tion with  another  at  a  different  potential,  the  electric  potentials 
of  the  two  conductors  become  equalised ;  and  if  the  two  bodies 
be  of  the  same  form,  size,  temperature,  and  chemical  nature,  and 
if  they  be  symmetrically  arranged,  they  will,  after  separation, 
each  bear  a  charge  equal  to  one-half  of  the  algebraical  sum  of 
the  original  charges  of  the  two  bodies. 

This  change  of  distribution  involves  a  readjustment  of  the  lines  of  force 
and  of  the  equipoteiitial  surfaces  throughout  the  surrounding  dielectric,  and 
an  alteration  of  the  distribution  of  the  complementary  charge  over  the  oppo- 
site boundary  of  the  field  of  force. 

When  the  bodies  are  unequal  in  size,  etc.,  or  are  unsymmet- 
rically  arranged,  the  division  of  the  charge  between  them  is  not 
equal.  Two  similar  but  unequal  spheres  are  found,  after  being 
brought  into  communication  by  a  long  thin  wire,  which  is  then 
removed,  to  bear  charges  proportioned  to  their  radii. 

Electrostatic  Capacity  or  Permittance.  —  When  a  con- 
ductor has  a  charge  of  electricity  imparted  to  it,  the  potential  of 
its  surface  and  of  its  whole  volume  is  raised,  positively  or  nega- 
tively (i.e.,  lowered),  as  the  case  may  be.  When  a  body,  insu- 
lated in  air,  requires  a  charge  of  C  units  of  electricity  to  be 
imparted  to  it  in  order  to  raise  its  potential  by  one  unit  — 
that  is,  from  zero  to  unity,  or  from  V  to  V  -f-  1  —  it  is  said  to 
have  a  Capacity  or  Permittance  of  C  units.  When  this  body, 
so  insulated,  has  its  potential  raised  by  the  amount  V, 
the  Charge  of  Electricity  imparted  to  the  conductor  is  Q  =  VC 
units. 

When  a  series  of  conductors  —  whose  electrostatic,  capacities  are  Cy,  Cyy, 
C/y/,  etc.,  and  which  are  charged  to  potentials  V,,  Vy/,  V/yy,  etc.,  so  that  their 
several  charges  are  respectively  CyVy,  CyyVyy,  Cy//V//y,  etc.  —  are  connected  by 
a  wire,  the  potential  thereupon  assumed  by  the  whole  system  is  equal  to  — 
The  whole  charge  =YyCy  +  Yy/Cy/  +  V^C,,,  •  •  • 
The  whole  capacity        Cy  +       C/y  +        C/yy ... 


XVI.] 


CAPACITY. 


593 


The  electrostatic  capacity  of  a  conductor  is  the  same  whether 
it  be  solid  or  hollow  :  the  merest  film  of  gold  leaf  supported  on 
a  wooden  ball  has  as  great  a  capacity  as  a  solid  metallic  sphere. 

Electrostatic  stress  can  only  persist  within  the  field  of  force,  the  die- 
lectric, which  is  limited  by  the  surface  of  the  conductor  ;  beneath  this  surface 
it  is  a  matter  of  indifference  what  the  metallic  thickness  may  be,  since 
within  a  conductor  there  can  be  no  permanent  difference  of  potential,  no  per- 
manent electrostatic  stress.  The  quantity  C  is  not  really  a  capacity  of  the 
conductor  for  electricity  at  all,  but  is  a  measure  of  the  elastic  yielding  of 
the  Field  of  Force  involved. 

The  Electrostatic  Capacity  of  a  Sphere.  —  A  sphere 
of  radius  r,  within  an  unlimited  air-space  containing  no  other 
charged  bodies,  is  charged  with  quantity  Q  ;  this  quantity,  uni- 
formly distributed  over  the  surface,  acts  as  if  it  were  gathered 
at  the  centre,  and  therefore  at  a  distance  r  from  the  surface. 
The  potential  at  the  surface  of  the  sphere  must  therefore  be 
V=Q/r.  The  capacity  C  =  Q/V;  this  is  Q/Q7r=r;  the 
Electrostatic  Capacity  of  a  Sphere,  insulated  in  air,  is  therefore, 
in  C.G.S.  electrostatic  units,  numerically  equal  to  its  Radius. 

The  Work  spent  in  charging  any  body  is  equal  to  half  the 
product  of  Q,  the  Charge  imparted  to  it,  into  V,  the  rise  of 
Potential  produced  in  it. 

To  bring  a  quantity,  Q,  of  electricity  from  a  place  of  zero  potential  to  a 
place  of  constant  potential,  V,  involves  the  expenditure  of  QY  units  of 
work,  by  our  definition  of  Potential.  To  bring  a  charge  Q  by  successive 
instalments  into  a  region  whose  potential  is  at  first  zero,  but  steadily  rises  as 
the  successive  instalments 
arrive,  work  must  be  done 
which  is  equal  to  half 
the  product  of  the  final  Fig.198. 

rise  of  potential  into  the 
whole  charge  brought  up. 
In  Fig.  198  the  small 
rectangles  represent  the 
work  done  in  bringing 
each  successive  instalment 
up  to  the  corresponding 
potential;  these  rectangles  increase,  and  their  sum,  which  represents  the 
total  work  done  in  bringing  up  the  whole  charge  OQ  to  the  final  poten- 
tial QP,  is  represented  by  the  triangle  OQP,  whose  area  =  £OQ  x  QP  = 
£  whole  quantity  x  final  rise  of  potential 


Quan 


tity 


The  work  done  in  charging  a  body  is  equal  to  the  electrical 
energy  stored  up  in  the  field  of  force  surrounding  that  body  ; 
and  since  this  is  equal  to  JQV,  we  see  that  the  energy  of  an 
electrified  body  depends  not  only  upon  the  Quantity  of 


594  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

electricity  borne  by  that  body,  but  also  upon  the  Potential; 
just  as  the  potential  energy  of  a  mill-pond  depends  not  only 
upon  the  Quantity  of  water  contained  in  it,  but  also  upon  the 
average  elevation  of  that  water  above  surrounding  objects. 
For  which  reason  a  mere  Quantity  of  Electricity  is  not,  in 
itself,  a  quantity  of  Energy ;  and  therefore  Electricity,  as  meas- 
ured by  Quantity  of  Electricity,  is  not,  like  Heat,  itself  a  form 
of  Energy. 

The  energy  of  a  charged  conductor  of  any  kind  is  measured  by  £QV ; 
but  this  is  equal  (since  Q  =  C V,  where  C  is  the  electrostatic  capacity  of  the 
conductor)  to  |CV2  or  to  |Q2/C. 

The  energy  of  a  system  of  connected  conductors  is  equal  to  ^V2-  2C,  or 
to  ^Q2  -4-  SC,  where  3C  is  the  aggregate  capacity  of  the  whole  system. 

Suppose  now  that  two  conductors,  of  which  the  one  is  charged  to  poten- 
tial V  while  the  other  is  at  zero  potential,  and  of  which  the  respective 
capacities  are  C,  and  C7/,  are  placed  in  metallic  communication ;  on  contact 
they  form  a  joint  conductor  whose  capacity  is  (C,  +  C;/).  The  energy  of 
the  single  charged  conductor  was  ^Q2/C7 ;  that  of  both  taken  together  is 
JQ2/C/  +  Cy/,  a  smaller  quantity.  There  is  therefore  an  apparent  Loss  of 
Energy  equal  to  &QYC,  -  $QyC,  +  C,,}  -  {ttQY^XC,,/^  +  C,,)},  or 
(C/yC,  +  C/7)  times  the  energy  of  the  original  charge.  If  Cy  =  C/y,  half 
the  energy  of  the  original  charged  conductor  is  apparently  lost  by  partial 
discharge,  being  transformed  into  Heat. 

Wherever  there  is  a  readjustment  of  electricity  in  the  form 
of  a  running-down  of  electricity  from  a  place  of  high  potential 
to  a  place  of  low  potential,  there  is  a  loss  of  energy  of  electrifi- 
cation ;  just  as  when  a  full  pond  is  allowed  partly  to  discharge 
itself  into  an  empty  one,  the  average  level  of  the  whole  is 
lowered,  and  the  energy  of  position  partly  disappears,  to  reap- 
pear in  the  form  of  Heat.  In  general,  where  electrified  conduc- 
tors are  connected  by  metallic  wires,  if  there  be  a  current,  the 
potential  energy  of  the  system  sinks  to  a  minimum ;  heat  and  — 
if  a  spark  pass — light,  sound,  and  mechanical  effect  being  pro- 
duced. Where  the  components  of  an  electrified  and  insulated 
system  are  allowed  to  approach  or  to  recede  from  one  another 
in  obedience  to  the  electric  forces,  the  energy  of  electrification 
becomes  in  part  converted  into  mechanical  work,  and  therefore 
falls  in  amount;  while  if  they  be  pulled  asunder  or  made  to 
approach  against  the  electric  forces,  the  mechanical  work  done 
upon  the  insulated  system  from  without  is  converted  into  energy 
of  electrification.  In  the  former  case  the  energy  left  in  the  sys- 
tem is  that  of  the  same  charge  at  a  lower  potential;  in  the  latter 
case  it  becomes  that  of  the  same  charge  at  a  higher  potential. 

Electrostatic  Induction.  —  When   an  electrified  body  or 


XVI.] 


ELECTROSTATIC   INDUCTION. 


595 


system  is  placed  within  a  hollow  metallic  shell  (Fig.  199),  with 
which  there  is  no  communication  except  through  non-conduc- 
tors, the  shell  becomes  charged  by  'Indue- 
ti o  11 '  across  the  intervening  dielectric.  If  Fie-199- 
the  bodies  placed  within  the  shell  be  posi- 
tively charged,  the  inner  surface  of  the  shell 
becomes  negatively,  the  outer  positively 
electrified.  The  opposite  charge  thus 
induced  on  the  inner  surface  of  the  shell, 
the  similar  charge  induced  upon  its 
outer  surface,  and  the  original  inducing 
charge  on  the  internally-suspended  system, 
are  all  equal  in  amount,  if  the  shell  completely  or,  practically, 
even  if  it  very  largely  surround  the  electrified  body  suspended 
within  it.  Thus  the  positive  and  the  negative  charges  called  into 
existence  by  Induction  are  together  algebraically  equal  to  zero. 

The  Difference  of  Potential  between  an  inducing  sphere  and  an  induced 
spherical  surrounding  shell  is  Q/r  —  Q/V,  where  Q  is  the  charge  on  the 
inducing  sphere,  r  its  radius,  and  r'  the  radius  of  the  hollow  spherical  shell. 

The  distribution  of  the  induced  charge  on  the  interior  sur- 
face of  a  completely-surrounding  shell  is  such  that  on  external 
points  it  produces  an  effect  equal  and  opposite  to  that  of  the 
interior  insulated  charge ;  the  two  inner  charges  therefore  pro- 
duce together  no  effect  upon  external  bodies,  and  the  induced 
charge  on  the  outer  surface  is  the  only  charge  which  can  affect 
particles  situated  in  the  outer  air.  The  two  interior  charges  are 
bound  to  each  other,  for  they  are  of  opposite 
character,  and  there  is  a  field  of  force  between 
them ;  the  outer  charge  is  said  to  be  free. 

The  distribution  of  electricity  over  the  inner  and 
outer  surfaces  of  the  shell  is,  if  the  shell  be  spherical, 
governed  by  the  law  that  the  superficial  density  cr  at  any 
point  E  is  a-  =  (CE2  ~  CM2)  .  Q/47T-  CE-  ME3,  where  C 
is  the  geometrical  centre  of  the  shell,  and  M  the  point 
at  which  the  charge  Q  is  situated.  Then  the  inner 
charge  Q  at  M,  and  the  'opposite  charge  -  Q  on  the  surface  (the  inner 
surface  of  the  shell),  produce  together  no  effect  on  surrounding  particles. 
The  charge  +  Q  on  the  outer  surface  acts  as  if  it  were  all  at  M. 

The  lines  of  force  or  of  induction  radiating  from  M  are  not  displaced 
by  the  interposition  of  the  insulated  shell ;  neither  are  the  equipotential  sur- 
faces :  but  the  shell  divides  the  field  of  force  round  M  into  two  regions,  of 
which  either  can  be  destroyed  without  affecting  the  stress  or  the  potential- 
slope  in  the  other. 

At  page  583  we  saw  that  the  number  of  lines  of  force  from  a  charge  Q 


Fig.200. 


596  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

was  4  TrQ.  Similarly,  the  total  induction  in  the  inductive  field  is  said  to 
be  also  4?rQ,  or  Q/rf2  per  sq.  cm.;  and  every  4?r  Lines  of  Induction 
passing  out  of  a  dielectric  into  a  conductor  are  said  to  be  able  to  induce  one 
unit  of  charge  on  the  bounding  surface  of  that  conductor;  so  that  the 
Induction  per  sq.  cm.,  i,  =  4-7nr,  where  cr  is  the  superficial  density  of  the  in- 
duced charge.  Then,  since  the  Electric  Force  <}>  is  also  equal  to  4^0-,  it  is 
said  that  the  Lines  of  Force  and  the  Lines  of  Induction  coincide  in  air,  the 
standard  medium. 

If  the  outer  surface  of  the  shell  be  connected  with  the  sur- 
face of  the  earth,  the  shell  and  the  earth  become  one  extended 
conductor,  and  the  positive  charge  on  the  outer  surface  of  the 
shell  is  repelled  to  the  earth's  surface  ;  it  now  blends  with  and 
neutralises  the  negative  charge  previously  borne  by  the  earth 
and  surrounding  objects  in  consequence  of  the  original  positive 
electrification  of  the  inducing  body.  As  a  result  of  this  we 
have,  within  the  shell,  a  purely  local  field  of  force,  restricted 
to  the  space  between  the  internally-suspended  body  and  the 
interior  surface  of  the  shell,  and  giving  rise  to  no  phenomena 
outside  that  cavity. 

The  potential  of  the  whole  system  is  lowered  by  this  annihilation  of  the 
outer  region  of  the  original  field  of  force ;  but  the  potential-slope  within 
the  interior  field  remains  unchanged. 

If,  on  the  other  hand,  the  insulated  body  within  be  made  to 
touch  the  enveloping  shell,  the  internal  field  of  force  will  be 
destroyed ;  but  the  outer  induced  charge  will  remain,  distrib- 
uted over  the  outer  surface  of  the  shell. 

Any  quantity  of  electricity  may  thus  be  wholly  transferred  to  the  outer 
surface  of  a  hollow  insulated  conductor,  if  a  charged  body  be  made  to 
touch  its  internal  surface. 

A  sheet  of  tinfoil  charged,  and  separated  from  a  second  sheet 
by  an  intervening  layer  of  air  or  glass  or  mica  or  waxed  paper, 
will  act  inductively  across  the  dielectric.  The  nearer  surface 
of  the  second  sheet  is  oppositely,  the  farther  surface  similarly 
charged;  and  if  the  second  sheet  of  tinfoil  be  connected  to 
earth,  the  similar  charge  escapes  and  the  field  of  force  is  now 
almost  wholly  limited  to  the  thin  space  between  the  two  sheets 
or  plates. 

A  layer  of  dielectric  intervening  between  two  conducting 
surfaces  constitutes  an  Electrostatic  Accumulator  or  Condenser. 
In  this  dielectric  layer,  a  limited  Field  of  Force  may  be  set  up, 
the  lines  of  force  through  which  stretch  across  from  one  con- 
ducting surface  to  the  other ;  and  the  Permittance  or  electro- 
static Capacity  of  such  a  field  is  greater  than  that  of  the  field 


xvi.]  CONDENSERS.  597 

set  up  when  either  of  the  two  conducting  surfaces  is  separately 
charged  in  the  open  air. 

From  the  four  equations,  Q  =  ACT  (i.)  ;  <|>  =  (V,  —  Vy/)/c?  (ii.)  ;  <|>  =  47r<r 
(iii.)  ;  and  Q  =  VC  (iv.)  ;  we  find  that  Q  =  (V,  -  V/y)  -  A/4mf,  and  C  = 
A /4  Trd,  where  A  is  the  opposed  area  of  the  plates,  and  d  the  thickness  of 
air  between  them. 

Suppose  a  condenser  to  be  made  up  of  an  inner  conducting  sphere 
(solid  or  hollow)  of  radius  r  and,  concentric  with  this  sphere  and  separated 
from  it  by  a  layer  of  air,  an  outer  shell  whose  inner  spherical  surface  has  a 
radius  r1.  Let  the  inner  electrified  sphere  bear  a  charge  Q;  this  charge 
acts  as  if  it  were  concentrated  at  the  centre.  The  potential-fall  or  -differ- 
ence across  the  field  of  force  is,  as  at  Fig.  199,  Q/r  —  Q/r'.  The  mean  area 
A  is  4?rrr'.  The  expression  C  =  K/^ird  becomes  C  =  rr'/(r'  —  r).  This 
is  greater  than  the  capacities  of  either  of  the  two  component  spherical  sur- 
faces, the  capacities  of  these  being  respectively  equal  to  r  and  r'. 

The  thinner  the  dielectric  between  the  two  metallic  surfaces  of 
a  condenser,  the  greater  is  its  electrostatic  capacity;  and  correspond- 
ingly, the  less  will  be  the  potential  to  which  a  given  charge  will  raise  it. 
Starting  from  a  given  original  charge,  with  its  corresponding  field  of  force, 
the  thinner  the  dielectric  of  the  condenser  the  smaller  will  be  the  part  of  the 
original  field  left  when  the  second  plate  of  the  condenser  is  put  to  earth ; 
but,  the  charge  011  the  first  plate  remaining  the  same,  the  potential-slope  in 
that  part  remains  unaffected  :  the  difference  of  potentials  between  the  two 
plates  is  therefore  less  than  the  original  potential-difference,  in  the  larger 
original  field,  under  the  same  potential-slope  <J>  due  to  the  given  charge.  If 
the  opposed  plates  be  made  to  approach  one  another,  the  potential- 
difference  between  the  opposed  plates  falls  proportionately,  because 
<J>  =  47TO-  remains  constant  if  o-  be  not  altered;  and  in  order  to  maintain  a 
given  potential-difference  constant,  it  would  be  necessary  to  increase  the 
charge  on  the  insulated  plate :  or  conversely,  if  the  potential-difference  be 
determinate  (e.g.,  where  the  opposed  plates  are  each  connected  with  one 
terminal  of  a  galvanic  battery),  on  mutual  approach  of  the  plates  the  charges 
upon  them  will  proportionately  increase. 

In  this,  it  is  assumed  that  any  "free  charge  "  on  the  farther  surface  of 
either  conducting  film  is  so  small  that  it  may  be  left  out  of  account.  But 
see  p.  630. 

The  nature  of  the  dielectric  between  the  plates  of  the  con- 
denser is  not  a  matter  of  indifference.  It  is  found  —  and  this 
proves  that  in  the  phenomena  of  electrical  force  the  dielectric 
plays  an  important  part  —  that  the  permittance  or  capacity  of  a 
condenser  varies  with  the  nature  of  the  interposed  dielectric, 
and  is  proportional  to  a  constant  special  to  each  substance  (or 
in  some  instances  even  to  particular  directions  within  the  same 
substance)  and  called  the  Specific  Inductive  Capacity  or  the 
Permittivity,  K,  of  that  substance.  The  sp.  ind.  cap.  of  air 
being  taken  as  a  standard  and  equal  to  unity,  that  of  sulphur 
is  3-2.  Sometimes  the  sp.  ind.  cap.  of  a  vacuum,  which  differs 
little  from  that  of  air,  is  taken  as  unity,  in  which  case  the 


598  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

dielectric  is  the  Ether  itself.  The  sp.  ind.  cap.  of  glass  rises 
slightly  when  the  temperature  is  increased  (between  12°  and 
83°  C.  a  rise  of  2J-  per  cent).  All  gases  have  very  nearly  the 
same  inductive  capacity,  whatever  their  chemical  constitution, 
their  temperature,  or  their  density.  If,  however,  their  pres- 
sure be  increased  or  diminished,  the  minute  difference  between 
their  sp.  ind.  cap.  and  that  of  a  vacuum  is  also  increased  or 
diminished  in  the  same  proportion ;  and  conversely,  when  a  gas 
is  employed  as  a  dielectric,  induction  across  it  diminishes  its 
pressure,  the  gas  then  adjusting  itself  so  as  to  become  rarer  and 
consequently  less  inductive. 

If  a  given  charge  Q,  will  raise  a  condenser  in  which  air  is  the  dielectric 
to  a  potential  V,  it  will  only  raise  a  similar  condenser  whose  dielectric  has 
sp.  ind.  cap.  =  K  to  the  potential  V/K ;  and  since  in  the  special  case  of  a 
conducting  material  used  in  the  place  of  a  dielectric  the  difference  between 
the  inner  and  outer  coats  is  zero,  the  sp.  ind.  cap.  of  a  conductor  may,  for 
some  purposes,  be  considered  infinite,  for  0  =  l/oo  . 

Similarly,  the  greater  the  sp.  ind.  cap.,  the  less  is  the  mechanical  Force 
across  the  dielectric  between  two  given  charges  at  a  given  distance; 
F  =  QQVKcP. 

This  is  the  most  general  formula:  the  formula  F  =  QQ'/d2  applies  only 
to  air  as  the  dielectric.  From  the  air-formula  we  get  the  following  Equation 
of  Dimensions:  [Q]  =  [Mechanical  Force  x  distance2] i=  [ML/T2-  L2] *  = 
[M*U/T]  ;  but  from  the  general  formula  we  get  [Q]  =  [KiM*Li/T].  In 
electrostatic  measurements  we  arbitrarily  assume  air  as  a  standard  and,  for 
air,  K  =  1,  a  Number  merely,  without  Dimensions ;  but  if  we  approach  the 
subject  from  any  other  point  of  view,  we  must  use  the  general  formula,  not 
the  air-formula,  in  order  to  ascertain  the  true  Dimensions  of  electric  quan- 
tities on  any  other  system  of  measurement. 

Whatever  be  the  medium,  the  quantity  of  induced  charge  remains  the 
same :  this  is  expressed  by  saying  that  the  same  number  of  Lines  of  Induc- 
tion pass  from  a  given  charge  whatever  be  the  medium,  and  if  the  given 
charge  be  Q,  measured  in  the  usual  air-units,  the  total  number  of  these  lines 
is  4?rQ  =  I,  the  Total  Induction ;  or,  at  distance  d  the  lines  of  induction  are 
Q/d2  per  sq.  cm.  =  i  the  induction  per  sq.  cm.,  =  47r<r  per  sq.  cm.,  where  cr 
is  the  superficial  density  of  the  induced  charge  on  any  bounding  conductor- 
surface.  But  in  any  medium  of  sp.  ind.  cap.  =  K,  say  higher  than  that  of 
air,  ,the  quantities  of  electricity  would  have  to  be  increased  in  the  ratio 
1  :  VK  in  order  to  give  the  same  mechanical  force  F  :  the  units  of  quantity 
are  thus  larger :  the  numerical  value  of  the  potential  due  to  a  given  charge 
thus  varies  inversely  as  VK,  and  <|>,  the  numerical  value  of  the  potential- 
slope  or  Electric  Force,  in  a  medium  of  sp.  ind.  cap.  K,  varies  inversely  as 
VK  for  charges  of  given  numerical  value,  measured  with  reference  to  that 
medium,  or  inversely  as  VK  x  VK  =  K  for  charges  of  an  equal  number  of 
air-units,  these  units  being  smaller.  Therefore  <|>,  —  47rcr  in  air,  becomes 
<}>  =  47rcr/K  in  any  other  medium,  when  a-  is  measured  in  the  usual  air-units; 
but  i  remains  equal  to  47r0-;  hence  i  =  K<|>,  and  in  a  medium  of  sp.  ind.  cap. 
K,  the  lines  of  induction  are  more  numerous  than  those  of  force  in  the 
ratio  K  :  1.  The  units  of  quantity  vary  pari  passu  in  the  terms  for  i  and  <|> : 
hence  i  is  always  equal  to  K<j> :  but  in  air  i  =  <|>. 


xvi.]  SPECIFIC   INDUCTIVE   CAPACITY.  599 

Given  charges,  Q  and  Q',  measured  in  air-units,  thus  produce  a 
mutual  force  F  =  QQ'/Kd2;  the  potential  is  Q/Kd:  the  potential-slope  or 
electric  force  <j>  varies  as  1/K  :  the  equipotential  surfaces  are  at  mutual  dis- 
tances K  times  as  great  as  in  air  :  the  capacity  of  a  given  conductor  varies 
as  K,  —  e.g.  ,  that  of  a  sphere  of  radius  r  is  Kr:  the  work  done  in  commu- 
nicating a  given  charge  to  a  given  conductor  varies  inversely  as  K,  for 


If  a  field  of  force  be  made  up  of  layers  of  different  dielectrics,  the 
potential-slope  in  each  is'inversely  proportional  to  K.  There  is  thus  a  kind 
of  refraction  of  the  potential-slope  at  the  bounding  surfaces  of  the  layers. 

The  sp.  ind.  cap.  of  a  dielectric  diminishes  with  the  time,  and  is  there- 
fore difficult  to  measure  directly  ;  and  when  a  condenser  is  discharged  by 
metallic  communication  set  up  between  its  two  coatings,  its  charge  does  not 
at  once  completely  vanish,  but  the  condition  of  the  dielectric  is  apparently 
very  similar  to  that  of  a  body  which,  being  imperfectly  elastic,  recovers 
slowly  and  irregularly  its  primitive  form  and  condition  after  deformation  ; 
and  it  is  curious  that  the  same  means  —  vibration,  shaking,  jarring,  etc.  — 
which  facilitate  the  return  of  such  a  body  to  its  normal  condition  after  a 
strain,  facilitate  the  prompt  and  complete  discharge  of  a  condenser  whose 
two  coatings  are  put  in  metallic  connection.  On  sending  alternate  charges 
into  a  condenser,  the  residual  discharge  liberates  them  in  the  reverse  order 
(Hopkinson)  ;  a  result  strikingly  like  that  of  Boltzmann  with  reference  to 
successive  torsions.  Quartz  employed  as  a  dielectric  has  one-ninth  the  resi- 
dual capacity  of  glass  ;  Iceland  spar  seems  to  have  no  residual  capacity  at 
all,  and  permits  prompt  discharge. 

The  dielectric  of  a  condenser  may  become  double-refracting  under  the 
influence  of  Electric  Stress,  which  tends,  without  altering  its  total  volume, 
to  dilate  it  at  right  angles  to  the  lines  of  force  :  its  optical  axis  is  parallel 
to  the  lines  of  force.  Glass  and  olive  oil  become  like  Iceland  spar  (nega- 
tive crystals,  p.  555)  ;  bisulphide  of  carbon,  paraffin,  resiri,  become  positive 
(Kerr).  Solids  slowly,  liquids  instantly,  acquire  or  lose  this  condition  of 
stress  :  and  when  an  air-condenser  is  released  from  its  stress  by  discharge, 
there  is  a  distinct  sound. 

Since  electrostatic  capacity  or  permittance  varies  as  K,  the  general  for- 
mula for  that  of  a  condenser,  such  as  a  Leyden  jar,  is  (K/d)-  (Surf  ace  /47r). 

The  form  of  condenser  known  as  a  Leyden  jar  usually  con- 
sists of  a  glass  vessel  lined  internally  and  externally  with  tinfoil. 
The  inner  coating  communicates  by  wire  with  a  smooth  metallic 
knob  projecting  externally  and  insulated  from  the  outer  coating. 
By  contact  between  the  knob  and  a  charged  conductor  "the 
inner  coating  is  charged.  By  induction  through  the 
glass  there  is  produced  -an  electrical  separation  in  the  external 
tinfoil.  The  external  surface  of  this  is  temporarily  connected 
with  the  earth.  Thereafter  there  remains  a  Field  of  Force  in  the 
glass  between  the  two  tinfoil  coatings.  This  may  be  discharged 
by  establishing  a  metallic  communication  between  the  two  coat- 
ings, the  outer  tinfoil  being  first  touched,  then  the  inner. 

A  Leyden  jar  when  charged  dilates  somewhat,  and  as  it  expands  its 
capacity  increases  ;  the  potential,  to  which  a  given  charge  is  competent  to 


600  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

raise  the  jar,  sinks  to  a  corresponding  degree.  When  discharged,  the  jar 
makes  a  dull  sound,  and  the  glass  glows  at  the  edges  of  the  tinfoil,  while  the 
internal  air  becomes  warm. 

A  submarine  telegraph  cable  is,  in  effect,  a  very  long  Leyden  jar.  The 
copper  core  is  the  inner  coating;  the  guttapercha  or  other  insulator  repre- 
sents the  glass ;  the  outer  coating  of  tinfoil  is  represented  by  the  protecting 
iron  wire  or  by  the  bounding  surface  of  the  sea-water.  When  a  charge  of 
electricity  is  passed  into  a  deep-sea  cable,  the  cable  takes  some  time  to  become 
fully  charged :  it  then  bears,  for  a  considerable  tinfe,  an  electrostatic  charge 
upon  the  surface  of  its  copper  core. 

An  ordinary  aerial  telegraph-wire  is  again,  but  to  a  less  marked  degree, 
a  Leyden  jar.  The  inner  coating  is  the  surface  of  the  wire  itself;  the  die- 
lectric is  the  air;  the  outer  coating  is  the  surface  of  the  earth.  The  elec- 
trostatic capacity  of  an  aerial  wire  is  small  in  comparison  with  that  of  a 
submarine  cable ;  but  it  is  not  insignificant. 

If  the  two  coatings  of  a  Leyden  jar  be  slid  past  one  another  so  as  to 
diminish  the  opposed  surfaces,  the  capacity  diminishes  and  the  potential  due 
to  the  actual  charge  of  the  jar  increases  :  the  potential  may  thus  be  adjusted 
(Sliding  Condenser). 

Batteries  of  Leyden  Jars.  — When  a  Leyden  jar  has  its  inner  coating 
placed  in  simultaneous  metallic  communication  with  the  i  n  n  e  r  coats  of  a 
series  of  uninsulated  jars,  the  whole  becomes  in  effect  one  great  Leyden  jar 
of  increased  surface,  and  the  jars  are  said  to  form  a  battery  connected  in 
Surface.  The  charge  of  the  first  jar,  being  then  distributed  throughout 
an  enlarged  conductor,  brings  it  to  a  reduced  potential ;  and  energy  is  lost 
in  the  production  of  sparks  when  the  battery  is  charged  by  the  first  jar. 

A  series  or  battery  of  Leyden  jars  is  said  to  be  charged  in  Cascade 
when  the  outer  coat  of  one  jar  is  connected  by  metal  with  the  inner  coat  of 
the  next,  and  so  forth,  while  a  charge  is  imparted  to  the  inner  coating  of  the 
first.  The  difference  of  potential  between  the  inner  coating  of  the  first  jar 
and  the  outer  coating  of  the  last  is  distributed  between  the  jars  of  the  bat- 
tery, and  thus  the  risk  is  diminished  of  any  of  the  jars  being  destroyed 
through  an  excessive  difference  of  potential  in  any  one  jar  causing  a  spark 
to  pass  and  perforate  the  glass.  The  charge  of  the  whole  system  is  only 
equal  to  that  of  a  single  jar,  and  the  difference  of  potential  in  each  of  n  jars 
is  (V,  —  V0)/n,  where  V,  and  V0  are  the  potentials  of  the  first  and  the  last 
coatings  ;  whence  the  energy  of  the  whole  ( =  half  the  whole  charge  x  the 
whole  potential-difference)  is  the  same  as  the  energy  of  a  single  jar  loaded 
with  the  same  charge  as  the  battery. 

If  the  conductors  surrounding  an  inducing  charge  do  not 
completely  enclose  it,  the  charges  induced  upon  them  are  each 
numerically  less  than  the  inducing  charge,  and  the  sum  of  those 
of  each  kind  is  also  numerically  less  than  that  charge.  In  no 
case  can  the  induced  charges  exceed  the  inducing  charge. 

Coefficient  of  Mutual  Induction.  —  The  Coefficient  of  Induction  of 
a  conductor  A  on  a  conductor  B  is  the  ratio  of  the  Charge  (or  change  of 
charge)  developed  in  B  to  the  Potential  (or  change  of  potential)  of  A.  It 
can  be  proved  that  the  coefficient  of  induction  of  A  on  B  is  always  equal  to 
the  converse  coefficient  of  B  on  A ;  and  this  reciprocally  valid  coefficient  is 


xvi.]  ELECTROSTATIC   INDUCTION.  601 

called  the  Coefficient  of  Mutual  Induction.     It  depends  upon  the 
relative  positions  of  A  and  B. 

Inverting  the  statement,  a  unit  charge  on  either  body  will,  by  induction, 
alter  the  potential  of  the  other  by  an  amount  equal  in  both  cases. 

The  effect  of  induction  is  seen  when  an  electrified  body — 
such  as  a  glass  rod  rubbed,  with  a  dry  silk-handkerchief  —  is 
brought  into  the  neighbourhood  of  light  bodies  suspended  or 
floating  in  the  air.  Over  each  of  these  bodies  there  is  a  separa- 
tion of  electricities ;  the  aspect  nearer  to  the  inducing  body  is 
charged  with  electricity  of  the  opposite  kind,  and  is  attracted ; 
the  farther  aspect  is  charged  with  electricity  of  the  same  kind, 
and  is  repelled,  but  to  a  less  extent,  because  it  is  more  distant ; 
its  charge  not  being  "  bound  "  is,  besides,  more  readily  dispersed 
into  the  surrounding  air.  On  the  whole,  these  light  bodies  are 
attracted.  If  they  come  in  contact  with  the  inducing  body, 
they  acquire  a  part  of  its  charge,  and  are  thereupon  repelled. 

As  another  effect  of  induction  we  find  that  while  two  similarly-charged 
bodies  at  the  same  potential  within  the  same  field  will  always  repel  one 
another,  yet  if  they  be  not  at  precisely  the  same  potential,  the  one  of  higher 
potential  will,  by  its  presence,  alter  the  distribution  of  electricity  over  the 
other,  the  weaker,  in  such  a  sense  that  the  weaker  one  may  even  become 
oppositely  charged  over  the  nearer  aspect,  and  the  attraction  of  the  more 
highly-charged  body  for  this  side  of  the  weaker  may  prevail  over  its  repul- 
sion of  the  farther  side ;  and  on  the  whole,  two  such  bodies  will,  if  they  be 
placed  at  a  sufficiently  small  distance  and  if  no  spark  pass  between  them, 
attract  one  another.  In  a  certain  intermediate  position  there  will  be  unstable 
equilibrium,  and  at  all  greater  distances  there  will  be  repulsion. 

When  a  conducting  body  is  brought  into  the  neighbourhood 
of  a  system  of  insulated  and  charged  conductors,  the  energy  of 
that  system  falls,  for  the  interposed  body  causes  by  its  presence 
a  redistribution  of  the  charge  of  the  system  ;  and  such  a  redistri- 
bution of  the  charge  causes  a  fall  of  the  potential  and  therefore 
of  the  energy  of  the  system.  If  the  body  introduced  be  a  die- 
lectric, the  effect  produced  is  similar  but  smaller. 

Electric  Screens.  —  A  conducting  sphere  surrounding  an 
insulated  electrified  body  and  connected  with  the  earth  will,  as 
we  have  seen,  shelter  an  external  particle  from  the  inductive 
action  of  the  enclosed  electrified  body;  and  conversely,  it  will 
shelter  the  internal  electrified  body  from  the  distribution-dis- 
turbing and  potential-lowering  influence  of  the  outside  particle. 
A  screen  of  perforated  tinfoil  or  a  cage  of  wire  gauze  has  nearly 
an  equal  effect :  such  screens  are  used  to  protect  delicate  instru- 
ments from  the  inductive  action  of  external  electrified  -bodies. 

The  place  of  an  enveloping  sphere  may  be  taken  by  a  plate 


602  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

of  metal  connected  with  the  earth.  If  the  diameter  of  this  be 
infinite  —  or  practically,  if  it  be  very  great  as  compared  with  the 
distance  between  the  electrified  and  the  protected  particle  —  the 
screening  action  will  be  perfect. 

In  Fig.  201,  A  is  an  insulated  body  positively  charged  by  a 
galvanic  battery  or  a  frictional  electric  machine ;  D  is  a  large 
Fig.2oi.  metallic  screen ;  B  is  a 

metallic  body  connected 
4-  >  by  a  wire  with  the  earth ; 

XN      this   wire  passes   round 
Earth  the  magnetic  needle  of 

>  a  Galvanometer,  G ;  the 

screen  D  is  suddenly  removed :  there  is  a  sudden  separation  of 
electricities  in  B  :  a  positive  charge  escapes  round  the  galvanom- 
eter and  deflects  its  needle  by  an  instantaneous  twitch. 

The  phenomena  of  electricity  in  equilibrium  are  very  similar  in  their 
mathematical  aspect  to  those  of  the  steady  flow  of  Heat;  equipotential  sur- 
faces represent  isothermal  surfaces ;  lines  of  force  represent  lines  of  flow  of 
heat ;  specific  inductive  capacity  takes  the  place  of  thermal  conductivity, 
and  potential-slope  that  of  temperature-gradient. 

Again,  the  calculation  of  the  variation  of  the  force  throughout  a  die- 
lectric field  resembles  very  closely  that  of  the  distribution  of  the  flow  in  a 
steadily-flowing  mass  of  incompressible  fluid;  just  as  the  stream  lines  in 
a  field  of  liquid-flow  may  be  held  to  exert  lateral  pressure  upon  one  another, 
so  do  the  lines  of  force  in  an  electric  field  laterally  repel  one  another,  as  is 
specially  manifest  at  the  surface  of  a  conductor,  where  the  "  elements  of 
charge  "  repel  one  another ;  within  the  same  medium,  in  each  of  the  tubes 
of  force,  or  tubes  of  flow,  the  product  of  the  force  or  of  the  flow  into  the 
cross-sectional  area  is  constant  (Law  of  Continuity  in  Hydrodynamics)  : 
and  the  Energy  per  unit  of  Volume  in  a  field  of  force  or  of  flow  is  at  each 
point  numerically  equal  to  the  electrostatic  or  hydrostatic  Pressure  per  unit 
of  Area  at  that  point. 

In  Faraday  and  Maxwell's  theory  of  Ether-stress,  the  flow  of  charge 
across  an  electrified  surface  is  insisted  on.  This  flow,  which  takes  place 
whenever  a  Separation  of  Electricities  occurs,  is  of  the  nature  of  a  Displace- 
ment in  the  Ether  permeating  the  Field  of  Force,  and  is  directed  along  the 
Lines  of  Force  at  every  point  in  that  field.  One  end  of  a  line  of  force  is  in 
a  condition  which  gives  rise  to  what  we  call  positive,  the  other  to  what  we 
call  negative  electrification  or  charge  at  the  surface  of  the  electrified  body. 
When  a  thin  insulated  sheet  of  tinfoil  is  exposed  to  the  inductive  influence 
of  a  charged  conductor,  there  is  a  separation  of  positive  from  negative 
charge  across  the  conductor  influenced,  and  on  each  side  there  is  a  charge 
induced  whose  density  is  ±  a:  This  means  that  the  lines  of  Induction  or  of 
Displacement,  across  which  the  tinfoil  lies,  are  directed  towards  the  tinfoil 
on  one  of  its  sides,  away  from  it  on  the  other.  The  Quantity  of  Electricity 
thus  induced  to  flow  in  either  direction  is,  over  any  given  area  A,  equal  to 
A  •  cr :  this  quantity  is  equal  to  A  •  <f>/4?r,  since  <}>  —  47rcr.  If  any  other  die- 
lectric than  air  intervene  between  the  inducing  charge  and  the  conductor 


xvi.]  ETHER-STRESS.  603 

acted  upon,  <J>  =  47rcr/K,  and  the  amount  of  the  induced  charge,  the  Quan- 
tity of  Flow,  the  so-called  Electric  Displacement,  is  Q  =  Acr=AK<}>/47r 
=  A/4:7r  x  i  the  Induction  per  sq.  cm.  A  conductor  offers  no  permanent 
resistance  to  the  displacement  of  an  electric  charge  through  it,  and  as  long 
as  there  is  maintained  between  the  extremities  of  a  conductor  a  permanent 
difference  of  potential,  so  long  will  the  electric  displacement  produced  be 
continuously  relieved  by  an  electric  flow  or  Current;  but  in  a  non-conductor 
or  dielectric  the  extremities  may  remain  under  a  permanent  difference  of 
potential,  a  permanent  stress  or  state  of  Polarisation,  for  the  Electric  Dis- 
placement, the  flow  set  up  in  it  during  the  first  instant  of  exposure  to  elec- 
tric stress,  is  arrested  by  a  certain  Electric  Elasticity  or  Elastivity 
(Heaviside)  of  the  Dielectric,  which,  being  represented  by  the  fraction 
electric  stress  electric  force  acting  across  each  unit  of  area  _  /j'A/4  "I  - 
electric  strain  ~~  quantity  of  flow  across  each  unit,  of  area  "  •  v  9/  "V  - 
4?r/K,  is  inversely  proportional  to  K,  the  specific  inductive  capacity  of  the 
dielectric. 

The  Electrostatic  Energy  of  the  dielectric  is  the  product  of  the  average 
displacing  electric  force,  ^<j>  per  sq.  cm.,  into  the  electric  displacement, 
K<|>A/47r  over  area  A,  effected  along  a  distance  d:  the  energy  of  volume 
Ad  is  thus  4<|>(K<|)A/47r)^,  and  that  of  unit  volume  is  K<|>2/87r;  while  the 
dynamical  Energy-Slope^  or  Electric  Tension  pQ  or  Traction  t,  in  the  direc- 
tion of  the  Lines  of  Force,  is  also  K«f>2/S7r  dynes  across  each  unit  of  area  of 
the  bounding  surface  of  the  dielectric. 

Dimensions  of  Electrostatic  Measures,  in  air.  —  The  Absolute  unit 
of  Quantity  of  electricity  in  electrostatic  measure  is  a  quantity  which, 
placed  at  a  certain  distance,  in  air,  from  a  similar  and  equal  charge,  repels 
it  with  a  certain  mechanical  Force.  The  Force  between  two  quantities  at  a 
given  distance  is  therefore  equal  to  (Product  of  Quantities)  +•  (Distance2). 
The  Dimensions  of  this  expression  are  [Q]  x  [Q]  -4-  [L2]  ;  but  the  dimen- 
sions of  a  Force  are  otherwise  known  to  be  [ML/T2]  ;  whence  [Q2/L2]  = 
[ML/T2]  and  [Q]  =  [M*Li/T]. 

Surface-Density  <r;  quantity  of  electricity  per  unit  of  area:  its 
dimensions  are  those  of  (Quantity)  -4-  (Area),  or  [or]  =  [Q]  •*-  [L2]  = 


Difference  of  Potential,  E:  quantity  of  Work  required  to  move  a 
unit  quantity  of  electricity  from  one  point  to  another  :  its  dimensions  are  those 
of  (Work  done)  H-(  Quantity  moved)  ;  whence  [E]  =  [ML2/T2J  * 


Electric  Force,  or  "  Electromotive  Intensity,"  $  :  the  Electric  Force 
at  any  point  in  a  field  is  the  mechanical  force  acting  upon  a  unit  quantity 
of  electricity  placed  there  :  Mechanical  Force  -f-  Electric  Quantity  =  [ML/T2] 
•4-  [M*U/T]  =  [Mi/L^T]  :  or,  Force  per  sq.  cm.  x  2  -  Surface-Density, 
=  [M  /  LT2]  +  [M*  /  IJT]  =  [M*  /  L*T]  .  Otherwise,  [Potential-Slope]  = 
[E  -=-  Distance]  =  [M*Li/T]  +-  [L]  =  [Mi/L*T].  Also,  Number  of  Lines  of 
Force  per  sq.  cm.  ;  [47rQ]  -  [4*r«]  -  [Q/L2]  -  [M*L?/T]  +  [L2]  =  [Ml/L*T]. 
Lastly,  [4™-]  =  [<r]  =  [M*/L*T]. 

Induction  per  sq.  cm.,  i:  same  as  [<{>],  in  air. 

Capacity,  C:  the  Quantity  necessary  to  produce  a  certain  rise  of 
Potential:  its  dimensions  are  those  of  (Quantity)  -4-  (Potential-difference); 
[C]  =  [M*L*/T]  -  [M*L*/T]  =  [L],  a  Length  simply.  The  relative 
capacities  of  conductors  of  similar  form  are  simply  proportional  to  their 
diameters. 


604  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

Specific  Inductive  Capacity,  K:  K  =  Quantity  displaced  x 
477  /"area  x  $,  see  equation,  p.  603:  [K]  =  [Q]  -s-  [<}>  x  area]  =  [M*jJ/T] 
-4-{[M5/L5T]  x  [L2]};  it  is  therefore  simply  a  Number,  when  air  is  taken 
as  the  standard  medium. 

Coefficient  of  Induction;  the  ratio  of  a  Charge  developed  to  a 
Potential  inducing;  Quantity  +  Potential;  [M*L*/T]  +  [MiL*/T]  =  [L]. 

Electrostatic  Dimensions  in  any  medium.  —  Let  K  be  the  sp.  ind. 
cap.  of  the  medium;  what  the  Dimensions  of  K  may  turn  out  to  be  we  do 
not  know  ;  but  F'  =  QQ'/Kd2,  whence  [Q]  =  [MiUKi/T].  Similarly,  [>] 
=  [MlKl/L*T];  [E]  =  [M4LVKiT];  [$]  =  fa*  /K]  =  [Potential-  Slope] 
=  [Mi/L4K*T]  ;  [i]=[MiK*/L4Tj;  [CJ  =  [KL];  [K]  =  [K]  ;  and  [Coeffi- 
cient of  Induction]  =  [KL]  . 

Relations  of  Electrostatic  Quantities,  in  any  medium.  —  Potential- 


Diiference  ;  V,  -  Vy/  =  E  =  dVSir-F/AK  =  dVS-n-f/K  =  d-^  =  ^Tr-d-  o-/K 
=  47r  •  d  •  Q/  AK  =  d  •  i/K.  Electric  Force,  <j>  =  E/d  =  V87r-F/AK  =  V8w//K 
=  47TO-/K  =  47r  •  Q/  AK  =  i/K  =  2//0-.  Surface-Density,  <r  =  K<|>/4fl- 
=  VKF/27rA  =  VKf/^r  =  KE/4ffrf  =  Q/A  =  1/47T.  Induction  per  sq.  cm., 
i  =  47TQ-  =  K<|>  =  V87rKF/A  =  v/87rK/  =  KE/d  =  47rQ/  A.  Quantity,  Q  =  A<r 
=  A  VKF/27rA  =  A  VK//27T  =  AK«j>/47r  =  ARE  /M  =  Ai  /  ±TT  =  I/4ir. 
Force  F  across  area  A  ;  F  =  A  •  KE2/87rd2  =  KA«|»2/8ir  =  27r  •  A<r2/K  = 
2irQ2/AK  =  A<}>cr/2  =  Aio-/2K  =  I<r/2K  =  A  .  <J>i/87r  =  Ai2/87rK.  Force  / 
per  sq.  cm.  ;  /=  KE2/8rf  =  K^/87r  =  27r(7z/K  =  2TrQJ2/A2K  =  ^(T/2  =  ia/'2K 
=  4>i/87r  =  i2/87rK.  Energy  of  Field  =  ^QV  =  J{A  VKF/2irA  x  d  •  VSirF/AK} 
=  d-F  =  Ad-f;  =%(AK$/4:Tr).d'~4>  =  Ad'K^/8Tr',  =  Ad  •  27ro-2  /  K  =. 
Ad  •  i2/  STJ-K  =  AKE2/  STT^  =  Ad  .  27rQ2  /  A2K  =  Ad  •  <|>i  /  STT  =  Ad  •  <Jxr/  2  = 
Ad  •  o-i/2K.  Energy  of  Field  per  cm.  cube  =/=  F/A  =  K«j>2/87r  =  2iro-2/K 
=  KE2/87rd2  =  27rQ2/A2K  =  <}>i/87r  =  «|>cr/2  =  ol/2K  =  i2/87rK.  Capacity  or 
Permittance  C  =  Q/V  =  KA/47rd.  Dielectric  Elasticity  =  <J>/o-  =  4v/K. 

The  fundamental  equations  for  the  above  relations  are:  <j>  =  E/d;  f= 
27r(72/K  ;  i  =  K<|>  =  47ro-  ;  where  cr  is  measured  in  the  usual  air-units.  If  in 
the  above  expressions  we  make  K  =  1,  we  obtain  the  ordinary  air-equations. 

OBSERVATION  OF  DIFFERENCES  OF  POTENTIAL. 

Observation  of  Differences  of  Potential  is  effected  by 
means  of  instruments  called  Electroscopes  and  Electrome- 
ters; the  former  indicate  the  nature,  the  latter  measure  the 
amount,  of  differences  of  potential. 

Gold-Leaf  Electroscope.  —  A  glass  flask  with  a  vulcanite  stopper  : 
through  the  stopper  passes  a  metal  rod,  surmounted  by  a  metallic  sphere  or 
plate,  and  terminated  below  by  a  pair  of  freely-suspended  strips  of  gold 
leaf.  If  the  metallic  part  of  the  electroscope  be  charged  by  contact  with 
an  electrified  body,  the  gold  leaves,  becoming  similarly  charged,  repel  one 
another,  and  diverge,  slightly  if  the  charge  be  feeble,  widely  if  it  be  great. 
The  electroscope  may  also  be  temporarily  charged  by  induction:  a 
4-  electrified  body  brought  into  the  neighbourhood  of  the  sphere  or  plate 
causes  that  sphere  or  plate  to  become  negatively,  while  the  more  distant 
gold  leaves  within  the  flask  are  positively  charged.  If,  while  the  electro- 
scope is  electrified  by  induction,  its  upper  extremity  be  momentarily  touched 
by  the  experimenter,  the  gold  leaves  collapse,  for  their  charge  escapes  to  the 
earth  :  the  plate  or  sphere,  however,  retains  its  charge,  and  when  the  indue- 


xvi.]  ELECTROSCOPES.  605 

ing  body  is  removed,  the  opposite  charge  borne  by  the  sphere  or  plate 
becomes  free  to  distribute  itself  over  all  the  metal  of  the  electroscope,  and 
the  leaves  again  diverge,  for  the  instrument  is  now  permanently  charged. 

If  the  electroscope  be  permanently  charged,  the  approach  of  a  body 
similarly  charged  will  cause  a  farther  divergence  of  the  leaves  :  the  approach 
of  a  body  oppositely  charged  will  cause  the  leaves  to  repel  each  other  with 
less  force  :  whence  the  nature  of  the  electrification  of  a  given  charged-body 
can  be  ascertained. 

The  deficiencies  of  the  electroscope  are :  that  its  indications  are  quali- 
tative, not  accurately  quantitative;  and  that  the  glass  does  not  thoroughly 
screen  the  gold  leaf  from  the  direct  inductive  action  of  external  charged- 
bodies.  In  order  to  obviate  the  latter  defect,  the  inner  surface  of  the  flask 
is  sometimes  lined  with  perforated  tinfoil,  or  the  whole  is  surrounded  by  a 
cage  of  wire  gauze. 

The  gold-leaf  electroscope  is  a  development  of  earlier  instruments,  in 
which  straws,  plain  balls  of  elder-pith,  or  gilt  pith-balls,  were  employed. 

In  the  discharging  electroscope  the  gold  leaves,  when  they 
diverge,  come  in  contact  with  two  metallic  uprights  which  communicate 
with  the  earth  :  they  are  thus  discharged  and  collapse,  again  to  be  charged  : 
the  number  of  oscillations  of  the  gold  leaves  affords  a  rough  measure  of  the 
quantity  of  electricity  borne  by  a  conductor  which  is  discharged  to  earth 
through  such  an  electroscope. 

Peltier's  Electroscope.  —  A  vertical  brass  ring,  insulated;  attached 
to  its  inner  circumference,  at  the  lowest  point,  a  vertical  pointed  rod ;  on 
the  pointed  rod  is  poised  either  a  magnetic  needle  or  else  a  metallic  rod 
whose  direction  is  determined  by  a  small  magnetic  needle  attached  to  it. 
The  whole  is  turned  round  a  vertical  axis  until  the  ring  and  the  poised 
metallic  rod  lie  in  the  same  plane.  If  the  ring  be  charged,  the  charge  is 
shared  with  the  poised  metallic  mass,  and  the  ring  and  the  poised  mass  repel 
one  another ;  the  latter  swings  round  until  the  force  of  electrical  repulsion 
is  balanced  by  the  tendency  of  the  magnet  to  point  to  the  magnetic  north 
and  south.  This  instrument  may,  by  imparting  to  it  a  series  of  successive 
known  charges,  be  so  graduated  as  to  act  as  an  electrometer. 

Bohnenberger's  Electroscope.  —  Two  vertical  dry  piles 
(p.  622),  the  one  with  its  +  pole,  the  other  with  its  —  pole  upper- 
most ;  between  these  oppositely-charged  uppermost  poles  there  is 
a  field  of  force,  within  which  a  strip  of  gold  leaf  is  suspended. 
If  uncharged,  this  strip  hangs  vertically ;  if  charged,  it  is  repelled 
by  one  pole  and  at- 
tracted towards  the 

Fig.202. 

other. 

Instead  of  two  piles, 
the  two  extremities  of 
one  and  the  same  dry 
pile  may  be  used  to 
make  such  an  electro- 
scope. In  Fig.  202,  AB  A  -  -he 
is  a  dry  pile  whose  poles  are  connected  with  the  metallic  plates 


606 


ELECTRICITY  AND  MAGNETISM. 


[CHAP. 


C  and  D,  between  which  there  is  thus  formed  a  field  of  force, 
in  which  the  gold  leaf  E  is  suspended. 

On  the  same  principle  the  Quadrant  Electrometer  of  Lord 
Kelvin  is  based.  In  Fig.  203  the  two  opposite  quadrants  A  and 
D  are  connected  with  one  another  by  wire,  but  are  insulated 
from  B  and  C.  A  and  D  are  thus  at  the  same  potential,  while 
B  and  C  are  also  at  the  same  potential,  —  a  potential  which  may 
differ  from  that  of  A  and  D.  A  and  D  may  be  brought  to  the 
potential  of  the  earth  by  means  of  a  wire  connected  with  gas  or 
water  pipes ;  B  and  C  may  be  brought  to  the  potential  of  any 
given  object  by  connecting  them  with  it  by  means  of  a  wire. 
The  quadrants  A,  D,  and  B,  C,  are  thus  at  different  potentials, 
and  a  metallic  needle — an  aluminium  needle  of  a  flat  dumb-bell 


Fig.203. 


shape  —  will,  if  it  be  suspended  symmetrically  over  the  quad- 
rants by  means  of  two  threads  arranged  parallel  to  one  another, 
and  if  it  be  kept  charged  by  constant  connection  with  one  coat- 
ing of  a  Leyden  jar  (which  may  be  replenished  when  necessary), 
impose  a  certain  amount  of  torsion  upon  those  two  suspending 
parallel  threads ;  the  amount  of  this  torsion  will  indicate  the 
nature  and  —  approximately  —  the  amount  of  the  difference  of 
potential  between  the  two  pairs  of  quadrants,  and  therefore 
between  the  earth  and  the  object  whose  Potential  is  to  be 
measured. 

If  the  quadrants  be  made  hollow,  and  the  needle  suspended  within  them, 
the  arrangement  is  better  adapted  for  electrometric  purposes. 

The  whole  arrangement  is  well  adapted  for  testing  the  adjustment  to 
equality  of  the  potentials  of  two  bodies. 


XVI.] 


ELECTROMETERS. 


607 


It  would  come  to  the  same  thing  if  the  potentials  really  measured  were 
those  of  the  air  in  the  neighbourhood  of  the  quadrants,  provided  that  the 
quadrants  be  all  of  the  same  metal,  or  that  the  potential  of  the  air  in  the 
neighbourhood  of  one  uncharged  metal  be  the  same  as  that  in  the  neighbour- 
hood of  another. 

The  amount  of  deflection  of  the  suspended  needle  may  be 
observed  by  connecting  with  it  a  very  light 
mirror,  upon  which  a  very  narrow  beam  of 
light  shines ;  as  the  needle  is  deflected,  the 
beam  of  light  reflected  from  the  mirror  is 
deflected  through  an  angle  twice  as  great  as 
that  of  the  deflection  of  the  mirror ;  and  the 
beam  of  light,  if  received  upon  a  distant  scale, 
thus  acts  as  a  weightless  pointer.  Fig.204. 

Upon  the  scale  the  deflection  of  the  spot  of  light 
may  be  read  off ;  that  deflection  is,  on  a  straight  scale, 
proportional  to  the  tangent  of  twice  the  angle  of  de- 
flection of  the  mirror :  for  small  angles  it  is  nearly 
proportional  directly  to  twice  the  angle  (Fig.  161). 

Coulomb's  Torsion  Balance.  —  A  long,  verti-    /^~~ 
cal,  slender,  hard-wire  or  silk-fibre  AB,  Fig.  204,  by 
which  there  is  suspended  in  a  horizontal  position  a 
thin  counterpoised  rod  of  glass  or  shellac,  CD,  which 
bears  at  one  of  its  extremities  a  little  gilt  sphere  D.  -L 

In  one  position  of  the  suspending  wire  the  gilt  sphere  >* 

D  comes  into  contact  with  a  sphere-ended  metal  rod 
EF :  this  rod  projects  through  the  walls  of  the  glass 
case  in  which  the  whole  is  encaged,  and  is  therefore 
insulated.  This  metal  rod  terminates  externally  in  a 
sphere  E,  which  may  be  charged  by  contact  with  an 
electrified  body,  such  as  a  proof -plane.  A  proof- 
plane  is  a  small  metallic  disc  provided  with  an  insulating  glass  or  ebonite 
handle.  It  is  used  by  laying  the  disc  upon  the  surface  of  an  electrified  body : 
when  the  disc  is  withdrawn,  it  bears  with  it  a  charge  proportional  to  the 
charge  previously  borne  by  that  part  of  the  surface  of  the  electrified  body 
with  which  it  had  been  placed  in  contact :  it  is  then  made  to  touch  the 
sphere  E  of  the  torsion  balance.  EF  being  charged,  the  two  spheres  F  and 
D,  when  they  come  in  contact,  become  charged  with  electricity  of  the  same 
kind,  and  repel  one  another :  they  do  this  until  there  is  equilibrium  between 
the  electric  repulsion  and  the  torsion  of  the  suspending  wire  AB.  The 
proof-plane  may  be  used  directly  in  the  place  of  EF ;  and  instead  of  a  proof- 
pi  a  ne  a  proof-sphere  maybe  used  when  the  curvature  of  the  body,  whose 
charge  is  to  be  examined,  is  but  small. 

Different  charges  may  be  compared  by  comparing  the  amounts  of  torsion 
necessary  to  bring  the  two  mutually-repellent  bodies,  D  and  F,  to  equal  dis- 
tances. A  preliminary  charge  is  given  to  the  ball  D ;  a  charge  Q  of  the 
same  kind  is  imparted  to  F,  or  brought  in  by  a  proof-plane  or  a  proof-sphere. 
Let  the  repulsion,  between  Q  and  the  charge  on  D,  be  such  that  the  suspended 


c\ 


_.  E 


608 


ELECTRICITY  AND   MAGNETISM. 


[CHAP. 


Fig.205. 


horizontal  fibre  makes  an  angle  FED  of  10°  with  that  position  in  which  D 
is  in  contact  with  F,  while  the  upper  end  A  is  twisted  in  the  contrary  direc- 
tion —  so  as,  as  it  were,  to  tend  to  force  F  and  D  together  —  through  an 
angle  of  410° ;  the  total  torsion  of  the  wire  AB  is  420°.  Now  remove  the 
charge  Q  and  substitute  a  charge  Q,' ;  the  index  at  A  indicates  95°  of  rota- 
tion there  when  D  is  in  its  former  position :  the  total  torsion  of  the  wire  is 
now  105°.  The  charges  Q  and  Q'  are  proportional  to  the  torsions  which 
their  repulsions  balance  ;  and  Q  :  Q'  : :  4  :  1 . 

Coulomb  also  made  use  of  the  method  of  oscillations  (p.  39)  :  he  swung 
an  electrified  needle  in  presence  of  an  electrified  ball ;  the  periods  of  the 
oscillations  varied  as  the  distance ;  but  the  period  varies  inversely  as  the 
square  root  of  the  force  acting  :  therefore  the  force  acting  varies  inversely 
as  the  square  of  the  distance.  When  the  distance  is  kept  fixed,  the  charges 
of  the  needle  or  ball  being  varied,  the  periods  of  the  oscillations  vary 
inversely  as  the  square  root  of  the  varied  charge. 

The  Absolute  Difference  of  Potential  between  two  bodies 
may  be  ascertained  by  measuring  the  attraction  between  two 
metallic  plates  which  are  respectively  connected  by  metallic 
wires  with  the  two  bodies  in  question.  In  Fig.  205  AB  is  a 

galvanic  battery,  the 
extremities  of  which 
are  permanently  at  dif- 
ferent potentials :  it  is 
desired  to  find  the  dif- 
ference between  these 
potentials.  Connect  A 
and  B  with  the  plates 
C  and  D.  The  field  of  force  between  C  and  D  is  uniform  at 
its  centre.  D  is  fixed ;  but  E,  the  central  part  of  C,  is  mov- 
able. The  attraction  between  E  and  D  may  be  measured  by 
observing  the  distortion  of  a  spring  which  tends  to  pull  E 
upwards  while  the  electrical  attraction  tends  to  pull  E  down- 
wards, this  observation  being  made  when  the  distance  of  D  is 
so  adjusted  that  the  lower  surface  of  E  is  flush  with  that  of  C. 
It  is  sometimes  found  advantageous  in  the  use  of  instruments  of 
this  kind  to  connect  D  alternately  with  B  and  with  the  earth : 
the  spring  tends  to  become  differently  distorted  in  the  two  cases, 
and  in  order  to  render  its  distortion  equal  in  both  cases  the  dis- 
tance of  D  must  be  varied.  The  amount  of  approximation  or 
retraction  of  D  may  be  measured  by  a  micrometer-screw. 

The  spring  which  keeps  up  E  against  the  attraction  of  D 
may  be  replaced  by  transforming  E  into  one  pan  of  a  delicate 
balance,  of  which  the  other  pan  may  be  loaded  with  known 
weights. 


I 


xvi.]  MEASUREMENT   OF  POTENTIAL.  609 

The  potential  of  E  is  VE ;  that  of  D  is  VD :  the  difference  of  potential 
to  be  measured  is  (VE  -  VD).  The  traction  across  the  field,  along  the  lines 
of  force,  or  the  pull  of  the  field  upon  the  plate  E,  is  44*°"  Per  S(l-  cm->  or 
^4>-Ao-  upon  its  whole  area  A.  This  is  equal  to  27ro-2-A:  and  it  is  bal- 
anced by  F,  the  stress  upon  the  spring ;  F  =  27rcr2  •  A.  Also,  kirv  =  $  = 
(VE  — VD)/<7.  Hence  the  Difference  of  Potential  in  absolute  measurement 
is  (VE  —  VD)  =  d  V8?rF/A,  in  which  expression  d,  A,  and  F  can  be  directly 
measured ;  d  being  the  distance  between  the  plates  E  and  D. 

Since  cr  =  (VE  —  VD)/47re?  per  unit  area,  the  charge  on  the  attracted 
circular  disc  of  radius  r  is  (VE  —  VD)r2/4d:  the  capacity  of  the  system  is 
therefore  r2/±d,  and  can  thus  be  measured  absolutely.  Standards  of 
Electrostatic  Capacity  can  thus  be  constructed. 

When  E  and  D  are  connected  with  A  and  B,  the  respective  potentials 
of  E  and  D  are  VA  and  VB ;  and  VA  -  VB,  the  difference  of  potential  between 
the  ends  of  the  pile,  =  o?V87rF/A.  When  D,  instead  of  being  connected 
with  B,  is  connected  with  the  earth,  its  potential  becomes  zero ;  and  when 
D  (movable  in  this  case)  is  brought  by  its  micrometer-screw  into  such  a 
position  that  the  plate  E  again  assumes  a  position  flush  with  the  fixed  guard- 
ring  C,  the  stress  F  upon  the  spring  is  the  same  as  before,  and  VA  —  V^^  = 
VA  —  0  =  e^VSTrF/A,  where  d,  is  the  new  distance  between  the  plates  E 
and  D.  Hence  VB  =  (d,  -  d)  VSTrF/A ;  and  the  Potential  of  B  is  easily 
measurable,  for  (dt  —  d),  the  change  of  distance  between  E  and  D,  is  much 
more  easily  measurable  than  d,  the  absolute  distance  between  them. 

Lord  Kelvin,  to  whom  the  above  method  is  due,  has  also  devised  instru- 
ments by  which  the  difference  of  potential  (or  voltage)  of  electric  light- 
ing currents  may  be  electrostatically  measured.  This  is  effected  by  observing 
the  extent  to  which  an  aluminium  strip,  charged,  succeeds  in  rising  up  from 
its  position  of  gravitational  equilibrium,  in  order  to  place  itself  immediately 
between  two  fixed  aluminium  plates,  oppositely  charged  (Kelvin's  Electro- 
static Volt-Meters).  If  the  electrifications  be  reversed,  the  attraction  is  the 
same ;  and  if  they  be  rapidly  alternated,  the  general  action  of  the  instru- 
ment remains  the  same.  These  instruments  are  graduated  so  as  to  measure 
the  potential-difference  in  Volts,  not  in  electrostatic  C.G.S.  units. 


PRODUCTION  OF  DIFFERENCE  OF 'POTENTIAL. 

The  principal  source  of  Difference  of  Potential  is  commonly 
stated  to  be  the  Contact  of  dissimilar  surfaces  —  that  is,  either 
of  different  substances  or  of  two  pieces  of  the  same  substance 
whose  surfaces  are  in  different  conditions.  A  piece  of  resin 
and  a  piece  of  glass  will,  upon  contact,  be  more  difficult  to  pull 
asunder  than  two  pieces  of  resin  or  two  pieces  of  glass :  and  if 
they  be  rubbed  together,  so  as  to  multiply  the  points  of  contact, 
the  effect  is  multiplied.  When  pulled  asunder,  two  such  bodies 
are  found  to  be  charged  equally  and  oppositely :  across  the  sur- 
face of  contact  there  has  been  a  Separation  of  positive  from 
negative  electricity.  The  development  of  electrical  condition 
is  thus  necessarily  a  phenomenon  of  continual  recurrence :  and 

2R 


610  ELECTKICITY   AND   MAGNETISM.  [CHAP. 

it  greatly  influences  the  adhesion  of  one  body  to  another.  In 
all  probability,  wherever  there  is  friction,  the  energy  ultimately 
converted  into  heat  is,  in  the  first  place,  converted  into  the 
energy  of  electrical  separation. 

When  two  substances  have  different  molecular  velocities  at  their  com- 
mon surface  of  mutual  contact,  the  molecules  hamper  one  another  and 
energy  is  lost :  this  energy,  formerly  that  of  molecular  motion,  now  takes 
the  form  of  the  energy  of  electrical  displacement.  Within  the  interior  of  a 
homogeneous  body  the  same  thing  must  happen  between  colliding  mole- 
cules whose  velocities  are  different ;  but,  all  being  alike,  and  the  average 
molecular  velocity  being  the  same  throughout  the  mass,  there  is  on  the 
whole  no  effect. 

Non-conductors  in  contact  become  electrified;  but  only 
on  their  surfaces  of  actual  contact.  When  they  are 
separated  their  final  discharge  is  incomplete,  and  the  residual 
charges  —  their  superficial  distribution  being  restricted  to  those 
parts  of  the  surfaces  which  have  been  most  nearly  in  actual 
contact  —  are  small  in  quantity  but  of  great  density,  and  there- 
fore of  high  potential ;  and  as  these  charges  are  not  diffused  by 
conduction  over  the  whole  surface,  their  potentials  remain  high 
after  separation. 

When  sulphur  is  melted  in  a  glass  test-tube,  after  cooling  the  sulphur 
is  found  to  bear  permanently  a  negative,  the  glass  a  positive  charge. 

In  the  following  series,  due  to  Faraday,  each  member  becomes  positively 
charged  when  rubbed  on  one  following  it,  negatively  when  rubbed  on  one 
preceding  it:  Cat  and  Bearskin  —  Flannel  —  Ivory  —  Feathers  —  Rock  Crys- 
tal —  Flint  Glass  —  Cotton  —  Linen  —  Canvas  —  White  Silk  —  the  Hand  — 
Wood  —  Shellac  —  the  Metals  (  +  Fe,  Cu,  Brass,  Sn,  Ag,  Pt)  —  Sulphur  — 
(Soapstone).  There  are  certain  irregularities  here  to  be  observed:  for 
example,  a  feather  lightly  drawn  over  a  piece  of  canvas  becomes  nega- 
tively electrified,  whereas  if  it  be  drawn  through  a  pressed  fold  of  canvas  it 
becomes  positively  charged. 

The  separation  of  electricities  by  contact  and  friction  is  utilised  in  the 
various  forms  of  electric  frictional  machines,  which  range  in  complexity 
from  a  simple  piece  of  sealing-wax  or  a  glass  rod  rubbed  with  a  catskin  or  a 
silk  handkerchief,  or  a  stout  glass  tube  rubbed  with  a  piece  of  dry  flannel, 
to  a  machine  in  which  a  glass  or  vulcanite  disc  or  cylinder,  set  in  rotation, 
rubs  against  silk  rubbers :  these  rubbers,  whose  conductivity  is  improved  by 
anointing  them  with  a  mixture  of  fat  and  mercury,  communicate  with  the 
ground,  and  their  negative  electricity  is  thus  carried  off  to  the  earth ;  the 
positive  charge,  borne  by  the  rotating  glass  or  vulcanite,  blends  with  a 
negative  charge  developed  by  induction  in  the  tips  of  a'  comb-like  series  of 
sharp  metallic  points  which  come  almost  in  contact  with  the  rotating  glass ; 
while  the  complementary  induced  positive-charge  is  conveyed  either  to  a 
large  insulated  Conductor  connected  with  these  points  by  a  metallic  chain 
or  wire,  or  to  the  surface  of  a  large  insulated  hollow  conductor  which  sur- 
rounds the  rubbing  parts  of  the  machine,  or  to  the  inner  coat  of  a  Leyden 


xvi.]  ELECTRIFICATION  ON  CONTACT.  611 

jar,  or  to  the  inner  coat  of  one  of  the  constituent  members  of  a  battery  of 
Leyden  jars.  A  charge  of  positive  electricity  may  be  thus  accumulated. 
If,  on  the  other  hand,  the  positive  charge  of  the  glass  be  conveyed  to  the 
earth,  while  the  insulated  conductor  is  metallically  connected  with  the 
rubbers,  a  charge  of  negative  electricity  may  be  accumulated  in  the  con- 
ductor. If  the  conductor,  in  which  positive  electricity  is  being  accumu- 
lated, be  connected  by  wire  with  the  negatively-charged  rubbers,  a  current 
of  electricity  will  pass  along  the  connecting  wire  so  long  as  the  machine 
is  worked,  and  that  wire  will  be  heated.  If  this  current  be  sent  through  a 
second  electric  machine  it  will  tend  to  cause  in  it  a  reversed  rotation.  It  is 
possible  (Gaugain)  thus  to  produce  continuous  currents  by  the  friction  even 
of  dissimilar  metals. 

When  two  metals  come  in  contact,  in  air  or  other  gas,  they 
at  once  become  electrified,  positively  and  negatively  respectively. 
The  amount  and  kind  of  charge  on  each  metal  depends  upon 
(1)  the  nature  of  the  metals,  (2)  the  condition  of  their  sur- 
faces, (3)  their  temperatures,  and  (4)  the  nature  of  the  sur- 
rounding or  intervening  gas,  if  there  be  any.  In  the  case  of 
copper  and  zinc  in  air,  the  copper  becomes  negatively  and  the 
zinc  positively  charged. 

The  older  view  as  to  this  was,  that  there  was  electrical  sepa- 
ration at  the  surface  of  contact  between  the  metals,  and  that 
each  metal  was  at  an  equal  potential  throughout:  then  the 
potential  of  the  air  in  the  immediate  neighbourhood  of  the 
metals  was  the  potential  of  the  metals  themselves.  The  newer 
view  is  that  the  metals  are,  before  contact,  by  reason  of  a  ten- 
dency to  chemical  action  (oxidation,  etc.)  each  at  a  potential 
different  from  that  of  the  surrounding  air  or  gas ;  that  their 
respective  potentials  are  more  or  less  different  from  one  another ; 
that  when  the  two  metals  are  brought  into  contact,  the  potential 
throughout  the  whole  conjoint  metal  becomes  uniform,  and  a 
momentary  current  runs  in  the  conjoint  metal  mass,  so  as  to 
charge  the  one  metal  (zinc)  positively  and  the  other  (copper) 
negatively ;  that  while  this  is  taking  place,  the  surrounding 
dielectric  is  being  electrically  displaced  so  as  to  produce  a 
Field  of  Force ;  that  the  Energy  necessary  for  this  is  derived 
from  a  trifling  amount  of  chemical  combination  (oxidation,  etc.) 
at  the  metal-gas  surface ;  that  in  the  case  of  a  zinc-copper 
couple,  before  contact  the  potential  of  the  air  is  say  V0  C.G.S. 
electrostatic  units,  that  of  the  zinc  is  V0  —  (x  -f-  0-0025),  and 
that  of  the  copper  is  V0  —  x  ;  that  after  contact  the  potential  of 
the  copper-zinc  couple  is  uniformly  V0  —  (x  +  0-00125),  that 
of  the  air  near  the  zinc  is  (V0  +  0-00125),  and  that  of  the  air 
near  the  copper  is  (V0  -  0-00125) ;  and  that  the  Field  of  Force 


612  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

is  maintained  in  the  air,  through  its  being  a  dielectric,  the  Ether 
in  which  offers  elastic  resistance  to  further  displacement. 

But  there  is  also  another  effect.  We  have  spoken  of  the 
two  metals  coming  to  the  same  potential  when  brought  into 
contact.  It  appears,  however,  that  they  cannot  perfectly  do 
this,  on  any  view  of  the  facts,  even  independently  of  the  air, 
except  at  a  particular  temperature.  There  is  almost  always  a 
slight  difference  of  potential,  a  true  contact-effect;  and  this 
varies  so  remarkably  with  the  temperature  as  to  give  rise  to  the 
phenomena  of  Thermo-electricity,  of  which  later  (p.  624). 

If  a  metallic  disc  be  composed  of  four  quadrants,  soldered  together  and 
consisting  alternately  of  zinc  and  copper  respectively,  and  if  the  disc  be 
arranged  horizontally,  a  needle  suspended  horizontally  over  the  centre  of 
the  disc  will,  if  that  needle  be  charged  with  positive  electricity,  be  repelled 
by  the  zincs  and  attracted  by  the  coppers,  and  it  will  therefore  swing  round 
so  as  to  lie  over  the  copper  quadrants ;  while  if  it  be  charged  negatively  it 
will  come  to  lie  over  the  zinc  quadrants.  The  needle  may  be  so  suspended 
by  two  threads  that,  when  uncharged,  it  lies  along  a  diameter  of  the  disc, 
a  diameter  which  coincides  with  a  line  of  junction  between  quadrants. 

Take  an  electroscope  surmounted  by  a  copper  plate,  varnished  on  its 
upper  side  ;  upon  this  plate  lay  a  zinc  plate  varnished  on  its  lower  side : 
these  plates,  separated  by  the  varnish,  act  as  a  condenser.  Bring  a  copper 
and  a  zinc  plate,  both  of  which  are  unvarnished  and  insulated,  into  contact : 
separate  them ;  with  the  zinc  touch  the  zinc,  with  the  copper  the  copper  of 
the  condenser.  Repeat  this  operation  several  times :  then  remove  the  zinc 
plate  of  the  condenser :  the  copper  is  found  to  be  strongly  charged  with 
negative  electricity,  while  the  zinc  plate  removed  is  positively  charged. 

Copper  filings  falling  through  an  insulated  zinc  funnel,  as  they  leave 
that  funnel  carry  with  them  a  negative  charge. 

These  experiments  may  be  interpreted  in  accordance  with  either  of  the 
above  views. 

In  Fig.  206  a  zinc  plate  Zn  and  a  copper  plate  Cu,  both  in 
air,  are  connected  by  a  copper  wire.     Then,  either  the  zinc  is  at 
Fig  206  potential  JE  =  0-00125 

_j_  I         electrostatic  units,  and 
—  the  copper  at  potential 
cu  _  |      '    -IE  =-0-00125;    or, 

according  to  the  newer 

,  x Zn +  E  .  ,,   6 

Cu  f  ' '  view,  they  are  at  equal 

"  potentials,  while  be- 
tween  them  there  is  a 
Field  of  Force,  the  potentials  within  which  present  a  potential- 
difference  of  0-0025  units.  If  one  of  the  plates  be  connected  to 
earth,  the  potentials  of  the  copper  and  zinc  are  altered :  but  the 
Field  of  Force,  though  its  terminal  potentials  are  altered,  remains 
constant  in  its  potential-fall. 


• 

xvi.]  ELECTRIFICATION  ON  CONTACT.  613 

Within  the  field  of  force  between  such  plates  arranged  with  an  inter- 
vening dielectric,  <f>  =  47rcr  =  const.  =  (VZn  —  VCu)/Krf,  where  d  is  the  thick- 
ness of  the  dielectric,  and  VCu,  VZn,  the  potentials  at  the  copper  and  the 
zinc  respectively.  Hence  the  superficial  density  a-  =  ( VZn  —  VCu)  /^irl^d ; 
but  if  the  numerator  of  this  fraction,  the  difference  of  potential,  be  con- 
stant, as  it  is  between  two  metals,  cr  =  const,  x  (!/</),  or  <r  <x  (1/d).  As 
the  thickness  d  diminishes,  the  electrostatic  capacity  or  Permittance  of  the 
field  of  force  increases,  arid  since  the  D.P.  remains  constant,  cr  increases. 
If  we  suppose  the  plates,  nominally  in  contact,  to  be  at  a  mean  molecular 
distance  of  about  20i0010000  cm.,  the  density  is  so  great  that  if  the  copper 
and  the  zinc  could  be  separated  from  one  another  before  their  charges  are 
allowed  to  recombine,  they  would  then  spark  across  20  feet  of  air.  But 
they  could  not  be  so  removed ;  in  air,  at  any  rate,  the  striking  distance  falls 
off  more  rapidly  than  the  potential-difference  does;  the  opposed  charges 
almost  wholly  discharge  themselves  when,  after  being  placed  in  contact, 
the  plates  are  pulled  asunder,  and  there  then  remain  in  these  only  residual 
charges  of  small  density,  which  vary  very  slightly  in  amount,  according  to 
the  mode  in  which  the  plates  are  pulled  asunder.  The  moving  molecules 
must  therefore,  even  though  the  masses  in  contact  seem  to  be  at  rest,  be 
constantly  discharging  and  renewing  the  separation  of  electricities. 

When  the  plates  of  copper  and  zinc,  of  Fig.  206,  are  con- 
nected not  by  a  copper  but  by  an  iron  wire,  we  have  three 
metals  and  two  contacts  ;  but  the  difference  of  potential  between 
the  terminal  copper  and  zinc  is  the  same  as  when  the  copper  and 
zinc  were  directly  in  contact ;  and,  more  generally,  if  any  num- 
ber, of  metals  be  arranged  in  linear  series,  in  Open  Circuit 
(say  n  metals  or  pieces  of  metal,  with  n  —  1  contacts),  there  is  a 
difference  of  potential  developed  (in  the  air  or  in  the  metals 
themselves,  according  to  the  view  adopted)  between  the  ter- 
minal metals ;  and  this  difference  of  potential  is  equal  to  that 
which  would  have  been  developed  between  these  two  terminal 
metals  if  they  had  touched  each  other  directly. 

If  we  make  both  terminals  of  the  same  metal,  there  will 
thus  be  no  difference  of  potential  between  them;  and  if  we 
connect  these  similar  terminals  with  one  another,  we  have  now 
a  Closed  Circuit  consisting  of  various  metals.  The  potential 
will  vary  from  part  to  part  of  the  circuit :  according  to  the  one 
view  it  will  vary  in  the  metal,  each  metal  being  equipotential 
throughout ;  according  to  the  other,  it  will  vary  only  in  the  sur- 
rounding air :  but  on  either  view,  taking  it  all  round  the  circuit, 
there  is  no  preponderance  of  potential-difference  in  the  one  direc- 
tion or  in  the  other,  and  there  is  no  current  round  the  circuit. 
The  closed  circuit  remains  in  electrostatic  equilibrium.  No  con- 
tinuous current  can,  therefore,  be  obtained  from  a  closed  metallic 
circuit,  or  indeed  from  any  closed  circuit  of  conductors  in  which 


614  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

the  material  of  the  circuit  suffers  no  alteration,  unless  Energy 
be  supplied,  either  from  without  or  else  at  the  expense  of  the 
energy  (e.g.,  the  Heat)  of  the  circuit  itself. 

If,  on  the  other  hand,  one  of  the  conductors  of  the  circuit 
suffer  a  chemical  change,  Energy  may  be  liberated,  which  may 
take  the  form  of  the  Energy  of  a  Continuous  Current.  Let  us 
consider  a  circuit  consisting  of  copper  —  hydrochloric  acid  — 
zinc  —  connecting  wire  —  copper.  Since  it  does  not  matter 
what  the  material  of  the  connecting  wire  may  be,  we  may  use 
copper;  the  circuit  is  then  Cu  —  HC1  —  Zn  —  Cu.  If  all  the 
members  of  this  series  were  mere  conductors,  the  circuit  as 
a  whole  would  attain  a  condition  of  electrostatic  equilibrium, 
and  there  would  be  no  current.  But  they  are  not  all  mere 
conductors ;  in  the  circuit  we  have  Chemical  Action,  which 
results  in  the  liberation  of  Energy;  and  this  energy  is  trans- 
formed into  that  of  a  continuous  current  round  the  circuit. 

If  a  piece  of  zinc  and  a  piece  of  copper  be  placed  in  hydro- 
chloric acid,  but  not  in  contact  with  one  another,  the  zinc 
becomes  charged,  negatively  with  respect  to  the  copper,  and 
the  copper  positively  with  respect  to  the  zinc. 

It  is  not  said,  observe,  that  either  of  the  metals  is  absolutely 
positive  or  negative  in  its  potential.  The  fact  appears  to  be 
that  the  potential  of  the  zinc  is  negative  to  that  of  the  acid-  by 
about  0-0060  electrostatic  units  of  potential,  while  that  of  the 
copper  is  also  negative,  but  to  a  less  extent,  namely,  about 
0-0035  units. 

This  phenomenon  has  been  thus  explained.  In  the  aqueous  solution  of 
hydrochloric  acid,  the  HC1  molecules  are  already  broken  up  into  H  and  Cl 
atoms  or  ions ;  these  atoms  are  permanently  charged,  the  hydrogen  posi- 
tively and  the  chlorine  negatively,  each  with  definite  quantities  of  elec- 
tricity. Then,  whatever  may  turn  out  to  be  the  cause  of  chemical  affinity, 
there  is  no  reason  to  doubt  that  chlorine  atoms  are  more  attracted  by  zinc 
than  hydrogen  atoms  are.  On  the  whole,  there  is  a  certain  amount  of 
combination  between  the  chlorine  and  the  zinc,  with  production  of  zinc 
chloride ;  but  the  negative  charge  of  the  chlorine  ions  is  communicated  to 
the  remainder  of  the  metallic  zinc,  which  thus  acquires  a  negative  electrifi- 
cation. There  thus  arrives  an  ultimate  condition  of  equilibrium,  in  which 
the  electrified  zinc  repels  the  similarly-charged  chlorine  atoms  as  much  as  it 
attracts  them  chemically.  Whether  this  be  a  correct  explanation  or  not,  it 
seems  clear  that  all  chemical  action  ceases  (if  the  zinc  be  perfectly  pure  and 
homogeneous)  when  the  zinc  is  at  a  potential  lower  than  that  of  the  acid 
by  0'0060  electrostatic  units.  Similarly,  the  copper  is  at  a  negative  poten- 
tial, and  for  the  same  reasons ;  but  the  chemical  affinity  is  less,  and  equi- 
librium is  reached  when  the  potential  of  the  coppsr  is  0-0035  units  below 
that  of  the  hydrochloric  acid. 


xvi.]  GALVANIC   CURRENT.  615 

It  has  been  considered  possible  that  a  similar  charging  of  metals  by 
dissociated  atoms  of  oxygen  may  account  for  the  phenomena  observed  in  air. 

The  acid  itself  is  at  zero  potential.  The  separation  of  elec- 
tricities takes  place  across  a  thin  film  of  liquid  between  the 
mass  of  the  metal  and  the  bulk  of  the  liquid;  and  electrostatic 
equilibrium  is  attained. 

Now  connect  the  zinc  and  the  copper  by  a  thick  wire  which 
offers  little  or  no  resistance  or  obstruction  to  the  equalisation 
of  potential  between  the  copper  and  the  zinc,  and  which  passes 
out  into  the  surrounding  air,  and  back ;  the  potential  in  the 
whole  connected  metal  becomes  approximately  uniform.  But 
when  this  happens,  the  circumstances  are  precisely  analogous 
to  those  presented  when  copper  and  zinc  are  brought  into  con- 
tact in  air ;  the  acid  or  electrolyte  here  tends  to  become  a  Field 
of  Force,  the  extremities  of  which  are  (in  the  instance  sup- 
posed) at  potentials  about  +0-00125  and  —0-00125  respec- 
tively. The  whole  potential-slope  now  tends  to  exist  within 
the  electrolytic  Field  of  Force,  that  is,  within  the  acid. 

If,  now,  the  electrolyte  had  been  a  dielectric  or  insulator, 
the  whole  arrangement  would  have  attained  a  condition  of 
electrostatic  equilibrium,  and  have  maintained  that ;  but  this  is 
not  the  case.  The  insulation  of  the  electrolyte  breaks  down ; 
but  its  Field  of  Force  tends  to  be  continually  restored  at  the 
expense  of  the  energy  of  chemical  combination.  This  tendency 
to  continual  breaking-down  and  re-formation  of  the  Field  of 
Force  results  in  the  continuous  passage  of  an  electric  current 
through  the  acid  in  the  direction  zinc  to  copper,  and  along  the 
connecting  wire  in  the  direction  copper  to  zinc ;  this  current 
is  kept  up  until  there  is  either  no  more  hydrochloric  acid  in  solu- 
tion or  no  more  zinc  to  be  dissolved  ;  and  the  total  Energy  of  the 
current  is  equal  to  the  Heat  which  would  have  been  evolved  if 
the  zinc  had  been  directly  dissolved  in  the  hydrochloric  acid. 

If  the  copper  had  been  associated  with  platinum  instead  of  zinc,  the 
copper  surface  would  have  been  more  negative  than  the  platinum  surface, 
and  chemical  action  would  have  been  manifest  at  the  copper  surface :  but 
when  copper  is  used  in  conjunction  with  zinc,  the  tendency  to  chemical 
action  at  its  surface  is  reversed  by  the  actual  current,  passing  in  a  direction 
opposed  to  that  of  the  current  which  chemical  action  on  the  copper  would 
itself  tend  to  produce.  So  far  is  this  the  case  that  Energy,  instead  of  being 
given  out  at  the  copper  surface,  is  absorbed  there,  and  appears  at  that 
surface  in  the  form  of  Heat ;  and  this  is,  like  the  rest  of  the  energy  of  the  cur- 
rent, ultimately  derived  from  the  energy  of  chemical  combination  of  the  zinc. 

The  galvanic  cell  and  circuit  is  thus  a  kind  of  engine,  in  which  Energy 
is  absorbed  from  a  Source,  partly  returned  to  a  condenser  or  Sink,  and  partly 
converted  into  the  Energy  of  Electric  Current. 


616  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

If  there  be  any  obstruction  or  Resistance,  or  any  absorption 
of  energy,  between  the  copper  and  the  zinc,  so  that  these  fail  to 
come  readily  to  the  same  potential,  then  the  fall  of  potential  in 
the  electrolyte  is  diminished  to  a  corresponding  extent ;  but  the 
difference  of  potential  between  the  zinc  and  the  acid,  and  that 
between  the  copper  and  the  acid,  remain  unaffected  by  this ;  and 
the  effective  potential-fall  in  the  electrolyte,  plus  that  along  the 
connecting  metal,  is  equal  to  the  difference  between  the  poten- 
tials at  the  copper  and  the  zinc,  when  these  are  in  the  acid  but 
not  in  contact  or  metallic  communication  with  one  another. 

The  potential-falls  and  differences  referred  to  are  slightly  modified  by 
the  minute  rise  and  fall  of  potential  which  occur  at  the  junction  of  dis- 
similar metals,  due  to  the  true  contact-effect. 

Different  metals  have  different  chemical  affinities  for  differ- 
ent chemical  fluids ;  and  consequently  the  amount  and  even  the 
direction  of  the  electromotive  difference  of  potential  within  a 
circuit  of  the  kind  described  depends  not  only  upon  the  nature 
of  the  metals,  but  also  upon  the  nature  of  the  fluid  or  electro- 
lyte employed.  Copper  and  iron  in  dilute  sulphuric  acid  give 
a  current  running  along  the  conducting  wire  from  copper  to 
iron,  and  the  iron  is  attacked,  not  the  copper:  in  a  solution  of 
sulphide  of  potassium  the  copper  is  attacked,  and  the  current 
runs  along  the  wire  from  iron  to  copper.  In  the  presence  of  facts 
of  this  order  the  theory  must  as  yet  be  considered  incomplete, 
for  chemical  affinity  remains  unexplained. 

For  each  liquid  it  is  possible  to  make  up  a  table  of  relative  potentials. 
In  dilute  sulphuric  acid  the  series  is,  commencing  with  the  most  negative  :  — 
Amalgamated  zinc  —  ordinary  zinc  —  cadmium  —  iron — tin  —  lead — alu- 
minium — nickel — antimony — bismuth — copper  —  silver — platinum . 

A  circuit  of  the  kind  just  described  is  a  Galvanic  Circuit. 
In  Fig.  207,  A  is  a  glass  vessel  containing  hydrochloric  acid  in 
n   207  solution ;  Cu  is  a  plate  of  copper,  Zn  a  plate 

of  zinc.  The  two  metals  are  connected  by  a 
wire  of  any  conducting  material :  the  current 
runs  in  the  direction  copper  —  conducting  wire 
—  zinc  —  acid  —  copper.  Excluding  the  con- 
necting wire,  such  an  arrangement  is  called  a 
Galvanic  Element  or  Cell:  while  a  num- 
ber of  such  cells  may  be  arranged  so  as  to 
form  a  Galvanic  Battery  or  Pile. 

The   total    E.M.D.P.    within    a    galvanic 
circuit  or  battery  is  measured  by  the  electro- 


XVI.] 


GALVANIC   CIRCUIT. 


617 


static  difference  of  potential  between  the  free  extremities  of  an 
open  circuit,  with  terminals  of  the  same  metal  at  the  same 
temperature ;  such  an  open  circuit  might  be  obtained  by  cutting 
through  the  conducting  wire  of  Fig.  207  (compare  p.  643). 

Sometimes  the  copper  end  of  a  cell  or  battery  is  said  to  be 
negative,  perhaps  because  copper  itself  is  "electronegative"  to 
zinc  in  contact  with  it  in  air;*  sometimes  it  is  said  to  be 
positive,  because  if  there  be  any  resistance  at  all  between  the 
copper  and  the  zinc  —  and  there  must  always  be  some  —  it  is 
positively  charged  relatively  to  the  zinc  end,  and  because 
the  current  flows  from  it  along  the  wire  to  the  zinc.  The 
reader  will  please  clearly  understand  that  in  this  volume  the 
latter  of  these  expressions  is  employed. 

If  two  equal  galvanic  cells  be  set  against  one  another,  as  in 
Fig.  208,  no  current  is  produced :  the  aggregate  E.M.D.P.  within 
the  entire  circuit  is  equal  to  zero. 


Fig.208. 


00000 


^n 


Cu 


\         1 

- 

"  *.  

. 

Fig.209. 

— 

Zn         Cu 

Zn 


If  there  be  the  slightest  difference  between  the  chemical  constitution 
of  the  two  liquids  in  the  two  cells,  a  feeble  current  will  pass.  (Gore's 
Voltaic  Balance.) 

If  a  cell  containing  copper  and  zinc  in  dilute  sulphuric  acid  be  set  in 
this  way  against  one  containing  platinum  and  zinc  in  the  same  liquid,  the 
E.M.D.P.  is  the  same  as  that  of  a  single  cell  containing  platinum  and  copper 
in  dilute  sulphuric  acid. 

If  two  cells  be  coupled  side  by  side,  copper  to  copper,  zinc 
to  zinc  (i.e.  "in  Surface"  or  "in  Parallel"),  and  if  a 
conducting  wire  be  led  from  any  one  of  the  coppers  to  any  one 
of  the  zincs,  the  whole  acts  like  one  cell  of  double  surface,  and 
the  E.M.D.P.  within  the  circuit  is  not  increased.  So  also  for  n 
cells  so  arranged. 

If  two  cells  be  set  behind  one  another,  as  in  Fig.  209,  copper 
being  connected  with  zinc,  the  difference  of  potential  between 
the  first  copper  and  the  last  zinc  is  twice  as  great  as  that 

*  Also  because  in  the  earliest  forms  of  Volta's  pile  there  was  a  superfluous  zinc 
at  the  copper  end,  and  vice  versa.  The  current  then  flowed  from  the  apparent  zinc 
end  to  the  apparent  copper  end. 


618  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

between  the  copper  and  the  zinc  in  a  single  cell :  or  if  n  cells  be 
arranged  one  behind  the  other  tandem-fashion,  in  Indian  file  or 
uin  Series,"  the  copper  of  each  being  connected  with  the  zinc 
of  the  next  in  regular  succession,  the  effective  difference  of 
potential  is  n  times  that  of  a  single  cell. 

If  the  battery  be  immersed  in  a  conducting  medium,  the  electricity 
escapes  by  its  sides  and  ends,  and  establishes  return-paths  through  the 
medium  surrounding  it. 

Principal  Forms  of  Galvanic  Cells  and  Batteries. — These  may  be 
divided  into  two  principal  classes :  (1)  those  which  have  in  each  cell  one 
fluid;  (2)  those  which  have  in  each  cell  two  fluids. 

One-fluid  cells  and  batteries.  —  Copper,  sulphuric  acid  (diluted), 
and  zinc,  form  the  most  commonly  used  triad  of  materials.  Volta's  pile; 
a  number  of  repetitions  of  the  sequence: — Copper  plate,  cloth  dipped  in 
water  or  acid,  zinc  plate :  the  terminal  copper  is  positive,  the  terminal  zinc 
negative.  Volta's  corona  di  tazze:  a  number  of  cups  containing 
dilute  sulphuric  acid,  in  each  of  which  are  placed  a  plate  of  copper  and  a 
plate  of  zinc,  not  in  contact  with  one  another:  each  copper  is  connected 
with  the  zinc  of  the  preceding  cup.  For  practical  purposes  this  is  made  in 
guttapercha-lined  boxes  divided  into  cells  by  partitions  which  are  themselves 
made  of  copper  on  one  side,  zinc  on  the  other ;  and  to  avoid  spilling,  the 
whole  may  be  filled  up  with  sand  or  stuffed  with  asbestos.  The  form  of  a 
single  cell  may  vary ;  a  cylinder  of  zinc  placed  within  an  open-ended  hollow 
cylinder  of  copper,  but  not  in  contact  with  it,  the  whole  being  immersed  in 
acid :  a  copper  cylinder  within  a  similar  hollow-cylinder  of  zinc  (Oersted)  ; 
a  sheet  of  copper  and  a  sheet  of  zinc  separated  by  flannel,  rolled  up  and 
immersed  in  acid  (Hare's  Deflagrate r);  a  larger  piece  of  copper,  bent 
so  as  to  face  both  sides  of  a  smaller  sheet  of  zinc,  and  thereby  to  diminish 
the  "  resistance  "  within  the  cell  (Wollaston).  In  all  these  cases  the  differ- 
ence of  potentials  between  the  free  extremities  of  an  open  circuit  with  similar 
terminations,  but  containing  a  battery  of  n  cells,  is,  when  we  employ  pure 
copper,  pure  zinc,  and  a  2%  solution  of  pure  sulphuric  acid  in  water,  about 
3-o9oVoow  C.G.S.  electrostatic  units,  or  -921n  "Volts;"  if  dilute  hydro- 
chloric acid  of  the  same  strength  be  used,  about  ^Woo7*  C.G.S.  units  or 
753n  Volts.* 

Instead  of  ordinary  zinc,  zinc  whose  surface  is  amalgamated  may  be 
employed  :  it  is  not  corroded  by  the  acid  except  while  the  current  is  passing, 
and  the  difference  of  potential  within  the  circuit  is  raised  about  -13  Volts 
for  each  cell  of  the  battery.  Ordinary  zinc  wastes  away  when  left  in  acid, 
because  it  is  not  homogeneous  ;  local  differences  of  potential  are  set  up  in  it, 
and  local  circuits  are  formed.  Zinc  may  be  amalgamated  by  setting  it  to 
stand  in  contact  partly  with  mercury,  partly  with  dilute  hydrochloric  acid, 
or  by  rubbing  mercury  into  it  with  a  rag  dipped  in  acid,  or  by  dipping  it  in 
a  liquid  prepared  by  dissolving  200  grms.  Hg  in  a  mixture  of  250  grms.  of 
nitric  and  750  grms.  of  hydrochloric  acid,  and,  when -the  solution  is  clear, 
adding  1000  grms.  of  hydrochloric  acid  ;  or  by  mixing  Hg,  4  %,  with  the 
melted  zinc  on  casting.  Iron  or  platinum  can  be  wetted  by  mercury  con- 
taining sodium. 

*  1  Volt  =  3fcj  C.G.S.  Electrostatic  Unit  of  Difference  of  Potential,  nearly. 


xvi.]  GALVANIC   CELLS. 

The  difference  of  potential  set  up  by  single-fluid  batteries  diminishes 
seriously  when  their  action  is  prolonged,  in  consequence  of  their  so-called 
Polarisation.  Hydrogen  is  liberated  at  the  copper  or  positive  plate,  and 
remains  there  as  a  film ;  this  hydrogen  is  positively  charged,  and  tends  to 
repel  all  other  atoms  of  hydrogen,  and  to  attract  the  negative  components  of 
the  fluid.  In  consequence  of  this  there  is  a  certain  tendency  towards  the 
production  of  a  current  opposed  to  the  main  galvanic  current :  and  if  a 
copper-zinc  couple,  which  has  been  for  some  time  in  action,  be  taken  out  of 
sulphuric  acid  and  immersed  in  water,  a  reverse  current,  comparatively 
feeble,  will  run  for  some  time  from  the  zinc  to  the  copper  through  the  con- 
ducting wire.  In  order  to  minimise  this  polarisation  various  devices  have 
been  resorted  to  :  the  hydrogen  has  been  swept  off  the  positive  plate  by  air 
from  a  bellows  (Grenet),  or  by  shaking  the  cell,  or  by  rapidly  rotating  the 
positive  plate  in  the  fluid  (Mocenigo) ;  or  it  has  been  removed  by  covering 
the  positive  plate  with  a  film  of  oxide  of  copper,  which  is  reduced  by  the 
hydrogen  (Becquerel),  or  by  covering  the  positive  plate  with  a  film  of  clay 
(Pulvermacher),  or  by  otherwise  roughening  its  surface  so  that  bubbles  of 
hydrogen  may  readily  form  and  rise  ;  this  was  done  by  Poggendorff,  who 
electrolytically  deposited  a  rough  film  of  copper  on  the  positive  copper 
plate,  and  by  Smee,  who  used  a  similarly-platinised  platinum  or  silver  or 
lead  plate  as  the  positive  plate.  Platinised  iron  (Paterson),  amalgamated 
iron  (Miinnich),  and  platinised  charcoal  (Walker),  have  been  recommended 
as  positive  plates.  Bunsen  used  gas  coke  with  dilute  sulphuric  acid  and 
amalgamated  zinc. 

For  the  negative  plate  zinc  is  used,  because  it  is  very  readily  oxidisable, 
convenient,  and  moderately  cheap.  Magnesium  would  give  a  higher  effec- 
tive difference  of  potential,  but  is  too  expensive  :  iron  gives  with  copper  too 
feeble  a  current,  but  may  be  in  some  cases  advantageous  as  compared  with 
the  more  expensive  zinc,  although  to  obtain  a  given  current  by  its  aid  a 
greater  number  of  cells  is  required. 

For  the  intervening  fluid  or  electrolyte,  instead  of  sulphuric  or  hydro- 
chloric acid  other  liquids  may  be  employed,  which  oxidise  the  hydrogen 
liberated  at  the  positive  plate.  Mtric  acid  oxidises  hydrogen,  being  itself 
reduced  to  nitrous  acid :  a  solution  of  iodine  with  iodide  of  potassium  in 
water  (Laurie)  forms  with  it  hydriodic  acid :  chromic  acid  is  reduced  by  the 
hydrogen  to  chromic  oxide  :  instead  of  chromic  acid  a  mixture  of  bichromate 
of  potash  and  sulphuric  acid  may  be  employed,  and  the  reaction  then  is 

3Zn  +  K2Cr2O7  +  7H2SO4  =  3ZnSO4  +  2KCr(SO4)2  +  7H2O 

Chrome-alum. 

A  common  bichromate-cell,  in  which  gas-carbon,  bichromate-mixture  — 
1000  water,  100  K0Cr2O7,  300  pts.  by  wt.  H2SO4  (Grenet)  ;  6-182  grms. 
K2Cr2O7,  6-282  cub.'cm.  strongest  H2SO4,  60-47  cub.  cm.  water  (Bunsen)  ; 
with  "the  addition  (Ducretet)  to  each  litre  of  about  2J  grms.  of  HgSO4  in 
order  to  keep  the  zinc  well  amalgamated  —  and  zinc  are  employed,  gives  a 
difference  of  potential  of  about  2  Volts,  which  remains  fairly  constant  when 
the  circuit  is  closed  for  about  three-quarters  of  an  hour,  but  which  in  an 
hour  and  a  half  sinks  to  about  1  Volt. 

Chloride  of  ammonium,  chloride  of  zinc,  used  as  exciting  fluids,  also  tend 
to  check  polarisation.  In  Leclanche's  cell  the  materials  are  zinc,  a  solution 
of  chloride  of  ammonium,  and  a  positive  plate  :  this  plate  consists  in  the 
older  Leclanche  cells  of  a  mixture  of  moistened  binoxide  of  manganese  and 


620  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

crushed  gas-coke  surrounding  a  central  rod  of  carbon,  all  in  a  porous  pot ; 
and  in  the  newer,  of  a  mixture  of  55  gas-coke  powder,  40  binoxide  of  manga- 
nese, and  5  shellac,  pressed  upon  a  carbon  core.  The  action  is  Zn  +  2NH4C1 
+  2MnO2  =  ZnCl2  +  2NH3  +  H2O  +  Mn2O3.  The  zincs  of  Lecl.anche  bat- 
teries are  very  little  corroded  when  they  are  not  in  use ;  hence  these  cells  are 
much  used  for  occasional  telegraphy  or  bell-ringing;  and  for  therapeutic 
purposes  a  large  number  of  such  elements,  each  the  size  of  a  small  test-tube, 
can  (Beetz)  be  packed  within  a  very  small  space  ;  but  elements  of  this  kind 
cannot  be  used  for  continuous  work,  unless  the  resistance  of  the  circuit  be 
extremely  high,  because  they  polarise  very  rapidly.  In  Gassner's  dry  cell, 
hydrated  ferric  oxide  is  used  as  a  depolariser ;  in  presence  of  the  NH4C1  it 
gives  up  oxygen. 

In  some  other  cells  the  electrolyte  is  somewhat  complex,  and  there  are 
differences  of  potential,  due  to  chemical  action,  set  up  even  within  it.  A 
cell  whose  metals  are  silver  and  zinc,  separated  by  an  intervening  mass  of 
chloride  •  of  silver  moistened  with  or  lying  within  a  solution  of  alkaline 
chloride  (Warren  de  la  Rue),  chloride  of  zinc  (Gaiffe),  or  alkali  (Scrivanoff), 
gives  a  very  constant  and,  relatively  to  the  bulk  of  the  cell,  a  powerful  cur- 
rent. The  difference  of  potential  in  a  silver  —  salt-water-and-chloride-of- 
silver  —  zinc  cell  is  (Warren  de  la  Rue)  1-065  Volts.  Becquerel  used  salt 
water  and  sulphate  of  lead  between  zinc  and  lead. 

Two-fluid  cells.  Daniell's  cell:  —  A  hollow  cylinder  or  cell  of 
porous  ware,  as  thin  as  practicable,  and  containing  dilute  sulphuric  acid 
and  a  rod  of  zinc,  is  surrounded  by  a  saturated  solution  of  sulphate  of  copper 
and  a  larger  cylinder  of  copper.  The  current  runs  through  the  fluid  in  the 
direction  Zn  —  H2SO4  ||  CuSO4  —  Cu  (where  the  symbol  ||  is  used  to  indicate 
the  porous  cell),  and  through  the  conducting  wire  as  usual  from  copper  to 
zinc.  The  chemical  action,  which  may  be  expressed  by  the  diagram 


-    |    Zn     SO4  H2  ||  SO4  Cu     Cu    |    + 

results  in  the  formation  of  sulphate  of  zinc  within  the  porous  cell,  sulphuric 
acid  near  or  within  the  walls  of  the  porous  cell,  and  the  deposition  of  copper 
upon  the  inner  surface  of  the  copper  cylinder.  There  is  thus  no  evolution 
of  hydrogen,  and  no  polarisation.  The  effective  difference  of  potentials  is 
1-124  Volts  when  the  liquids  employed  are  a  neutral  saturated  solution  of 
sulphate  of  zinc  and  a  saturated  solution  of  copper  sulphate,  and  when  the 
zinc  is  amalgamated  and  the  copper  electrolytically  deposited ;  and  this  does 
not  vary  very  much  with  the  strength  of  the  solution  of  sulphate  of  zinc, 
nor  does  it  do  so  to  any  great  extent  though  the  temperature  of  the  cell  rise 
from  3°  to  70°  C. ;  and  it  is  equal  to  almost  exactly  1  Volt  when  the  liquids 
used  are,  the  one  a  solution  of  sulphuric  acid,  1  vol.  to  water  22  vols.,  and 
the  other  a  saturated  solution  of  nitrate  of  copper.  The  "  internal  resist- 
ance "  sinks  (Preece)  to  one-third  when  the  cell  is  heated  to  100°  C.  Bat- 
teries of  this  construction  were  originally  due  to  Becquerel ;  and  they  are 
very  constant,  lasting  even  for  months  if  the  resistance  in  the  circuit  be 
kept  very  great ;  but  if  the  external  resistance  be  small,  as  where  the  copper 
and  the  zinc  are  connected  by  a  short  piece  of  wire,  the  current  produced 
rapidly  falls  off. 

The  solution  of  sulphate  of  copper  is  kept  saturated  by  crystals  placed 
in  it.  Any  metal,  if  "  electronegative  "  to  zinc,  can  be  used  as  a  positive 
plate,  for  it  soon  becomes  covered  with  copper. 

In  some  forms  of  Daniell's  cell  the  porous  cell,  which  is  fragile,  and 


xvi.]  GALVANIC   CELLS.  621 

which  tends  to  have  its  pores  blocked  up,  is  dispensed  with :  in  gravity  bat- 
teries —  e.g.,  Callaud's  —  a  stratum  of  acidulated  water  or  of  a  solution  of 
sulphate  of  zinc  floats  upon  a  denser  solution  of  sulphate  of  copper :  in  the 
former  stratum  the  zinc  is  suspended ;  in  the  latter  the  copper  lies.  Some- 
times, as  in  M  i  n  o  1 1  o '  s  battery,  the  copper  is  protected  by  sand  or  sawdust, 
beneath  which  a  layer  of  copper-sulphate-crystals  rests  upon  the  copper. 

In  Maidinger's  cell  the  crystals  lie  in  a  special  inverted  flask  filled 
with  zinc-sulphate  solution;  the  heavy  solution  in  this  flask  sinks  down 
whenever  the  density  of  the  lowest  layer,  the  solution  of  sulphate  of  copper, 
diminishes  in  consequence  of  the  deposition  of  its  copper  upon  the  positive 
plate. 

These  cells  without  porous  diaphragms  are  liable  to  diffusion  of  the 
copper-sulphate-solution  upwards  into  the  upper  layer,  the  solution  of  sul- 
phate of  zinc :  the  zinc  suspended  in  this  is  attacked,  and  a  film  of  copper 
is  deposited  on  it,  which  interferes  with  the  efficiency  of  the  cell.  Cells  of 
this  kind  are  therefore  good  only  for  frequent  use,  such  as  tends  to  exhaust 
the  layer  of  sulphate  of  copper  solution. 

The  copper  plates  submerged  in  the  lower  layer  of  liquid  are  connected 
with  the  external  circuit  by  wires  passing  down  through  the  whole  liquid, 
and  protected  by  an  insulating  covering  of  guttapercha. 

In  Remak's  portable  form  of  Daniell's  battery,  discs  are  arranged  in 
the  following  sequence :  —  Copper  plate,  cloth  dipped  in  solution  of  copper 
sulphate,  porous  earthenware  disc,  cloth  dipped  in  dilute  sulphuric  acid, 
zinc  plate,  copper  plate,  etc. 

In  Beetz's  dry  Daniell-cell,  which  is  exceedingly  constant  even  when 
the  circuit  is  kept  closed,  a  U-tube  has  one  limb  filled  with  plaster-of-Paris 
or  gelatine  made  up  with  sat.  ZnSO4  soln.  and  containing  a  Zn  wire ;  the 
other  limb  similarly  with  CuSO4  and  Cu  wire.  The  E.M.D.R  is  1-04  Volts. 

In  Grove's  cell  the  current  passes  through  Zn  —  H2SO4  ||  HNO3  —  Pt. 
The  nitric  acid  dissolves  the  hydrogen  liberated  by  the  sulphuric  acid,  and 
is  itself  reduced  to  nitric  peroxide  or  to  nitrous  acid ;  these,  if  not  too  abun- 
dant, are  dissolved  by  the  remaining  nitric  acid.  The  difference  of  poten- 
tial maintained  by  a  Grove's  cell  is  equal  to  about  1-92  Volts.  This  is 
1-708  x  that  of  a  Daniell,  and  the  internal  resistance  of  a  Grove  is  much  less ; 
for  a  short  time,  and  against  a  small  resistance,  a  Grove  can  produce  a  much 
stronger  current  than  a  Daniell  of  the  same  size ;  but  its  fumes  are  unwhole- 
some, noxious  in  a  laboratory,  and  destructive  to  the  binding-screws  of  the 
Grove  cell  itself. 

Grove's  cell,  like  Daniell's.  may  be  made  either  cylindrical  or  flat-plated: 
the  former  is  preferable,  because  cylindrical  porous-cells  are  not  so  liable  to 
break  as  flat  ones. 

The  difference  of  potential  maintained  by  a  Grove  mounts  from  170-8 
to  240  (Daniell  =  100),  when  the  dilute  sulphuric  acid  surrounding  the  zinc 
is  replaced  by  a  concentrated  solution  of  caustic  potash. 

The  nitric  acid  surrounding  the  platinum  is  often  mixed  with  strong 
sulphuric  acid,  which  exercises  a  dehydrating  action,  takes  water  to  itself, 
and  keeps  the  nitric  acid  concentrated. 

Instead  of  platinum,  carbon  may  be  used,  as  in  that  modification  of 
Grove's  cell  known  as  Bunsen's  cell,  originally  due  to  Grove;  or  iron, 
which  becomes  what  the  chemists  call  "  positive,"  and  is  not  dissolved  by 
strong  nitric  acid;  or,  as  in  Callan's  cell,  platinised  lead.  , 

The  nitric  acid  of  Grove's  cell  may  be  replaced  by  bichromate-of-potash- 


622  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

and-sulphuric-acid  mixture.  In  the  place  of  nitric  acid  a  saturated  solution 
of  ferric  chloride,  to  which  4  per  cent  of  nitric  acid  has  been  added,  forms 
an  excellent  liquid :  when  it  is  used,  the  total  difference  of  potential  kept 
up  by  the  cell  is  about  midway  between  that  of  a  Daniell  and  that  of  an 
ordinary  Grove  :  this  liquid  is  readily  renovated  by  boiling  it  with  a  little 
nitric  and  hydrochloric  acid.  In  Fuller's  cell  the  components  are  Zn  — 
H2SO4  dil.  ||  bichromate  solution  —  carbon;  the  zinc  rod  stands  in  mer- 
cury, which  creeps  up  the  zinc  and  keeps  it  amalgamated. 

In  Marie-Davy's  cell  the  current  runs  through  Zn  —  pure  water  ||  paste 
of  Hg2SO4  with  water  —  carbon.  Any  mercury  passing  through  the  porous 
cell  merely  amalgamates  the  zinc  and  does  no  harm.  Polarisation  is  great 
in  this  cell,  but  it  is  very  convenient  because  very  portable. 

In  Latimer  Clark's  Standard  Cell  the  current  runs  through  Zn  — 
pure  concent.  ZnSO4  soln.  ||  Hg2SO4-and-water-paste  —  Hg.  This  cell  is  very 
constant,  and  its  difference  of  potential  =  1-434  Volts  at  15°  C.  It  may  be 
used  as  a  standard  for  comparative  electrostatic  measurements  of  difference 
of  potential,  but  it  is  greatly  lacking  in  constancy  if  it  be  allowed  to  send  a 
sensible  current.  Such  cells  differ  among  themselves  by  about  ±  0-2  per 
cent,  mainly  because  some  cells  take  longer  than  others  to  reach  their  con- 
dition of  equilibrium.  If  the  paste  be  made  of  subsulphate  of  mercury  and 
a  concentrated  solution  of  sulphate  of  zinc,  the  constancy  is  even  greater, 
and  the  difference  of  potential  is  1-4455  Volts. 

The  greatest  difference  of  potential  yet  observed  as  having  been  pro- 
duced and  maintained  by  a  single  cell  is  that  of  a  combination  devised  by 
Goodman  in  1847.  The  current  in  this  runs  successively  through  potas- 
sium-amalgam —  inner  porous  cell  —  solution  of  caustic  potash  —  outer 
porous  cell  —  solution  of  permanganate  of  potash  —  and  lastly,  as  a  posi- 
tive plate,  stick-sulphur.  The  difference  of  potential  is  (Beetz)  302-3  (Dan- 
iel! =  100). 

Two  fluids  or  melted  substances  separated  by  a  porous  diaphragm  will 
give  a  current  even  though  plates  of  the  same  metal  be  immersed  in  both. 
Iron  in  nitric  acid  and  iron  in  sulphuric  acid  (Grove),  or  copper  in  dilute 
sulphuric  acid  and  copper  in  dilute  nitric  acid  (Napoleon  III.),  aluminium  in 
dilute  caustic  soda  and  aluminium  in  dilute  hydrochloric  acid  (Wohler),  or 
platinum  in  caustic-potash-solution  and  platinum  in  nitric  acid  (Becquerel), 
will  give  a  current,  and  as  the  one  metallic  plate  is  dissolved  away  the  other 
is  thickened.  In  a  flask  set  aside  and  containing  a  lower  layer  of  solution 
of  sulphate  of  copper  and  an  upper  layer  of  acidulated  water,  together  with 
a  copper  wire  set  to  stand  in  the  liquid,  it  will  be  found  that  the  part  of  the 
copper  wire  which  is  within  the  acidulated  water  becomes  thinned  away, 
while  that  part  which  is  within  the  solution  of  sulphate  of  copper  becomes 
thickened.  Further,  two  plates  of  the  same  metal,  immersed  in  acids  or 
alkalies  of  different  degrees  of  concentration,  will  give  a  current  which,  in 
the  case  of  sulphuric  and  hydrochloric  acids,  flows  from  the  stronger  through 
the  porous  diaphragm  into  the  weaker  acid,  but  which,  in  the  case  of  caustic 
alkalies,  flows  towards  the  stronger  solution. 

In  Shelford  Bidwell's  cell  the  materials  are  sulphurised  silver  —  com- 
pressed Ag2S  —  compressed  CuS  —  copper.  The  sulphides  of  silver  and  cop- 
per, though  solid,  conduct  electrolytically.  The  E.M.D.P.  is  0-053  Volt,  and 
the  resistance  about  7  Ohms. 

Dry  piles  may  be  constructed  as  follows  :  —  Pieces  of  "  gold  paper  "  and 
of  "  silver  paper  "  may  be  pasted  back  to  back  and  cut  into  small  discs : 


xvi.]  DRY   PILES.  623 

these  discs  are  then  piled  up  and  pressed  into  a  glass  tube,  or,  better,  strung 
upon  a  silk  thread,  their  similar  faces  all  looking  in  the  same  direction. 
Such  a  pile  developes  a  considerable  difference  between  the  potentials  of  its 
extremities,  and  it  remains  thus  charged  for  apparently  indefinite  periods. 
In  principle  a  dry  pile  resembles  a  Volta's  pile,  the  discs  of  wet  cloth  in 
which  have  almost  dried  up.  Paper  is  never  perfectly  dry :  the  paper 
between  the  metallic  faces  of  each  disc  takes  the  place  of  the  moist  discs  of 
cloth ;  and  besides  this,  the  air  acts  more  on  the  one  metallic  face  of  each 
disc  than  on  the  other.  In  consequence  the  chemical  action  is  not  nil;  and 
a  definite  difference  of  potential  is  set  up,  by  which  chemical  change,  other- 
wise too  feeble  to  be  detected  within  any  reasonably  short  period  of  time,  is 
rendered  strikingly  manifest.  The  quantity  of  energy  liberated  by  a  dry 
pile  is  very  small,  and  little  work  can  be  done  by  it ;  but  one  extremity  of  a 
dry  pile  can  keep  a  charged  gold-leaf  steadily  repelled  for  a  long  time.  If 
the  two  ends  or  poles  of  a  dry  pile  be  brought  near  one  another,  an  insulated 
strip  of  gold  leaf  suspended  between  them  and  alternately  attracted  by, 
coming  in  contact  with,  and  repelled  from,  each  pole,  may  oscillate  between 
the  poles  for  a  very  long  time,  but  only  so  long  as  the  chemical  decomposi- 
tions going  on  within  the  pile  can  furnish  the  energy  requisite  to  overcome 
the  friction  of  the  air  and  the  small  rigidity  of  the  gold  leaf. 

Difference  of  potentials  is  the  most  delicate  test  that  we 
possess  for  chemical  action. 

The  chemical  action  set  up  under  the  influence  of  actinic 
rays  also  produces  difference  of  potential,  which  may  serve 
(Becquerel)  to  measure  the  chemical  energy  of  sunlight. 

Difference  of  potential  is  also  produced  by  friction  of 
water  against  steam  or  air,  as  where  a  jet  of  partly-con- 
densed steam  or  of  suddenly-expanding  un dried  air  is  driven 
through  a  conical  nozzle  of  metal  or  glass  or  wood :  the  steam 
or  air  becomes  positively,  the  vessel  from  which  it  is  driven 
becomes  negatively  charged.  If  the  nozzle  be  of  ivory  there  is 
no  charge.  If  the  vessel  contain  some  turpentine-oil  the  charges 
are  reversed. 

When  a  liquid  is  brought  into  the  spheroidal  state  it  assumes 
an  electrical  condition,  which  varies  with  the  nature  of  the  liquid 
and  with  that  of  the  hot  surface  on  which  it  lies. 

When  saline  solutions  are  evaporated,  the  vapour  and  the 
liquid  assume  different  electrical  conditions  if  there  be  either 
friction  of  the  crystals  on  the  vessel,  as  when  the  crystals  crackle, 
or  friction  of  the  heated  water  upon  the  salt.  In  the  evaporation 
of  water  there  is  no  difference  of  potential  set  up  unless  there  be 
friction  between  the  water  and  the  vapour :  if  there  be  friction, 
the  steam  becomes  positively  charged. 

Pressure  or  traction  applied  to  tourmaline  crystals,  if  the 
force  applied  have  a  component  parallel  to  the  crystallographic 


624  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

axis,  causes  a  separation  of  electricities ;  opposite  extremities  of 
the  axis  become  oppositely  electrified,  and  the  amount  of  differ- 
ence of  potential  produced  depends  only  on  the  amount  of  force 
applied.  The  same  result  follows,  whether  the  alteration  of 
form  of  the  crystal  affected  be  the  result  of  force  or  of  the  appli- 
cation of  heat  or  cold. 

Such  crystals  behave  as  if  they  were  always  in  a  state  of  electric  stress 
interiorly,  with  a  neutralising  surface-charge ;  then  on  altering  the  interior 
stress  by  any  means,  the  surface-charge  no  longer  neutralises  the  stress,  and 
the  electrification  becomes  apparent. 

Electro-capillarity.  —  Mercury  standing  under  water  has  a  convex 
surface  and  a  definite  surface-tension.  The  water  and  the  mercury  are  at 
different  potentials.  The  surface  of  the  water  and  the  surface  of  the  mer- 
cury, though  nominally  in  contact,  are  at  a  mean  distance  of  about  one 
twenty-millionth  of  a  centimetre.  The  two  surfaces  therefore  act  as  an 
accumulator  which  has  a  definite  capacity.  The  surface  of  contact  between 
mercury  and  water  has  thus  three  properties  —  Surf  ace-Tension,  Difference 
of  Potential,  and  Electrostatic  Capacity ;  and  these  depend  upon  one  another, 
so  that  if  one  be  varied  the  other  two  will  vary  (Lippmann).  Thus  if  we 
vary  the  surface-curvature  of  mercury,  as  by  setting  it  in  vibration  in  a 
conical  tube,  and  thus  altering  the  area  and  the  amount  of  tension  of  the 
surface ;  or  if  we  heat  the  mercury  or  the  water  and  thus  again  alter  the 
surface-tension,  the  capacity  and  the  difference  of  potential  will  also  vary. 
Part  of  the  work  done  upon  the  mercury  in  setting  it  in  vibration,  or  of  the 
heat  supplied,  is  spent  in  setting  up  a  difference  of  potential,  the  very  exist- 
ence of  which  causes  a  tendency  to  restitution  of  the  original  surface-tension  ; 
for  if  we  vary  the  difference  between  the  potentials  of  the  water  and  the 
mercury  by  charging  either  the  one  or  the  other,  the  surface-tension,  and 
consequently  the  surface-form,  of  the  mercury  varies  also. 

Thermo-electricity.  —  The  difference  of  potential  set  up 
between  two  metals  by  their  mere  contact  —  that  is,  the  true 
contact-effect — depends  upon  their  temperature  as  well  as  upon 
their  chemical  nature  and  state  of  purity  or  their  physical  state 
—  their  hardness,  their  tension,  and  so  forth.  If  bismuth  and 
antimony  (in  the  form  of  commercial  pressed  wire)  develope,  on 
contact  at  19°  C.,  a  difference  of  potential  of  V  Volts,  the  same 
materials  develope,  on  contact  at  20°  C.,  a  difference  of  potential 
of  ( V—  -000103)  Volts.  If  a  semicircle  of  bismuth  wire  and 
one  of  antimony  wire  be  joined  so  as  to  form  a  circle,  and  if 
one  of  the  two  junctions  be  maintained  at  19°  C.,  while  the 
other  is  kept  at  20°  C.,  then,  since  the  colder  junction  presents 
a  greater  difference  of  potential  than  the  hotter,  the  aggre- 
gate difference  of  potential  within  the  circuit  is  not  zero,  but 
is  equal  to  \  V-  (V -  -000103) |  Volts  =  -000103  Volts,  or  103 
microvolts  (millionths  of  a  Volt).  This  difference  of  potential 


xvi.]  THERMO-ELECTRICITY.  625 

within  the  circuit  is  maintained  as  an  electromotive  difference 
of  potential,  and  there  is  therefore  a  constant  current  round  the 
circuit,  so  long  as  the  junctions  are  kept  at  these  fixed  tempera- 
tures ;  and  the  energy  of  this  current  is  partly  (but  in  part 
only)  derived  from  the  heat  supplied  at  the  hotter  thermo-elec- 
tric junction,  and  is  due  to  transformation  of  the  energy  of 
molecular  motion.  Across  the  colder  junction  the  current  runs 
from  the  antimony  to  the  bismuth :  but  across  the  hotter  junc- 
tion it  runs  from  bismuth  to  antimony. 

The  bismuth  is  accordingly  said  to  be  " thermo-electrically  positive"  to 
the  antimony,  though  it  is  really  negative  to  it  at  both  junctions. 

The  D.P.  at  the  colder  junction  is  only  partly  neutralised  by  the  counter 
D.P.  at  the  hotter ;  adifferential  effect  is  produced.  On  the  whole,  the 
cold  junction  is  analogous  to  a  galvanic  cell,  and  the  remainder  of  the  circuit, 
including  the  hot  junction,  to  the  wire  connecting  the  terminals  of  that 
cell ;  but,  as  will  be  seen  later,  the  current  is  kept  up  by  means  of  energy 
absorbed,  as  Heat,  both  at  the  hot  junction  and  in  the  circuit  itself. 

Antimony  and  bismuth  are  the  extremes  of  a  thermo-electric  series, 
which  is  at  ordinary  temperatures,  according  to  Becquerel,  the  following :  — 
Bi,  Pt,  Pb,  Sn,  Cu,  Au,  Ag,  Zn,  Fe,  Sb. 

The  electromotive  difference  of  potential  produced  and 
maintained  within  the  closed  circuit  is  approximately  propor- 
tional to  the  difference  between  the  temperatures  of  the  two 
junctions,  if  this  difference  be  very  small ;  and  it  is  therefore, 
when  measured  in  microvolts,  equal  to  the  product  of  the  differ- 
ence of  temperatures  into  a  Number.  When  the  E.M.D.P.  is 
measured  in  microvolts  (one  microvolt  being  Ii00  *000  Volt  or 
300,000000  electrostatic  unit  of  D.P.),  this  number  is  called  the 
thermo-electric  power  between  the  two  given  metals  at  the 
given  mean  temperature.  For  Bismuth  and  Antimony,  at  a 
mean  temperature  of  19J6  C.,  it  is  103 ;  for  E.M.D.P.  =  103 
microvolts  =  103  x  (20°  C.  -  19°  C.).  If  the  E.M.D.P.  be  meas- 
ured in  C.G.S.  electromagnetic  units,  of  which  100  make  a 
microvolt,  the  thermo-electric  power  in  this  case  is  10,300. 

The  thermo-electric  power  between  any  two  metals  is  not  a 
constant  number,  but  varies  with  the  temperature.  In  Fig.  210 
it  may  be  seen  that  near  the  freezing-point  of  water  a  difference 
of  one  degree  between  the  temperatures  of  two  junctions  of  a 
lead-iron  circuit  makes  between  the  two  junctions  a  potential- 
difference  of  17-34  microvolts,  or  1734  electromagnetic  units, 
while  at  higher  mean-temperatures  the  thermo-electric  power  is 
progressively  less,  becomes  m7,  and  ultimately  changes  its  sense. 
The  thermo-electric  power  between  copper  and  Lead,  on  the 
other  hand,  increases. 

2s 


626 


ELECTRICITY  AND   MAGNETISM. 


[CHAP. 


A  diagram  of  this  kind  is  called  a  Thermo-electric  Diagram, 
and  indicates  the  Thermo-electric  Power  between  its  met- 
als at  any  mean  temperature  within  its  range. 

The  lines  of  iron  and  copper  cross  one  another  at  274°-5  C. 
An  iron-copper  couple,  one  of  whose  junctions  is  at  a  tempera- 


Fig.210. 

200°C  26 


400°C 


Lead 


563-5 


ture  slightly  over,  the  other  at  a  temperature  equally  under 
274°-5,  will  develope  within  its  circuit  no  current.  That  mean 
temperature,  274°-5  C.,  is  for  iron  and  copper  the  so-called 
neutral  point. 

If  one  copper-iron  junction  be  at  150°  C.,  at  what  temperature  must  the 
other  be  in  order  that  there  may  be  no  current  ?  The  temperature  required 
is  399°  C.,  which  lies  as  far  beyond  the  neutral  point,  274°-5  C.,  as  274°-5 
does  beyond  150°  C.  The  triangle  ABN  (Fig.  210)  represents  the  total 
E.M.D.P.  when  the  junctions  are  respectively  at  150°  C.  and  274°-5  C. ;  *  the 
triangle  NA'B'  represents  the  total  and  opposite  E.M.D.P.  which  would  be 
developed  if  the  junctions  were  at  274°-5  and  399°  respectively :  these  tri- 
angles are  equal :  their  sum  is  nil:  the  total  electromotive  potential-difference 
between  150°  and  399°  is  nil:  there  is  consequently  no  current. 

If  one  copper-iron  junction  be  maintained  at  the  constant 
temperature  of  100°  C.,  and  the  other  be  successively  exposed 
to  temperatures  101°,  102°,  103°,  and  so  forth,  each  step  in  the 
temperature  of  the  hotter  junction  produces  an  increment  of  the 
effective  E.M.D.P.  within  the  circuit;  but  each  successive 
increment  is  smaller  than  its  predecessor :  as  the  temperature 
of  the  hotter  junction  nears  274°-5,  the  successive  increments  of 

*  Since,  for  a  small  difference  of  temperature,  E.M.D.P-.  =  Therm.-elect.  power 
X  diffce.  of  temp,  measured  in  °C.,  each  step  in  temperature  multiplied  by  its  corre- 
sponding thermo-electric  power  forms  in  the  thermo-electric  diagram  a  small  rectan- 
gle, which  represents  the  E.M.D.P.  developed  by  each  difference  of  temperature  : 
the  sum  of  all  these  rectangles  between  150°  and  274°'5  represents  the  total  E.M. 
difference  of  potential  set  up  when  these  are  the  temperatures  of  the  two  junctions  : 
this  sum  is  equal  to  the  triangle  ABN. 


xvi.]  THERMO-ELECTRICITY.  627 

E.M.D.P.  become  less  and  less :  when  the  hotter  junction  is  at 
274°-5,  the  neutral  point,  the  increment  is  m7,  and  the  electro- 
motive difference  of  potential  and  the  current  which  it  causes  to 
run  round  the  circuit  are  at  their  maximum.  Thereafter,  as  the 
hotter  junction  is  still  more  strongly  heated,  the  E.M.D.P.  at 
first  gradually  and  then  more  rapidly  sinks.  When  at  length 
the  hotter  junction  is  at  449°  (the  colder  one  Still  remaining  at 
100°)  there  is  no  E.M.D.P.,  and  no  current  round  the  circuit: 
and  when  the  temperature  of  the  hotter  junction  exceeds  449°, 
the  direction  of  the  current  is  reversed,  being  now  from  iron 
to  copper  across  the  hotter  junction ;  and  thereafter,  successive 
increasing  differences  of  temperature  develope  successive  numer- 
ically greater  negative  E.M.D.P.'s.  The  Neutral  Point  is  thus 
a  fixed  temperature  for  each  pair  of  metals ;  at  that  tempera- 
ture there  is  no  (true)  contact-effect,  and  the  temperature  of 
the  colder  junction  on  the  one  hand  (whatever  that  temperature 
may  be)  and  the  corresponding  Temperature  of  Reversal  on 
the  other,  are  equidistant  on  either  side  of  it,  so  long  as  the 
lines  in  the  diagram  are  straight,  which  they  are  in  most  cases 
within  pretty  wide  limits. 

Curves  indicating  the  relation  between  the  differences  of  Temperature 
between  two  junctions  and  the  electromotive  differences  of  Potential  devel- 
oped in  consequence  of  them  (sometimes  called  Gaugain's  curves), 


Fig.211. 
OL 

q 

aj 

u 


f^1CO"C  200°C  300°C  400"C 

Temperature 

have  a  form  which,  for  most  pairs  of  metals,  is  that  of  a  parabola :  and  the 
numerical  value  of  the  tangent  of  the  angle  made  by  this  curve  with  a  line 
parallel  to  the  axis  of  a;,  and  cutting  the  curve  at  that  point  of  it  which 
corresponds  to  any  given  temperature,  x°  C.,  is  a  numerical  measure  of  the 
thermo-electric  power  at  that  mean-temperature :  for  both  the  tangent  and 
the  thermo-electric  power  are  numerically  equal  to  the  fraction 

Increment  of  E.M.D.P. Change  of  ordinate 

Increment  of  mean  temperature  Change  of  abscissa 
Even  within  one  and  the  same  bar,  except,  apparently,  in  lead,  differ- 
ences of  potential  are  set  up  when  a  bar  is  unequally  heated,  and  some  of 
the  heat  supplied  is  expended  in  setting  up  this  electrically-stressed  condi- 
tion; but  in  a  homogeneous  metallic  ring,  however  irregularly  heated  it 
may  be,  there  is  no  current.  The  metal  on  either  side  of  a  hot  or  cold  June- 


628  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

tion  of  two  metals  is,  on  the  other  hand,  like  a  single  bar,  and  differences 
of  potential  are  set  up  within  it ;  these  may  modify  the  amount  of  the  effec- 
tive difference  of  potential  within  the  whole  circuit,  and  are  found  to  sup- 
ply an  explanation  of  the  phenomena  of  inversion  (p.  651). 

Metals  interpolated  in  the  circuit  produce  no  effect  on  the 
amount  of  the  effective  difference  of  potential  within  the  circuit, 
unless,  indeed,  their  junctions  be  at  different  temperatures.  If 
that  be  the  case,  their  thermo-electric  effects  form  a  part  of  the 
general  thermo-electric  effect  of  the  circuit. 

When  a  number  of  pieces  of  bismuth  and  antimony  are 
arranged  end  to  end,  their  alternate  junctions  being  hotter 
and  colder  respectively,  the  E.M.D.P.  maintained  between  the 
extremities  of  a  pile  of  n  pairs  of  elements  is  n  times  that  found 
to  be  due  to  one  such  pair. 

This  principle  is  utilised  in  the  Thermo-electric  P i  1  e,  which  con- 
sists of  a  number  of  pieces  of  bismuth  and  antimony  (or,  better,  of  an  alloy 
Fig.  213  of  10  parts  by  wt.  of  bismuth  and  1  of  anti- 

H  H  H  H  H  mony>  an(i  an  alloy  of  15  parts  of  antimony 

and  7  of  cadmium),  arranged  after  the  fash- 
ion of  Fig.  212.  When  the  face  of  this  pile 
marked  HHH  is  exposed  to  heat,  the  junc- 
tions H,  H,  H  become  warmer  than  the  junc- 
tions C,  C,  C,  and  a  current  passes  through 
the  circuit  O  from  the  antimony  to  the  bis- 
muth terminal.  If  it  be  placed  opposite  a 
piece  of  ice,  the  face  HHH  will  cool  itself  by  radiation,  and  the  current  is 
now  in  the  reverse  direction. 

When  a  junction  of  two  metals  is  connected,  by  a  pair  of  copper  wires, 
with  a  similar  junction  at  the  same  temperature,  no  current  passes,  whatever 
be  the  length  of  the  intervening  cable  or  its  local  variations  of  temperature ; 
but  if  one  of  the  junctions  assume  a  different  temperature  from  the  other, 
then  a  current  passes,  and  the  temperature  of  the  distant  junction  may  be 
inferred  from  the  strength  of  the  current  which  passes,  this  being  measured 
by  a  galvanometer,  or  directly  determined  by  heating  or  cooling  the  similar 
junction  situated  under  the  observer's  control  until  its  temperature  becomes 
the  same  as  that  of  the  distant  one :  this  is  known  to  have  occurred  when 
the  current  through  the  galvanometer  ceases.  A  couple  of  junctions  of  this 


Fig.213. 


kind,  with  an  intervening  double  wire  and  galvanometer,  form  a  differen- 
tial thermometer,  the  indications  of  which  must  be  interpreted  with  refer- 
ence to  the  thermo-electric  diagram  of  the  two  metals  iised. 

As  sources  of  electricity,  thermo-electric  piles  are  not  much  in  use. 
BecquerePs  thermo-electric  piles,  made  of  thirty  pairs  of  blocks  or  rods 
of  artificial  sulphide  of  copper  (which  fuses  only  at  about  1000°  C. )  and  of 


xvi.]  THERMO-ELECTRIC 

German-silver,  can  decompose  .water  when  the  differences  of  temperature 
employed  are  from  250°  to  300°.  In  Rebicek's  form  of  Noe's  thermo-elec- 
tric pile,  twenty-five  pairs  of  plates  of  German-silver  and  of  an  alloy  of  zinc 
and  antimony  are  ranged  round  a  Bunsen  gas-burner:  each  such  pile 
maintains  an  effective  difference  of  potential  of  from  2  to  2-75  Volts,  so  long 
as  the  Bunsen  burner  is  kept  lighted,  while  the  internal  resistance  is  0-75 
Ohms.  In  (Diamond's  pile,  about  6000  couples  of  iron  and  of  bismuth- 
antimony  alloy  are  ranged  round  a  coke  fire,  and  the  E.M.D.P.  produced  is, 
if  the  couples  be  arranged  in  file,  about  218  Volts.  The  disadvantages  of 
thermic  piles  as  sources  of  electricity  are  that,  in  general,  the  E.M.D.P.  pro- 
duced is  so  extremely  small  that  moderately-slight  external  resistances  make 
the  current  extremely  weak,  and  that  it  is  difficult  to  keep  the  cold  junction 
cool ;  and  even  in  Clamond's  pile,  which  is  able  to  keep  a  pair  of  electric 
arc-lamps  in  action,  about  95  or  96  per  cent  of  the  heat  of  the  fire  is  not 
converted  into  the  energy  of  a  current,  and  is  thereby  practically  wasted. 
For  many  purposes,  such  as  electroplating  on  the  small  scale,  ]SToe's  bat- 
teries, three  of  which  produce  an  E.M.D.P.  nearly  equal  to  that  produced 
by  seven  DanielFs  cells,  are  very  useful,  for  when  they  are  once  built  up 
their  current  can  be  produced  or  arrested  at  will.  An  arrangement  like 
that  of  Fig.  213  has  been  used  as  a  self-acting  source  of  electrical  currents, 
and  therefore  of  energy,  sufficient  to  maintain  in  action  a  self-winding  clock. 

The  most  important  source  of  electricity  is  the  transform- 
ation of  the  energy  of  work  into  that  of  electrical  separation 
by  means  of  magneto-electric  and  dynamo-electric 
machines,  the  action  of  which  will  be  explained  in  the  sequel. 

Atmospheric  Electricity.  —  The  atmosphere  in  different  regions  is 
often  found  to  be  at  different  local  potentials,  which  differ  from  that  of  the 
earth  sometimes  even  by  as  much  as  3000  Volts  within  100  feet.  This  is 
possibly  (Tait)  due  to  a  contact-effect  between  air  and  aqueous  vapour; 
and  it  is  possibly  necessary  that  there  should  be  at  least  traces  of  dust 
present,  as  well  as  water-vapour.  A  conductor  insulated  from  the  earth 
may  be  brought  to  the  same  potential  as  any  point  in  the  air,  by  leading  to 
that  point  a  metallic  wire,  and  by  furnishing  this  exploring  wire  with 
an  extremely  fine  point,  or,  better,  by  fixing  at  its  extremity  a  sponge 
dipped  in  spirit  and  set  on  fire,  or  a  little  cistern  from  which  a  quantity  of 
water  is  allowed  to  drop.  In  the  former  case  the  flame  continuously  con- 
veys masses  of  gas  away  from  the  end  of  the  exploring  wire ;  and  so  long  as 
there  is  any  difference  of  potential  between  the  region  of  the  air  explored 
and  the  conducting  system  of  which  the  exploring  wire  forms  a  part,  there 
will  be  a  current  along  the  wire,  and  finally  the  whole  conducting  system 
will  come  to  the  same  potential  as  the  air  around  the  flame.  Similarly, 
waterdrops,  on  falling  from  an  insulated  cistern,  bring  the  cistern  to  the 
same  potential  as  the  air  around  it:  each  drop,  just  before  falling  off, 
becomes  electrified  with  a  charge  opposite,  while  the  nozzle,  the  cistern, 
and  the  main  mass  of  water  are  electrified  with  a  charge  similar  to  that  of 
the  air  in  the  neighbourhood  of  the  falling  drop.  As  the  drop  is  in  the  act 
of  falling  off,  it  is  attracted  by  the  cistern :  it  is  held  back  as  it  falls :  it 
falls  down  with  less  speed  than  it  would  have  assumed  if  it  had  fallen  from 
an  uninsulated  cistern;  and  when  it  reaches  the  ground  it  produces  less 
heat.  The  energy  of  the  electrification  acquired  by  the  cistern  is  equal  to 
the  missing  kinetic  energy  of  the  falling  drops. 


630  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

In  an  analogous  way  the  air  within  a  room  may  be  strongly  electrified ; 
connect  a  flame  with  the  conductor  of  an  electrical  machine,  and  work  the 
machine :  in  one  minute  a  Holtz  machine  will  raise  the  potential  of  the  air 
of  a  room  by  2000  Volts.  Combustion  alone  will  effect  electrification  of  the 
air,  which  is  negatively  charged  by  burning  coal-gas,  positively  by  burning 
charcoal. 

When  difference  of  potential  has  once  been  produced,  it 
may  be  turned  to  account  for  the  conversion  of  Work  into 
Electrical  Energy. 

Charge  a  plate,  whose  free  capacity  is  C,  to  potential  V ;  its  charge  will 
be  Q  =  CV.  Bring  up  to  it  a  second  plate,  parallel  and  at  a  distance  d; 
the  two  plates  now  form  a  condenser.  The  capacity  of  this  condenser  is 
C'  =  K  x  surface/47rc?  (p.  599).  Connect  the  second  plate  to  earth.  Its 
potential  becomes  zero,  and  on  its  outer  surface  it  has  no  charge.  The 
capacity  of  each  plate  for  "  free  charge  "  may  be  reckoned  as  having  been 
reduced  to  £C  by  the  mutual  approach,  for  lines  of  force  can  now  only 
pass  from  one  face  of  each  plate  towards  surrounding  objects.  The  poten- 
tial of  the  inducing  plate  will  now  be  V  =  CV/C'  +  |C,  which  is  less 
than  V.  The  original  charge  CV  on  the  inducing  plate  is  now  divided  into 
two  parts.  Of  these,  one  is  "free  charge  "  =  |C2V/C'  +  -£-C;  the  other 
is  "bound  charge,"  C'CV/C'  +  ^C,  which  faces  an  equal  and  opposite 
charge  on  the  second  plate  of  the  condenser.  Now  insulate  the  second 
plate  and  remove  it  to  a  distance  d' :  the  capacity  of  the  condenser  decreases 
in  the  ratio  d/d' ;  then  as  the  capacity  of  the  condenser  decreases,  the 
potential  of  the  second  plate  tends  to  fall  from  zero  to  —  VC'/C'  +  -£C, 
while  that  of  the  first  tends  to  return  to  V. 

In  the  case  of  conducting  plates  in  free  air,  the  negative  potential 
imparted  to  the  second  plate  by  this  method  cannot,  therefore,  become 
numerically  quite  equal  to  the  original  potential  of  the  first  plate;  but  in 
the  Electrophorus,  next  to  be  considered,  the  charge  cannot  travel  along  the 
surface  of  the  original  charged  plate.  In  that  case,  the  whole  charge  remains 
"bound";  and  since  E  =  47rQe?/A,  and  Q  is  constant,  then  on  increasing  d, 
the  potential-difference  will  rise. 

Difference  of  potential  may  also  be  increased  by  the  expen- 
diture of  Work,  with  the  assistance  of  induction  (Holtz,  Voss, 
Wimshurst). 

The  work  is  done  in  stretching  the  Field  of  Force  against  the  mechanical 
traction  t  across  the  field ;  t  =  E2/87re?2,  in  dynes  per  sq.  cm.,  where  E  is  the 
rise  in  potential  occasioned  by  pulling  the  plates  apart  through  a  distance  d ; 
whence  E  =  d  V87rt. 

The  Electrophorus  consists  of  a  cake  of  resin  or  vulcanite  and  an 
insulated  metallic  plate.  The  former  is  slightly  charged  by  being  rubbed 
with  a  catskin  or  a  dry  silk-handkerchief :  the  metallic  plate  is  then  laid 
upon  it.  The  contact  between  the  two  can  never  be  perfect  at  all  points ; 
practically  there  is  an  intervening  film  of  air  between  the  resin  and  the 
metallic  plate,  and  the  latter  is  charged  by  induction  with  an  attracted  and 
a  repelled  charge.  The  latter  charge  may  be  withdrawn  by  touching  the 
metallic  plate  with  the  finger,  or  by  making  metallic  communication 
between  the  metallic  plate  and  the  earth ;  the  former  remains,  facing  and 


XVI.] 


ELECTROPHOKUS. 


631 


attracted  by  the  original  charge  on  the  resin.  Work  is  now  done  from 
without  in  pulling  the  metallic  plate  away  from  the  resin;  as  the  distance 
between  the  metal  and  the  resin  increases,  the  electrostatic  capacity  of  the 
electrophorus,  considered  as  a  condenser,  diminishes;  the  potential  there- 
fore increases,  both  on  the  metal  and  over  the  resin :  the  knuckle  applied 
to  the  edge  of  the  insulated  metallic  plate  may  now  receive  a  spark.  When 
the  metallic  plate  is  next  laid  on  the  resin  a  new  charge  is  induced  in  it, 
which  may  again  be  withdrawn  in  the  same  way  when  the  plate  is  removed. 
Small  original  charges  may  thus  induce  successive  charges  of  high  potential. 

Sir  William  Thomson's  (Lord  Kelvin's)  Replenishes  —  This 
instrument,  which  is  used  as  a  means  of  keeping  the  Leyden  jar  connected 
with  the  suspended  needle  of  Kelvin's  electrometer  at  a  constant  poten- 
tial, is  sketched  in  the  accompanying  diagram  (Fig.  213 a).  A,  B,  two 
metal  half-cylinders,  insulated  from  one  another ;  C,  D,  two  metallic  plates 
insulated  from  one  another  and  capable  of 
rotation  round  the  axis  O;  E,  F,  an  insulated  Fig.  213  a. 
spring  capable  of  touching  both  C  and  D  when 
they  are  in  the  position  shown  in  the  figure ; 
G,  H,  two  springs  connected  with  A  and  B  and 
capable  of  being  pressed  upon  by  C  and  D  as 
they  rotate.  Start  from  the  position  shown  in 
the  figure.  B  is  positively  charged  by  contact 
with  one  of  the  plates  of  the  Leyden  jar ;  C 
becomes  negatively,  and  D  positively,  charged. 
Rotate  C  to  the  left,  D  to  the  right.  Their 
metallic  connection  with  one  another  is  broken 
and  they  remain  oppositely  charged.  As  they 
pass  G  and  H,  C's  —  charge  escapes  to  A,  and 
D's  +  charge  to  B  ;  and  thereafter  D  and  C 
respectively  acquire  —  and  +  charges,  and  stand 
in  the  former  positions  of  C  and  D.  The  + 
charge  of  the  -j-  plate  of  the  Leyden  jar  may 
thus  by  continuous  rotation  of  CD  be  con- 
tinuously increased.  If,  on  the  other  hand,  C  be 
rotated  to  the  right,  D  to  the  left,  C's  negative 
charge  is  conveyed  to  B,  and  the  positive  charge  of  the  -f-  plate  of  the  con- 
denser may,  by  continuous  rotation  in  this  sense,  be  reduced  to  any  desired 
extent,  or  even  reversed.  The  potential  of  the  Leyden  jar  may  thus  be 
adjusted  to  any  desired  amount,  which  may  be  determined  by  a  subsidiary 
pair  of  plates,  connected  with  the  inner  and  outer  coatings  respectively  and 
separated  by  springs,  coming  to  assume  a  position  at  any  pre-arranged  fixed 
distance  from  one  another. 

Sir  William  Thomson's  (Lord  Kelvin's)  Water-Gravity  Electric 
Machine.  —  In  Fig.  214,  A  and  B  are  two  Leyden  jars,  whose  inner  coatings 
consist  of  sulphuric  acid  and  are  connected  with  the  metal  tubes  C,  F  and 
E,  D  respectively.  C  and  D  are  co-axial :  so  are  E  and  F.  Water  falls  in 
drops  from  the  bifurcated  metal-tube  G,  which,  being  connected  with  the  ordi- 
nary water  supply,  is  in  communication  with  the  earth,  and  is  therefore  at 
zero  potential.  A  small  initial  charge,  consisting  (say)  of  positive  electricity, 
is  imparted  to  one  of  the  Leyden  jars,  say  A.  Water  is  made  te  flow  from 
G  in  streams  so  thin  as  to  break  up  into  drops  within  the  tubes  C  and  E. 
Just  before  these  drops  break  off  from  the  stream,  they  are  by  induction 


632 


ELECTRICITY  AND  MAGNETISM. 


[CHAP. 


Fig.214. 


within  C  charged  with  negative  electricity,  while  the  complementary  posi- 
tive charge  is  conveyed   along   G  to  the  earth.     When   the   drops   have 

become  separate,  they  fall 
down  charged  negatively. 
They  then  fall  upon  a  me- 
tallic funnel  placed  in  the 
tube  D,  and  charge  the  ex- 
terior of  that  tube  nega- 
tively :  this  charge  is  shared 
with  the  Ley  den  jar  B. 
This  Ley  den  jar,  thus  neg- 
atively charged,  by  a  cor- 
responding inductive  action 
causes  the  drops  which  fall 
through  E  to  become  posi- 
tively charged.  When  these 
drops  fall  upon  F  they  in- 
crease the  positive  charge 
of  the  Leyden  jar  A.  Thus 
the  Leyden  jars  A  and  B 
become  more  and  more  highly  charged,  the  one  with  positive,  the  other  with 
negative  electricity  on  its  inner  coat.  The  energy  of  their  electrification  is 
derived  from  the  work  done  by  gravity  upon  the  falling  water ;  and  thus  this 
contrivance  is  an  electrical  machine  worked  by  gravity. 


Earth 


Earth- 


STEADY  ELECTRICAL  CURRENTS. 

If  the  plates  of  a  charged  condenser  be  connected  by  a  wire, 
the  condenser  will  be  discharged ;  the  quantity  Q  of  electricity 
disappears  in  a  time  £,  and  during  that  time  t  certain  phenomena 
occur,  with  waning  vigour,  which  are  spoken  of  as  those  of  a 
Current  of  Electricity.  It  is  as  if  electricity  ran  out  of  a  place 
where  it  was  stored  up  under  a  potential-difference  E,  down  to 
the  ordinary  potential-level  V  =  0  •;  and  as  if  its  path  were  along 
the  wire.  In  truth,  the  phenomenon  is  one  of  the  release  of  the 
Field  of  Force  from  constraint,  and  of  transmission  of  energy 
through  that  field,  not  through  the  connecting  wire.  Still,  with 
this  reservation,  it  is  convenient  to  adhere  to  the  terminology 
according  to  which  such  a  discharge  is  spoken  of  as  a  Current 
in  the  wire. 

If  a  very  large  condenser  be  discharged  through  a  very  long 
and  thin  wire,  the  phenomena  of  Current  may  remain  fairly 
steady  for  any  short  interval  of  time ;  and  if  the  E.M.D.P.  be 
that  between  the  terminals  of  a  galvanic  cell,  these  phenomena 
are,  apart  from  Polarisation,  etc.,  continuously  uniform  or  steady, 
as  if  there  were  a  practically  inexhaustible  reservoir  of  electric 
quantity  to  draw  upon,  so  long  as  the  cell  holds  out. 


xvi.]  STEADY   CURRENTS.  633 

The  Intensity  or  Strength  of  a  current  —  i.e.,  the  Quantity 
of  electricity  which  passes  any  cross-section  of  the  conductor 
during  one  second  of  time  —  depends,  on  the  one  hand,  upon 
the  effective  difference  between  the  potentials  at  different  parts 
of  the  conductor,  and,  on  the  other,  upon  the  nature  of  the  con- 
ductor—  that  is  to  say,  upon  its  size  and  its  substance.  A  long 
or  thin  wire  is  a  worse  conductor  —  has  less  Conductance 
and  offers  more  Resistance — than  a  thick  or  short  one:  a  silver 
wire  conducts  better  than  a  copper  one  of  the  same  size. 

The  Density  of  a  Current,  A,  is  the  quantity  of  electricity  passing  per 
sq.  cm.  of  cross-section  of  the  conductor.  It  is  therefore  equal  to  Intensity 
-r-  Cross-Section  =  I/o. 

The  relation  between  E  the  electromotive  difference  of  poten- 
tial, I  the  Intensity  of  the  current, and  R  the  Resistance  of  a 
uniform  conductor,  is,  when  the  flow  is  steady,  expressed  by  the 
equation,  I  =  E/R.  When  there  are  several  sources  of  difference 
of  potential  within  the  circuit,  or  several  successive  conduc- 
tors, each  of  which  offers  its  own  resistance  to  the  onward  flow 
of  the  current,  the  law  assumes  the  generalised  form  that 
.,  _  £E  _  effective  sum  of  all  the  differences  of  potential  „,  . 

SR  sum  of  all  the  successive  resistances 

is  Ohm's  Law. 

The  Resistance,  as  defined  by  Ohm's  Law,  is  E/I  or  2E/I, 
and  it  must  be  specially  noted,  as  an  experimental  fact,  that  for 
any  given  conductor  or  set  of  conductors,  this  fraction,  once 
found,  remains  almost  absolutely  constant,  whatever  may  be 
the  value  of  E  or  2E,  provided  that  the  temperature  and  the 
structure  of  the  conductor  remain  unchanged.  The  Measure- 
ment of  the  Resistance  of  a  conductor  is  the  experimental  deter- 
mination of  this  fraction. 

Ohm's  Law  may  also  be  written  I  =  E/Z  -4-  R/Z;  E/Z  is  the  "electro- 
motive force  "  or  Potential-Slope  <|> ;  R//  is  the  Resistance  per  linear  centi- 
metre. Also,  if  A  stand  for  current-density,  A  =  E//  -*•  R  —  <J>/R  =  <|>D, 
where  R  and  D  are  the  Resistivity  and  Conductivity,  as  defined  below. 

The  C.G.S.  Electrostatic  Unit  of  Intensity  is  the  intensity  of  a  current 
in  which  one  C.G.S.  electrdstatic  unit  of  quantity  passes  a  given  section  of 
the  conductor  during  one  second.  It  is  the  current  which  passes  when  the 
difference  of  potential  E  =  1  C.G.S.  electrostatic  unit,  and  the  total  resistance 
is  also  R  =  1  C.G.S.  electrostatic  unit  of  resistance. 

The  C.G.S.  Electrostatic  Unit  of  Resistance  is  the  resistance  offered  by 
a  conductor  which,  when  it  is  interposed  between  two  bodies  whose 
potentials  are  maintained  at  a  constant  difference  of  one  C.G.S.  electrostatic 
unit,  allows  one  C.G.S.  Electrostatic  unit  of  Quantity  to  pass  along  it,  per 
second. 


634  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

These  units  are  inconvenient  for  practical  purposes,  and  electricians  use 
as  their  practical  units  certain  fractional  or  integral  multiples  of  these. 

The  Resistance  of  a  uniformly-cylindrical  conductor,  such  as 
a  wire,  depends  upon  three  things :  (1)  its  length  Z,  directly ; 
(2)  its  cross-section  0,  inversely;  (3)  its  Conductivity  D, 
inversely.  It  is  therefore  equal  to  Z/0D  =  R. 

The  Conductance  of  a  conductor  is  the  reciprocal  of  its 
Resistance.  In  a  wire  it  is  therefore  equal  to  ov/L  Ohm's  La\v 
may  therefore  be  written  I  =  DE,  E  =  I/D,  or  D  =  I/E,  where  D 
is  the  Conductance. 

The  C.G.S.  Unit  of  Conductance  is  that  of  a  conductor  of  unit  resist- 
ance. 

The  specific  Conductivity,  D,  of  any  substance  is  a  constant, 
special  to  each  substance,  and  even  found  to  differ  from  sample 
to  sample  of  that  which  is  nominally  the  same  substance.  It 
represents  the  number  of  units  of  electricity  which  can  pass,  per 
second,  between  two  bodies  kept  at  a  constant  potential-difference 
of  one  unit,  when  the  conductor  interposed  between  these  bodies 
has  a  length  of  1  cm.  and  a  cross-section  of  1  sq.  cm.  It  varies 
very  greatly  from  one  substance  to  another. 

The  reciprocal  of  D,  (!/D)  =  R,  the  Resistivity  of  a  substance.  The 
Resistance  of  a  conductor  of  length  I  and  cross-section  o  is  therefore  equal  to 
IR/O  =  R. 

In  the  following  table  the  first  column  of  figures  gives  the 
Resistivities,  the  next  column  the  Conductivities  of  a  certain 
number  of  substances,  in  electrostatic  measure  ;  while  the  third 
column  gives  the  numbers  which  denote  their  Relative  Conduc- 
tivities when  the  conductivity  of  mercury  is  taken  as  a  standard 
and  called  unity.  It  is  very  usual  to  take  the  conductivity  of 
silver  as  a  standard  =  100. 

The  numbers  in  the  following  table  have  (with  the  exception  of  those 
for  the  last  four  substances)  been  calculated  from  the  data  of  the  authorities 
named,  on  the  assumption  that  144521  grammes  of  mercury  (sp.  gr.  13-5955), 
in  the  form  of  a  column  of  uniform  cross-section  (1  sq.  mm.)  and  106-3  cm. 
in  length,  has  a  resistance  equal  to  the  900,000,000000th  part  of  an  electro- 
static unit  of  resistance,  that  is,  equal  to  one  Ohm  or  practical  unit  (see 
p.  711)  of  Resistance. 

An  Ohm-coil  is  a  coil  of  wire  whose  resistance  is  one  Ohm. 

The  conductivity  of  an  Ohm-Coil  is  called  a  Mho. 

If  the  following  table  be  read  without  the  multiplier  or  divisor,  V2,  it 
then  expresses  the  specific  resistivities  and  conductivities  in  another  system 
—  the  Magnetic  or  Electromagnetic  system  of  C.G.S.  units,  from  which  the 
Ohm  and  the  Volt  are  primarily  derived,  the  Ohm  being  109  electromagnetic 
units  of  resistance,  and  the  Volt  108  electromagnetic  units  of  potential- 
difference.  This  system  depends  upon  the  laws  of  Magnetism,  afterwards 
to  be  explained. 


XVI.] 


RESISTIVITIES   AND   CONDUCTIVITIES. 


635 


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636  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

The  conductivity  of  metals  decreases,  that  of  most  bad  con- 
ductors (including  carbon)  increases,  with  their  temperatures: 
a  heated  wire  or  dynamo-electric  machine  increases  the  resist- 
ance in  the  circuit  of  which  it  forms  a  part ;  while  at  the  tem- 
perature of  the  electric  arc,  carbon  appears  to  offer  no  resistance. 
Very  roughly,  and  with  well-marked  exceptions  in  the  cases  of 
iron  and  mercury,  the  resistivity  of  a  pure  metal  is  proportional 
to  its  absolute  temperature :  but  pure  metals  appear  (Dewar) 
to  have  no  resistivity  when  exceedingly  cold,  whereas  in  alloys 
there  is  no  such  result. 

When  metals  melt,  their  conductivities  fall  suddenly.  Alloys 
are  in  general  worse  conductors  than  the  arithmetical  considera- 
tion of  their  percentage  composition  and  the  conductivities  of 
their  component  metals  would  lead  us  to  expect;  but  with 
changes  of  temperature,  their  conductivity  varies  less  than  that 
of  pure  metals  does. 

There  is  a  broad  resemblance  between  the  conductivities  of 
metals  for  electricity  and  for  heat :  the  best  conductors  of  the 
one  are  in  general  the  best  conductors  of  the  other ;  and  in  both 
cases  alloys  offer  a  relatively  high  resistance.  The  series  are, 
however,  not  identical.  The  velocities  of  light  through  films 
of  the  metals  are  also  (Kundt)  closely  related  to  their  conduc- 
tivities. 

Variable  Conductivity.  —  Conductivity  varies  not  only  with  varying 
temperature,  but  also  with  varying  magnetisation,  tension,  torsion,  or  pres- 
sure. It  increases  with  longitudinal  stretching,  diminishes  with  longitu- 
dinal compression  of  a  wire,  and  diminishes  in  iron,  but  increases  in  tin 
and  zinc,  when  the  stress,  being  transverse,  tends  to  widen  the  wire  (Tom- 
linson).  In  powders  or  porous  material,  such  as  metal  filings,  platinum 
sponge,  charcoal,  it  increases  with  the  pressure ;  and  if  the  pressure  vary, 
within  small  limits,  the  variations  of  conductivity  follow  and  are  propor- 
tional to  the  variations  of  pressure.  This  is  the  principle  of  the  Micro- 
phone. In  such  materials  Heat  raises  the  internal  pressure  and  therefore 
the  contact,  and  this  modifies  the  amount  of  resistance  and  the  heat  pro- 
duced within  the  conductor :  this  last  itself  affects  the  conductivity,  as  in 
the  Tasimeter,  which  detects  changes  in  temperature  by  the  variation  of 
a  current  passing  through  a  rod  of  carbon  fixed  between  metallic  supports, 
and  exposed  to  varying  temperatures.  Selenium,  which  in  the  amorphous 
form  is  a  non-conductor,  but  in  the  crystalline  form  is  a  conductor,  varies 
in  conductivity  with  its  state  of  aggregation,  its  temperature,  the  length  of 
time  during  which  a  current  has  been  passing  through  it;  and  crystalline 
selenium,  when  acted  upon  by  .light  (especially  the  yellow  and  the  red), 
and  to  a  less  extent  when  acted  upon  by  dark  rays,  increases  in  conduc- 
tivity: in  the  case  of  very  bright  sunlight  this  increase  being  sometimes 
even  tenfold.  Light  of  variable  intensity  produces  corresponding  and 
rapidly-responding  variations  in  the  conductivity  of  the  crystalline  selenium 


xvi.]  VAKIABLE   CONDUCTIVITY.  637 

upon  which  it  may  fall  —  a  fact  utilised  in  the  construction  of  the  Photo- 
phone.  Metals,  unlike  selenium,  become  worse  conductors  as  the  temper- 
ature rises;  but  (Siemens)  at  210°  C.  selenium  changes  its  character  and 
comes  to  act  like  a  metal. 

Reduced  Resistance  and  Reduced  Length  of  a  Conductor. 

—  This  may  be  explained  by  a  few  numerical  examples.  We 
suppose  the  unit  of  resistance  to  be  the  Ohm,  as  above  denned, 
the  resistance  of  freshly-distilled  mercury  in  a  column  of  1  sq. 
mm.  section  and  1-0(33  metres  in  length. 

1.  What  length  of  soft-copper  wire  of  1  sq.  mm.  sectional 
area  will  give  a  resistance  equal  to  one  Ohm?     1-063  x  61-70 
=  65-5871  metres.     The  figure  61-70  is  taken  from  the  table 
of  conductivities  above.     The  Resistance  of  65-5871  metres  of 
copper  is  thus  equal  to  that  of  1-063  metres  of  mercury:  the 
Reduced  Length  of  65-5871  metres  of  copper  is  1-063  metres 
of  mercury. 

2.  What  will  be  the  resistance  of  a  column  of  mercury  100 
metres  long  and  1  sq.  cm.  in  section  ?     It  will  be  equal  to  that 
of  a  column  of  mercury  1  sq.  mm.  in  section  x  1  metre  in  length 
multiplied  by  I  =  100,  and  divided  by  o  =  102.     It  is  therefore 
0-940734   Ohm.      Its  reduced  length  is  1  metre  of  standard 
mercury-column,  1  sq.  mm.  in  cross-section. 

3.  WThat  will  be  the  absolute  resistance,  and  what  the  resist- 
ance in  Ohms,  of  1000  metres  of  platinum  wire  whose  diameter 
is  \  millimetre  ?     Its  sectional  area  o  ==  irr2  =  TT  (¥V)2  S(l-  cm-  = 
__^_  Sq.  cm. ;    its  length  I  —  100,000  cm. ;    its  resistivity  R  is 
14562-13  -f-  V2;  the  Resistance  of  the  wire  is 

T.      I*     1Annnn     14562-13  „  6400  _  2,966560,000000 
K  =  -  =      JU,Ul          -^yT~       ~^~  ~^~ 

C.G.S.  electrostatic  units,  or  2966-6  Ohms. 

The  strength  or  intensity  of  a  steady  current  is  measured 
by  a  Galvanometer  (p.  713),  round  the  magnetic  needle  of  which 
the  current  is  passed :  in  the  Tangent  Galvanometer  the  tangent 
of  the  angle  of  deflection  of  the  needle  is  proportional  to  the 
intensity  of  the  current. 

The  strength-  of  a  current  is  equal  throughout 
all  parts  of  a  circuit  in  which  there  is  a  steady  flow.  A 
magnetic  needle  is  equally  deflected  when  brought  into  the 
•neighbourhood  of  any  part  of  the  circuit,  whether  the  circuit  be 
locally  composed  of  solid,  of  liquid,  or  of  heated  or  rarefied  gas. 

The  Practical  Unit  of  Intensity  is  the  intensity  of  that  current  which  is 
produced  in  a  conductor  whose  total  resistance  is  1  Ohm  (  =  1^900,000,000000 


638  ELECTKICITY  AND  MAGNETISM.  [CHAP. 

C.G.S.  electrostatic  unit),  when  there  is  kept  up  between  its  extremities  a 
potential-difference  which  constantly  amounts  to  one  Volt,  or  1/300  C.G.S. 
electrostatic  unit. 


Since  I  =      =  =  __  __  C.G.S.E.S.unit^ 

R      1  Ohrn      1/900,000,000000  C.G.S.E.S.  unit 
the  practical  unit  of  intensity,  the  Ampere,  is  equal  to  3000,000000  C.G.S. 
electrostatic  units  of  intensity. 

In  a  current  whose  Intensity  is  one  Ampere,  the  practical  unit  of 
quantity,  one  Coulomb,  passes  any  given  section  during  each  second  :  the 
Coulomb  is  thus  equal  to  3000,000000  C.G.S.  electrostatic  units  of  quantity. 

Electrical  engineers  have  adopted  the  Ohm,  the  Volt,  etc.,  as  means  of 
practical  measurement.  The  Ohm  and  the  Volt  in  electrical  workshops  are 
not  abstract  calculations,  but  standard  wires  and  standard  batteries  (or 
multiples  or  fractions  of  these),  by  comparison  with  which  the  resistance  or 
the  E.M.D.P.,  the  so-called  electromotive  force,  of  any  given  combination  of 
materials  may  be  relatively  measured  ;  e.g.,  an  average  pint  Daniell  cell  will 
deliver  a  maximum  current  of  about  \  Ampere,  its  D.P.  being  about  1  Volt, 
and  its  internal  resistance  about  4  Ohms  ;  an  average  pint  Grove  cell  will 
deliver  a  maximum  current  of  ten  Amperes,  its  D.P.  being  about  2  Volts, 
and  its  internal  resistance  about  i  Ohm. 

Dimensions  of  Electrostatic  Measure  in  Air.  —  Current-Inten- 
sity— a  quantity  passing  per  second:  [I]  =  [Q/T]  =  [MiLfyT]  -j-  [T]  = 


Resistance:  [R]  =  [E/I]  =  [MiLl/T]  +  [MsU/T2]  -  [T/L]  ;  Con- 
ductance =  [1/R]'=  [L/T],  a  Velocity.* 

Resistivity:   [R]  =  [R]  x  [o/fj  =  [T/L]  x  [L2/L]  =  [T]. 

Conductivity:  [D]  =  [1/n]  =  [1/T]. 

In  any  medium  of  sp.  ind.  cap.  K,  the  Dimensions  in  E.-S.  measure 
are:  —  Current-strength,  [MlL3Ki/T]  ;  Resistance,  [T/LK]  ;  Conductance, 
[LK/T]  ;  Resistivity,  [T/KJ  ;  Conductivity,  [K/T]. 

The  above  dimensions  are  based  on  the  assumption  that  the  Quantity 
of  electricity  in  a  current  is  the  same  thing  as  the  Quantity  of  an  electro- 
static charge  :  they  are  therefore  called  Dimensions  in  Electrostatic  Measure. 

Fall  of  Potential  in  a  Homogeneous  Conductor  of  uniform 
thickness.  —  During  the  maintenance  of  a  steady  current  one 
end  of  a  homogeneous  conductor  is  at  a  higher,  the  other  at  a 
lower  potential,  and  between  these  points  the  fall  is  gradual,  so 
that  intermediate  points  are  at  intermediate  potentials.  Fig. 
215  shows  that  if  the  length  of  the  uniform  conductor  be  repre- 

*  Suppose  a  sphere  of  radius  r  and  therefore  of  capacity  C  =  r  to  he  charged,  in 
air,  with  quantity  Q  ;  the  potential  will  be  V  =  Q/r  ;  and  Q  =  Vr.  If  this  sphere  be 
connected  with  the  earth  by  a  wire,  whose  resistance  is  R,  for  a  short  time  t,  along 
that  wire  a  current  will  run,  whose  mean  intensity  is  I;  the  quantity  conveyed 
by  that  current  in  time  t  is  It  ;  and  this  is  lost  by  the  sphere,  whose  charge  sinks  to 
Q'.  Hence  Q  —  Q'  =  It.  If  the  potential  of  the  sphere  is  not  to  sink,  the  radius  must 
diminish.  If  the  radius  shrink  to  r'  in  time  t,  the  velocity  of  its  contraction  is 
(r  —  r')/t:  and  Q  =  rV  as  before;  and  also,  Q'  =  r'V,  V  being  unchanged.  From 
these  we  find  that  (r  —  r')  /t  =  I/V  =  1  /R  =  D,  the  conductance  of  the  wire.  But 
(r  —  r')  /t  is  a  Velocity;  whence,  in  electrostatic  measure,  the  Conductance  of  a  wire 
is  a  Velocity. 


xvi.]  FALL  OF  POTENTIAL   ALONG  CONDUCTORS.  639 

sented  by  CZ,  the  end  C  connected  with  the  positive  terminal 
of  the  battery  is  at  a  potential  which  differs  by  (P)P'  from  the 
potential  of  the  end  Z,  connected  with  the  negative  terminal. 
The  Fall  is  steady,  and  depends  (1)  upon  the  difference  of  poten- 
tial between  the  ends  of  the  conductor,  arid  (2)  upon  the  length 
of  the  conductor ;  it  is  meas- 
ured  by  the  Slope  of  the  line  ppcc^-- 

PP',  the  amount  of  fall  of  4  | 

potential  per  unit  of  length,  o1 

If  the  battery  be  con- 
nected at  its  midpoint  with 
the  earth,  the  conductor  CZ  is  near  C  at  a  positive  potential; 
towards  the  midpoint  of  CZ  this  diminishes;  the  midpoint  of 
the  conductor  is  a  point  of  zero  potential  (the  potential  of  the 
earth);  and  as  we  approach  Z  we  find  the  potential  increasingly 
negative. 

Necessarily,  the  Potential-Fall  or  -Slope  along  the  conductor 
depends  upon  the  difference  between  the  potentials  of  its 
extremities,  not  upon  the  values  or  signs  of  these  in  relation  to 
the  arbitrary  earth-zero  of  potential. 

Resistance  in  a  Heterogeneous  Conductor.  —  When  a  con- 
ductor is  made  up  of  a  succession  of  conductors  which,  on 
account  of  their  differing  materials  or  conditions  or  thicknesses, 
present  different  resistances  to  the  current,  it  may  become 
necessary  to  consider  each  conductor  as  reduced  to  an  equiva- 
lent length  of  a  standard  conductor,  such  as  a  column  of  mer- 
cury 1  sq.  mm.  in  cross-section.  For  example  :  a  current  passes 
successively  along  (1)  a  metre  of  mercury  1  sq.  mm.  in  section ; 
(2)  10  metres  of  mercury  1  sq.  cm.  in  section ;  (3)  1  mm.  of 
pure  water  1  cm.  in  section ;  (4)  61-70  metres  of  soft  copper 
wire  4  sq.  mm.  in  cross-section :  what  is  the  total  resistance  of 
this  combination  ?  We  must  reduce  all  to  a  common  term,  to  Re- 
duced Lengths  of  our  standard  mercury  column.  Then,  above, 
(1)  is  equivalent  to  a  metre  of  such  a  column,  (2)  is  equivalent 
to  -jig-  metre,  (3)  to  400,000  metres,  and  (4)  to  ^  metre  of  such 
a  mercury  column  ;  and  the  whole  resistance  is  that  of  400,001-35 
metres  of  the  standard  conductor. 

Resistance  in  a  Galvanic  Circuit.  —  In  a  galvanic  circuit  we  have  to 
consider  two  sets  of  resistances :  those  internal  to  the  cells,  the  internal 
resistance,  Rf ;  those  in  the  conducting  media,  the  external  resistance,  Re. 
Then  Ohm's  Law  is  I  =  E/Rt  +  Re. 

Let  n  cells  be  arranged  side  by  side,  copper  to  copper,  zinc  £o  zinc  ;  the 
E.M.D.P.  of  the  combination  is  the  same  as  that  of  one  cell,  and  =  E  Volts ; 


640  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

the  internal  resistance  (the  combination  being  virtually  one  cell  of  w-fold 
surface)  is  Rt-/n  Ohms ;  the  external  resistance  is  unaltered.  The  intensity 
or  current-strength  is  therefore 


I  =  (E/(Ri/»)  +  Re}  =  (nE/R*  +  nRe}  Amperes. 

If  the  internal  resistance  be  extremely  small  in  comparison  with  the 
external,  R,  may  vanish  from  this  expression;  then  I  =  {nE/nRe}  =  E/Re 
Amperes,  and  the  current-strength  is  little  increased  by  the  use  of  many 
cells  ;  but  if  the  external  resistance  be  extremely  small,  the  current-intensity 
becomes  {nE/Rf},  and  the  side-by-side  arrangement  in  Surface*  secures 
the  highest  strength  of  current. 

If  n  cells  be  arranged  in  file,  copper  to  zinc,  the  E.M.D.P.  is  nE  Volts  ; 
the  internal  resistance  is  nRf  Ohms  ;  and  the  external,  as  before,  Re  Ohms. 


The  current-intensity  is  now  I  =  {wE/nRj  +  Re}  Amperes.  This  arrange- 
ment of  cells  behind  one  another  in  Indian  file  or  in  Seriesf  is  the  best 
for  securing  the  highest  attainable  strength  of  current  when  the  internal 
resistance  is  extremely  small  in  comparison  with  the  external;  for  then, 
R<  vanishing,  the  current-intensity  is  {nE/Re}  Amperes  ;  while  if  the  external 
resistance,  on  the  other  hand,  be  exceedingly  small  in  comparison  with  the 
internal,  the  intensity  is  {nE/rcRf}  =  E/Rf,  which  differs  but  little  from 


-)-  Re},  the  strength  of  the  current  produced  by  one  cell.  • 
For  extremely  great  external  resistances,  then,  arrange  in  Series,  if  the 
highest  attainable  strength  of  current  be  aimed  at  ;   for  extremely  small 
external  resistances,  arrange  in  Surface. 

When  neither  the  internal  resistance  nor  the  external  can  be  considered 
as  vanishingly  small  the  one  in  comparison  with  the  other,  the  best  arrange- 
ment, for  high  intensity,  is  to  unite  cells,  ab  in  number,  into  a  series  of  b 
each  :  in  each  series  of  6,  the  b  cells  are  placed  side  by  side,  copper  to  copper, 
zinc  to  zinc  ;  then  a  such  series  are  arranged  in  file,  the  copper  terminal  of 
each  series  being  connected  with  the  zinc  of  the  next.  In  this  way  we 
virtually  make  up  a  large  cells,  each  of  6-fold  surface,  and  we  arrange  these 
in  file  or  series. 

In  each  of  these  virtual  large  cells  the  E.M.D.P.  is  E  Volts  ;  the  resist- 
ance is  1-frth  of  r  Ohms,  the  resistance  of  a  single  cell.  Now  couple  a  such 
large  cells  in  Series;  the  E.M.D.P.  of  the  combination  is  aE  Volts;  the 
internal  resistance  of  the  whole,  R£,  is  equal  to  a  x  (r/6)  Ohms  ;  the  inten- 
sity or  strength  of  the  current  produced  is 

aE  aE  E 

1  =  -I  -  =  ^  -  =  —  —  BT  Amperes, 


a'+Re 
b  n  n        a 

where  n  =  ab.  The  denominator  of  the  last  fraction  is  the  least  possible, 
and  the  strength  of  the  current  consequently  the  greatest,  when  Re/r  =  a/6. 
When  the  current  is  strongest,  Re  is  thus  equal  to  ar/b,  or  the  external 
resistance  is  equal  to  Rf,  the  internal.  If  the  external  resistance  be  equal  to 
nr,  and  still  more  if  it  be  greater  than  nr,  the  problem  of  the  most  advan- 
tageous arrangement  of  the  cells  in  rank  and  file  becomes  an  insoluble  one, 
and  the  cells  must  be  arranged  in  series. 

*  Obsolete  synonym  —  "  in  Quantity." 
t  Obsolete  synonym  —  "  in  Tension." 


xvi.]  RESISTANCE   IN  GALVANIC   CIRCUIT.  641 

Problem.  —  Sixty  Grove  cells,  in  each  of  which  the  resistance  is  -6  Ohm, 
are  at  disposal :  a  resistance  of  10  kilometres  of  soft  copper  wire  of  4  mm. 
diameter  is  to  be  encountered;  what  is  the  best  arrangement  of  the  cells  ? 
The  external  resistance,  Re,  is  that  of  1,000,000  cm.  of  copper  wire  of  cross- 
section  -125664  sq.  cm.  and  relative  conductivity  61-70:  this  is  equal  to 
12-13308  Ohms.  Now  in  the  equation  Re  =  ar/b  =  a2r/n,  Re  =  12-13308, 
r  =  0-6,  n  =  60 ;  whence  a  =  34-83.  The  nearest  feasible  number  correspond- 
ing to  this  value  of  a  is  30 ;  and  the  best  arrangement  is  the  division  of  the 
60  cells  into  30  virtual  double-surface  cells,  arranged  in  Series. 

If  the  external  resistance  be  that  of  one  kilometre  of  such  wire,  a  being 
found  equal  to  11-015,  the  best  arrangement  is  12  sets  of  cells,  each  contain- 
ing 5  cells  joined  in  surface,  and  these  sets  joined  to  one  another  in  Series. 

To  obtain  maximum  current-strength  is  not,  however,  the  most  economi- 
cal way  of  using  a  battery;  half  the  energy  is  wasted  in  overcoming 
internal  resistance :  this  internal  resistance  must  be  proportionally  reduced 
in  order  to  reduce  this  waste ;  and  if  this  be  done,  then,  though  the  current 
is  not  the  maximum  obtainable,  the  amount  of  zinc  consumed  is  reduced  in 
a  still  greater  ratio.  For  economical  working,  therefore,  keep  the  resistance 
of  the  cell  or  battery  as  low  as  possible  compared  with  that  of  the  general 
circuit ;  that  is,  work  with  high  external  resistances. 

The  Fall  of  Potential  in  a  Heterogeneous  Conductor.  —  If 

we  draw  a  diagram,  setting  out  on  a  base-line  and  using  as 
abscissse  the  Reduced  Lengths  of  the  several  successive  conduc- 
tors which  make  up  a  heterogeneous  conductor,  and  if  for  a 
moment  we  let  drop  from  view  any  local  differences  of  poten- 
tial set  up  by  contact  of  different  materials,  then  the  line  of 
potentials  slopes  uniformly  down  from  one  end  of  the  hetero- 
geneous conductor  to  the  other  end,  and  from  such  a  diagram 
we  may  find  the  total  fall  of  potential  along  each  component 


Fig.216. 


0AThln  T'hi*ek  Water 

Mercury 


conductor.  Fig.  216  very  diagrammatically  represents  the  fall 
of  potential  in  the  composite  conductor  specified  in  the  preced- 
ing large-type  paragraph,  p.  639.  A  A'  is  the  potential  at  the 
junction  of  the  slender  and  the  thicker  column  of  mercury,  BB' 
that  at  the  one  surface,  CC'  that  at  the  other  surface  of  the 
water,  OO'and  DD'  the  terminal  potentials. 

If  now  we  follow  this  up  with  another  diagram  in  which  the 
real  lengths  of  the  conductors  are  supposed  to  be,  represented, 
we  find  a  remarkable  appearance  presented  by  it.  The  poten- 

2T 


642 


ELECTRICITY  AND  MAGNETISM. 


[CHAP. 


tial-line,  which  indicates  the  successive  falls  of  potential,  is  repre- 
sented by  the  line  O  A'B'C'D'  in  Fig.  217.  The  fall  of  potential 
is  exceedingly  rapid  along  the  bad  conductors,  for  bad  conduct- 
ors keep  up  a  great  difference  of  potential ;  and  the  whole  fall 


Fig.217. 


c'  D 

of  potential  is  distributed  among  the  component  conductors,  to 
each  according  to  its  Reduced  Resistance. 

If  there  be  local  differences  of  potential,  these  must  be  added 
to  or  subtracted  from  the  total  fall  of  potential  for  which  the 
conductor  has  to  provide.  By  way  of  illustration,  let  AB  be  a 
conductor,  of  which  one-half  consists  of  copper  wire,  the  other 
half  of  zinc  wire,  of  an  equal  thickness,  and  let  its  extremities  be 
kept  at  potentials  which  differ  by  2  AX.  In  Fig.  218,  AC  is  the 
reduced  length  of  the  copper  wire,  and  CB  (=  f^Jf  AC)  the 
reduced  length  of  the  zinc  wire.  Between  the  copper  and  the 
zinc  there  is  (on  the  older  view  discussed  on  p.  611)  a  rise  of 
potential  represented  by  DE,  which  would  make  the  slope  of 
the  line  of  potentials  steeper  throughout  the  conductor.  To 
BX'  add  XrF,  which  is  equal  to  DE :  connect  X  and  F  by  a 


Fig.218. 


dotted  line.  Of  this  the  portion  XD  would  represent  the  fall 
of  potential  along  the  copper :  the  sudden  rise  of  potential  at  D 
would  bring  the  line  of  potentials  up  to  E,  whence  it  would  be 
continued  parallel  to  XF,  along  the  course  EX',  arriving  at  the 
terminal  potential  X'.  From  this  diagram  another  might  be 
constructed  in  which,  instead  of  the  reduced  lengths  AC  and 
CB,  the  corresponding  true  lengths  would  be  represented  and 


XVI.] 


FALL  OF  POTENTIAL   IN  CONDUCTORS. 


643 


the  corresponding  true  slope  of  the  potential-line  found  for 
each. 

In  Fig.  219,  the  line  ZAA'CZ 
shows  the  slope  of  potentials  in  a 
galvanic  circuit^  the  various  parts 
being  supposed  to  be  brought  to 
their  reduced  lengths.  We  mio-ht 

O  '         t> 

again     reduce     this     diagram     to 

another,    in    which    the    reduced  Zn  cu  zn 

lengths  of  the  different  parts  of  the  circuit  would  be  replaced 
by  their  true  lengths,  and  the  true  slope  of  the  potential-line 
found  for  each. 

From  this  we  find  that  the  actual  difference  of  potentials 
between  the  plates  of  a  battery  in  a  closed  circuit  depends  upon 
the  relation  between  the  internal  and  the  external  resistance. 

Thus  if  R*  be  the  internal  and  Re  the  external  resistance,  the  difference 
of  potentials  between  the  plates,  available  for  the  service  of  the  part  of  the 
circuit  external  to  the  plates,  is  to  the  whole  E.M.D.P.  of  the  battery 
(measured  between  terminals  on  open  circuit)  as  Re :  R»  +  Re.  It  is  there- 
fore equal  to  ERe/(Rt  -f  Re)  and  we  cannot  assume  E,  the  whole  E.M.D.P. 
of  the  battery,  to  be  available  unless  Re  be  so  great  in  comparison  with  Rf 
that  the  latter  may  be  neglected. 

Flow  along  large  Conductors.  —  If  a  conductor  be  very 
wide  in  comparison  with  the  wires  leading  to  and  from  it,  the 
current  widens  out,  and  no  part  of  the  conductor  is  free  from 
equipotential  surfaces  and  lines  of  flow.  If  it  be  practically 
infinite,  the  resistance  offered  by  it  depends  on  the  radius  of  the 
wires  or  plates  connecting  it  with  the  battery,  and  on  the  resist- 
ivity of  the  conductor  itself:  not  on  the  distance  traversed  by 
the  current  in  the  wide  conductor. 

In  Fig.  220  the  battery  B  is  connected  with  two  galvanometers,  Gr  and 
G',  by  a  long  telegraphic  wire  interrupted  at  A :  at  K  there  is  a  key  by 

Fig.220. 


which  contact  may  be  made  or  broken.     CD  is  a  wire,  the^  continuity  of 
which  may  be  broken  by  another  key.     When  C  and  D  are  connected 


644  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

through  the  wire  CD,  and  connection  made  at  K,  both  galvanometers  are 
deflected.  If,  however,  the  connection  CD  be  broken,  and  connection  be 
suddenly  made  at  K,  the  galvanometer  G  is  alone  deflected:  the  earth 
between  E  and  E'  does  not  simply  replace  the  wire  CD  between  C  and  D. 
On  the  other  hand,  it  is  beyond  doubt  that  currents  do  run  in  the  earth's 
crust.  A  telephone,  part  of  whose  circuit  runs  in  the  straight  line  joining 
two  telegraph  stations,  will  pick  up  signals  from  the  earth-currents  :  and  the 
use  of  the  earth  in  place  of  a  return  wire  is  familiar  in  telegraphy. 

Simultaneous  Currents.  —  Any  number  of  currents  may 
co-exist  on  the  same  wire,  and  the  resultant  at  any  point  is  the 
algebraic  sum  of  the  separate  currents,  positive  or  negative 
respectively.  Thus  two  currents  in  opposite  directions  and  of 
equal  strengths  may  produce  no  effect  in  a  single  wire  which  is 
made  a  part  common  to  two  circuits ;  and  if  this  wire  be  led 
round  a  magnetic  needle,  no  deflection  will  be  produced.  If  the 
one  of  these  two  currents  be  stronger  than  the  other,  the  effect 
will  be  that  corresponding  to  their  difference. 

Derived  Currents.  —  When  a  steady  current  finds  the  con- 
ductor  to  divide  and  then  to  reunite,  it 
divides  into  portions  which  run  along 
the  several  paths  open  to  it.  In  Fig. 
221  the  current  arriving  at  A  divides 
into  two  moieties  ;  if  the  two  paths  be 
equal  in  their  resistance,  these  moieties 
will  be  equal.  If  the  resistances  be  not 
equal,  the  current  passing  along  each  branch  will  be  inversely 
proportional  to  its  resistance,  for  the  difference  of  poten- 
tial between  the  extremities  is  the  same  for  every  branch,  and 
in  each  branch  the  product  of  the  current-strength  into  the 
resistance  is  equal  to  the  difference  of  potential. 

The  double  path  acts  like  a  single  conductor  whose  resistance  is  equal 
to  1/{1/R'  +  1/R"},  where  R'  and  R"  are  the  resistances  of  the  two 
branches.  The  conducting  power  or  Conductance  of  the  double  path  is 
the  sum  of  the  conductances  of  the  two  branches;  these  are  respectively 
1/R'  and  1/R";  their  sum  is  {1/R' +  1/R"},  and  the  Resistance  of  the 
double  path  is  the  reciprocal  of  this  sum. 

Kirchhoff 's  Laws.  —  I.  Where  a  steady  current  branches, 
the  quantity  of  electricity  arriving  by  the  single  wire  is  equal  to 
the  quantity  leaving  the  junction  by  the  branches.  The  alge- 
braical sum  of  the  currents  passing  towards  (or  passing  from) 
the  junction  is  equal  to  zero ;  5)1  =  0. 

II.  In  a  metallic  circuit  comprising  within  it  a  source  of 
permanent  difference  of  potential  E,  the  products  of  the  intensity 


xvi.]  KIRCHHOFF'S  LAWS.  645 

of  the  current,  within  each  part  of  the  circuit,  into  the  corre- 
sponding resistance  are,  if  the  elements  of  current  be  all  taken 
in  cyclical  order,  together  equal  to  E  ;  2  (IR)  =  E.  In  a  metal- 
lic circuit  in  which  there  is  no  source  of  permanent  difference  of 
potential,  E  =  0,  and  S  (IR)  =  0. 

This  law  applies  to  each  several  mesh  of  a  wire  network  as 
well  as  to  a  single  metallic  loop,  and  it  holds  good  even  when 
an  extraneous  current  is  passed  through  the  loop. 

Shunts.  —  If  between  A  and  B  (Fig.  221)  a  single  wire  run  whose 
resistance  is  R,  a  certain  current  I  will  pass;  if  a  lateral  path  or  Shunt 
be  made  available,  the  resistance  in  which  is  ^V^>  the  current  in  the  shunt 
is  I//}  and  the  current  in  the  original  wire  will  sink  to  Iy,  l-100th  of  its 
former  intensity.  This  result  may  be  found  from  the  equations  I  =  Iy  +  Iy/ 
and  IyR  —  (I/y  x  ^V^1)  —  0-  If  the  original  wire  contain  a  galvanometer 
which  would  suffer  risk  of  damage  if  the  whole  original  current  were  sent 
through  it,  the  current-intensity  I  can  thus,  by  the  use  of  shunts,  be  mod- 
erated to  any  desired  degree. 

If  the  shunt  have  a  very  high  resistance  the  current  running  in  it  is 
proportionately  very  small,  and  the  distribution  of  potential  in  the  circuit, 
as  well  as  the  intensity  of  the  current  in  the  original  path,  is  very  little 
interfered  with  by  the  interposition  of  this  new  path.  If  in  this  new  path, 
with  its  known  resistance,  there  be  arranged  a  galvanometer  or  a  sensitive 
current-measurer  of  any  other  kind,  the  indications  of  this  instrument  will 
measure  the  intensity  of  the  current,  and  therefore  the  difference  of  poten- 
tial between  A  and  B.  This  is  realised  in  Lord  Kelvin's  Voltmeter. 

The  Resistance  of  two  conductors  may  be  compared,  by  means  of  a 
Voltmeter,  by  observing  the  relative  differences  of  potentials  between 
pairs  of  equidistant  points  in  the  two  conductors,  when  these  conductors 
are  successively,  in  the  same  circuit,  traversed  by  one  and  the  same  current. 

Wheatstone's  Bridge.  —  In  Fig.  222  there  is  represented  an  arrange- 
ment of  conductors  known  by  this  name.  The  respective  resistances,  inten- 
sities, and  directions  of  the  current  are  indicated  in  that  figure.  Kirchhoif's 
Laws  give  us  the  relations  p. 

between  these.     Law  I.  shows  B 

that  at  A,  I  (the  intensity  in 

the  wire  <CuA)  =  IT  +  I4  (1)  ; 

that  at  B,  IL  =  ±  I5  +  I2  (2)  ; 

and  that  at  C,  I2  +  I3  =  I  (3). 

Law  II.  shows  that  within  the 

loop    CZnCuABC,    in    which 

there  is  a  source  of  difference 

of  potential  E,  the  E.M.D.P.  of 

the  battery,  E  =  IR  +  1^  + 

I2R2    (4)  ;    while   within  the 

loop  ABD  there  is  no  galvanic 

cell,  E  =  0,  and  I.R,  ±  I.R.- 

I4R4  =  0  (5)  ;    and    similarly 

within  the  loop  BCD,  I2R2  —  I3R3  ±  I5R5  =  0  (6).     From  these  equations  the 

value  of  I5  may  be  shown  to  be  equal  to  zero,  when  Rx :  R£ : :  R4 :  R3,  or 

as  regards  resistances,  AB  :  BC  : :  AD  :  DC. 


646  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

To  compare  the  resistances  of  two  conductors,  one  is  placed  between 
A  and  B,  the  other  between  B  and  C  ;  then  the  relative  resistances  of  AD 
and  DC  are  altered  by  moving  the  end  of  the  wire  BD  along  AC  (or 
otherwise  by  means  of  resistance-coils)  until  the  galvanometer  G  gives  110 
deflection.  At  that  moment  the  resistance  of  the  conductor  in  AB  is  to  the 
resistance  of  the  conductor  in  BC  as  the  length  of  AD  is  to  the  length  of 
DC.  A  scale  under  the  wire  AC  enables  this  last  ratio  to  be  read  off.  If 
one  of  the  conductors  compared  —  say  that  in  AB  —  be  a  wire  of  known 
resistance,  a  Standard  Resistance-Coil,  the  resistance  of  the  other  may  be 
absolutely  measured.  If  the  resistance  in  AB  be  exchanged  for  a  tenfold 
resistance,  the  value  of  the  resistance  in  BC  will  seem  to  be  numerically 
diminished  to  a  tenth,  and  thus  resistances  ten  times  as  great  as  before  can 
be  measured  by  being  placed  between  B  and  C.  In  this  way  the  range  of 
the  instrument  can  be  increased. 

When  the  above  ratio  obtains  between  the  several  resistances,  the  cur- 
rents will  remain  unchanged  whether  BD  remain  open  or  closed,  or  be 
closed  intermittently  by  a  key. 

Sometimes  the  battery  is  not  kept  continuously  in  action,  and  a  key  is 
interposed  in  BD  :  the  resistances  are  adjusted  until  closing  both  the  bat- 
tery-circuit and  the  galvanometer-circuit  produces  no  deflection  in  G ;  but  to 
avoid  complications  due  to  Self -Induction  (p.  704),  the  battery-circuit  must 
be  closed  first  and  then  the  galvanometer-key  pressed  down.  One  key  is  so 
arranged  as  to  perform  these  operations  successively. 

The  Resistance  of  an  Electrolyte  may  be  measured  (Kohlrausch's 
Method)  by  making  AB  (Fig.  222)  the  adjustable  resistance,  BC  the  elec- 
trolyte in  question  ;  instead  of  a  battery  insert  in  AGC  the  secondary  coil 
of  an  Induction-Coil,  which  delivers  alternating  currents  :  in  BD  interpose 
a  Telephone.  When  nothing  is  heard  in  the  telephone,  RAB  :  RBC  : :  AD  :  DC, 
as  before. 

Measurement  of  the  Constants  of  a  Galvanic  Cell  or  Battery.  — 
The  E.M.D.P.  —  1 .  W  i  e  d  e  m  a  n  n '  s  M  e  t  h  o  d.  Two  batteries  or  cells,  the 
one  a  standard  ;  connect  in  series  :  the  joint  E.M.D.P.  is  E,  +  Ey/;  the  total 
resistance  is  R/  +  R/'  +  Re ;  the  intensity  of  the  current  produced  is  I  =  (E7 
-f  E/y)  /R/  +  R*"  +  Re.  Now  turn  one  of  the  batteries  round,  and  connect 
so  that  the  two  now  oppose  one  another;  the  joint  E.M.D.P.  is  E,  —  Ey/ :  the 
total  resistance  is  as  before :  the  intensity  is  I'  =  (E,  —  Ey/)  /R/  +  R*"  +  Re. 
Hence  E, :  Ey/ : :  (I  -  P)  :  (I  +  P).  E,,  the  E.M.D.P.  of  the  standard  cell, 
is  known  ;  I  and  P,  the  intensities,  can  be  observed :  whence  E/;  can  be 
calculated. 

2.  Potentiometer-Method.  —  Connect  a  low-resistance  battery  of 
fair  constancy  with  the  extremities  of  a  stretched  wire  AB,  one  metre  long, 
divided  into  millimetres.  To  one  end  A  of  this  conducting  wire  connect,  in 
series,  a  resistance-coil,  a  galvanometer,  and  a  Clark's  standard  cell :  the 
free  terminal  of  the  Clark's  cell  is  connected  with  a  wire  or  "  slider  "  which 
may  be  made  to  touch  any  point  of  the  stretched  wire  AB.  Adjust  the 
position  of  the  slider  on  AB,  gradually  diminishing  the  resistance,  until 
there  is  no  deflection  in  the  galvanometer.  The  D.P.  between  A  and  the 
point  touched,  due  to  the  main  current  along  the  wire,  is  then  equal  and 
opposite  to  the  E.M.D.P.  of  the  Clark's  cell.  Do  the  same  thing  with  the  cell 
or  battery  to  be  tested.  For  convenience  the  Clark's  cell  and  the  battery  to 
be  tested  should  both  be  connected  with  the  galvanometer,  and  the  adjust- 
ments made  with  each  in  succession :  and  then  the  chance  of  variation  in 


xvi.]  BATTERY   MEASUREMENTS.  Qtf 

the  main  battery  may  be  eliminated  by  touching  the  stretched  wire  with 
the  two  sliders  alternately  at  the  points  determined,  and  thus  checking  the 
results.  This  method  eliminates  polarisation  and  the  effect  of  the  internal 
resistance  of  the  battery  to  be  tested. 

The  Internal  Resistance  of  a  Battery  may  be  measured  (by  Mance's 
method)  by  making  it  one  of  the  four  resistances  within  a  Wheatstone's 
bridge  (Fig.  223);  one  of  the  other  resistances,  say  AB,  being  rendered 
adjustable  either  by  making 

AB  consist  of  standard  re-  JL  Fig.223. 

sistance-coils  or  by  the  use 
of  a  Rheostat  or  Rheo- 
chord,  by  which  variable 
quantities  of  wire  or  mer- 
cury, or  fluids  of  various 
kinds  equivalent  to  so  many 
Ohms  resistance,  may  be  in- 
troduced into  AB.  A  galva- 
nometer is  placed  in  AGC ; 
a  key  in  BD.  The  adjustable 
resistance  in  AB  is  varied  in 
amount  until  the  deflection  of  the  galvanometer  becomes  unaffected  by 
making  or  breaking  contact  in  BD.  The  relation  RAB  :  RBC  : :  AD  :  DC  again 
holds  good. 

If  we  make  the  galvanometer  G  and  the  battery  between  B  and  C 
exchange  places,  we  have  (Lord  Kelvin)  a  very  easy  method  of  finding  the 
resistance  of  a  galvanometer  coil. 

The  Energy  of  a  Steady  Current.  —  In  a  steady  current 
of  intensity  I,  a  quantity  I  of  electricity  passes  during  each 
second  from  a  place  where  the  potential  is  V;  to  a  place  where 
it  is  V0  ;  but  V,  —  V0,  the  fall  in  its  potential,  is  E,  the  electro- 
motive difference  of  potential  within  the  circuit.  This  fall  is 
constant,  for  the  electromotive  difference  of  potential  is  kept 
up :  the  Energy  transmitted  by  the  current  is  therefore  I  x 
( V,  —  V0)  =  IE  per  second ;  or,  in  the  course  of  that  period  of 
time  during  which  a  quantity  Q  passes,  the  Total  Energy 
transmitted  is  equal  to  QE  ;  all  in  C.G.S.  units  or  ergs. 

Since  by  Ohm's  law  I  =  E/R,  we  find  that  the  Energy  per 
second,  IE,  =  E2/R  ;  and  that  it  is  also  equal  to  PR,  per  second. 

The  energy  transmitted  by  a  steady  current  of  one  Ampere- 
intensity  under  an  electromotive  potential-difference  of  one  Volt 
is  equal,  since  Energy  per  second  =  El,  to  (3--^  x  3000,000000) 
=  10,000000  C.G.S.  units  or  ergs  per  second.  This  Rate  of 
Transference  of  Energy,  an  Activity  of  10  megergs  (one  Joule) 
per  second,  is  called  an  Ampere- Volt  or  a  Watt ;  and  it  is  equal 
to  1/746  Horse-power  nearly,  or  to  1/735-75  Cheval-vapeur. 

A  steady  current  of  V  Volts  and  A  Amperes  ihus  repre- 
sents A  F/746  horse-power. 


648  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

This  applies  to  any  selected  part  of  the  circuit.  Let  this  part  be  the 
battery  or  cell  itself.  If  the  resistance  of  this  be  Rf  and  if  the  actual  cur- 
rent-strength be  I,  then  the  energy  used  up  in  the  battery  is  PRt- :  and  E^,  the 
potential-fall  in  the  battery,  is  E{  =  IR*.  But  the  whole  potential-fall  in  the 
whole  circuit  is  E  =  I(RZ-  +  Re).  Therefore  the  fraction  Rf  -=-(11,  +  Re)  of 
the  total  energy  is  absorbed  in  the  battery  or  cell  itself ;  and  the  remain- 
der is  at  the  disposal  of  the  external  circuit,  whose  resistance  is  Re. 

Energy  in  charging  a  condenser.  —  A  condenser  may  be  charged  by 
having  its  opposite  plates  put  in  metallic  connection  with  the  opposite 
terminals  of  a  battery.  The  opposed  plates  acquire  a  potential-difference 
equal  to  the  E.M.D.P.  of  the  battery  as  measured  on  open  circuit.  When 
this  is  done,  the  energy  stored  up  in  the  condenser  is  equal  to  the  total 
energy  of  the  current  during  the  charging;  but  the  latter  has  been  con- 
verted into  Heat ;  therefore  the  battery  has  had  to  evolve  a  double  amount 
of  energy  during  the  accumulation  of  electrostatic  energy  in  the  condenser. 

Energy  stored  in  the  Ether.  —  If  there  be  anything  of 
the  nature  of  a  Displacement-Current  in  the  dielectric,  as  there 
is  during  the  brief  period  in  which  the  steady  current  is  first 
being  set  up,  the  energy  of  that  displacement-current  is  stored 
up  in  the  dielectric,  and  only  appears  as  Heat  when  it  is  restored 
from  the  Ether  to  the  wire,  on  the  cessation  of  the  conduc- 
tion-current. 

Transmission  of  Energy  by  a  Steady  Current.  —  During 
the  passage  of  a  steady  current,  the  Lines  of  Force  in  the  Field 
are  directed,  not  at  right  angles  to  the  wire,  but  approxi- 
mately parallel  to  it.  They  travel  inwards,  moving  broadside- 
on,  towards  the  wire  ;  and  when  they  reach  it,  they  give  up 
their  momentum  (J.  J.  Thomson).  Energy  is  thus  conveyed, 
not  through  the  wire,  but  through  the  Field,  the  dielectric. 

The  direction  of  transmission  of  energy,  being  at  right  angles  to  the 
lines  of  force,  is  always  somewhere  along  the  electric  equipotential  surfaces 
in  the  field.  These  surfaces  radiate  out  sideways  from  a  cell  (Poynting) , 
from  the  margin  of  the  zinc  plate ;  and  those  which  do  not  cross  over  to  the 
copper,  and  thus  provide  a  path  for  the  absorption  of  energy  by  the  copper, 
each  cut  the  wire  once,  returning  to  the  opposite  margin  of  the  zinc  plate. 
Where  the  equipotential  surfaces  are  most  crowded  together  in  the  field,  as 
where  there  is  the  steepest  potential-gradient  along  the  circuit,  the  trans- 
mission of  energy  through  the  dielectric  is  most  concentrated ;  but  the  energy 
supplied  from  any  part  of  the  field  to  the  circuit  is  never  any  more  than  is 
necessary  to  make  up  for  the  local  transformations  of  energy  into  Heat, 
WTork,  etc.  The  energy  is  gradually  absorbed  or  transformed  as  it  passes 
through  the  wire  from  circumference  to  axis,  and  at  the  axis  it  has  become 
wholly  transformed.  Prof.  O.  J.  Lodge  has  worked  out  a  scheme  of  illustra- 
tive models  to  indicate  how  the  Ether  may  thus  transmit  Energy ;  see  his 
Modern  Views  of  Electricity.  The  quantity  of  energy  flowing,  at  any  point 
in  the  field,  per  sq.  cm.  per  second,  is  (Oliver  Heaviside)  <j>  •  h/4?r  ergs,  where 
<j>  and  h  are  respectively  the  local  electric  and  magnetic  forces  per  sq.  cm.  in 
the  field,  both  at  right  angles  to  the  direction  of  transmission  of  energy. 


xvi.]  EFFECTS   OF  A  STEADY  CURRENT.  649 


EFFECTS  OF  A  STEADY  CURRENT. 

Production  of  Heat.  —  If  a  galvanic  circuit  be  completed 
and  allowed,  as  it  were,  steadily  to  run  to  waste,  no  external 
work  being  done  by  it,  heat  is  developed  within  the  cell  and 
in  the  conducting  wire.  The  Heat  produced  represents  the 
total  Energy  of  the  steady  current,  and  is  equal,  like  that 
energy,  to  PR  ergs  per  second  (Joule's  Law),  or  to  E2/R  or  El 
ergs  per  second,  where  R  is  the  total  resistance  of  the  circuit, 
and  I  the  intensity  of  the  current  actually  passing. 

The  heat  per  cub.  cm.  of  the  wire  is  EI/vol.  =  E//  x  I/o  =  <j>A. 

Problem.  —  A  uniform  copper-wire  whose  cross-section  is  4  sq.  mm., 
and  whose  length  is  106-3  metres,  connects  the  poles  of  a  cell  whose  effective 
difference  of  potential  is  one  Volt,  and  whose  internal  resistance  is  4  Ohms. 
How  much  heat  will  be  developed  during  one  minute  ? 

E    is    one    Volt  =  ?fa    C.G.S.    electrostatic    unit    of    E.M.D.P.      The 


total   resistance,   R,   is   4    Ohms   internal  +  x  i  x  -—      Ohms 

external  =  4-405   Ohms  =  -  ^^  -   C.G.S.  electrostatic  units.      The 
900000,000000 

Heat  =  Energy  =  ^  per  second  =  —  +  -  ^5  -  =  2,270148   ergs 
R  3002      900000,000000 

per  second  =  136,208880  ergs  per  minute  =  about  3-3  ca  per  minute. 

If  a  current  be  maintained  in  a  wire,  the  temperature  rises  until  a  point 
is  reached  at  which  the  loss  by  radiation  and  convection,  on  the  one  hand, 
and  the  heat  supplied  at  the  expense  of  the  energy  of  the  current,  on  the 
other,  exactly  balance  one  another.  Thereafter  the  temperature  remains 
constant.  The  wire,  being  warm,  expands.  This  expansion  may  be  meas- 
ured. It  varies  as  the  intensity  varies.  If  the  current-intensity  do  not 
materially  vary,  the  amount  of  expansion  may  be  utilised  as  a  means  of 
ascertaining  E,  the  difference  of  potential  between  the  ends  of  the  wire. 
This  principle  is  applied  in  Major  Cardew's  and  Ayrton  and  Perry's  Volt- 
meters. 

When  a  current  is  made  to  pass  through  a  heterogeneous 
conductor  composed  of  different  metals,  between  which  there  is 
developed  a  difference  of  electric  potential  (true  contact- 
effect,  p.  624),  the  energy,  which  is  wholly  converted  into 
heat  when  no  work  is  done  by  the  current,  is  divided  into  two 
parts.  Of  these  one  part  obeys  Joule's  Law,  and  is  equal,  per 
second,  to  PR,  the  product  of  the  total  resistance  into  the  square 
of  the  actual  intensity:  the  other,  which  may  be  positive  or 
negative,  goes  to  produce  Peltier's  effect,  which  is  the  follow- 
ing: —  Consider  a  junction  of  metals,  A  and  B,  such  that  when 
this  junction  is  made  the  hot  junction  of  a  thermo-electric  cir- 
cuit a  current  passes  through  it  from  A  to  B  :  let  a  current  be 


650 


ELECTRICITY  AND   MAGNETISM. 


[CHAP. 


made  to  run  ab  externo  through  that  junction  in  the  same  direc- 
tion, A  to  B ;  that  junction  will  under  such  circumstances  be 
cooled,  while  if,  on  the  other  hand,  the  current  be  made  to  flow 
from  B  to  A,  the  junction  will  be  heated. 

In  Fig.  224,  OL  is  a  conductor  composed  partly  of  iron,  partly  of  cop- 
per ;  a  steady  current  is  made  to  flow  through  the  conductor  from  O  to  L, 


V, 


Fig.224. 


(Iron,  R,) 


(Copper,  R,,) 


between  the  potentials  V,  and  VIV.    At  the  junction  J  there  is  a  sudden  fall  of 

potential  (V,,  -  V,,,).     In  OJ  the  intensity  =  ™1  of  potential  =  V.-V, 

Resistance  of  OJ  R, 

• 

in  JL  the  intensity  is 


•y      _ 


T  = 


V,  -  V, 
R 


Resistance  of  OJ 
In  both  it  is  equal  :  hence 


(  V,  ~  Viv)  ~  (V,,  ~ 


where  EOL  is  the  total  fall  of  potential,  Ej  the  fall  at  J,  and  R  the  total 
resistance.  Hence  EOL  =  RI  +  Ej.  The  Energy  of  the  current  is  EOL  x  I  = 
Rl2  +  EjL  The  first  part  of  this  expression,  RI2,  represents  heat  distributed 
over  the  whole  conductor  ;  the  second  part,  Ejl,  represents  heat  locally  devel- 
oped at  J,  and  proportional  to  the  fail  of  potential  there.  If  the  current  be 
made  to  pass  from  copper  to  iron  there  will  be  a  rise,  a  negative  fall  :  the 
heat  developed  at  the  junction  will  be  a  negative  quantity,  and  the  junction 
will  be  cooled.* 

In  a  thermo-electric  circuit  of  copper  and  iron,  the  current  flows  from 
the  copper  to  the  iron  across  the  hot  junction.  At  the  hot  junction  the 
current  passes  through  a  rise  of  potential  (copper-iron)  ;  the  current  there- 
fore tends  to  cool  the  hot  junction.  At  the  cold  junction  the  current  passes 
through  a  fall  of  potential  (iron-copper)  ;  it  therefore  tends  also  to  heat  the 
cool  junction.  This  cooling  of  the  hot  and  heating  of  the  cool  junction  is 
Peltier's  Effect. 

When  a  current  is  passed  ab  externo  through  iron,  copper,  iron  succes- 
sively, it  again  heats  the  iron-copper  and  cools  the  copper-iron  junction. 
The  main  current  is  weakened  by  the  reverse  thermo-electric  current  sec- 
ondarily produced,  and  when  the  main  current  is  cut  off,  the  latter  acts 
alone  until  the  junctions  come  to  the  same  temperature. 

Thomson's  Effect  (Lord  Kelvin's).  —  The  same  thing  may  occur  even 
within  a  single  metal.  Hot  iron  is  negative  to  colder  iron  ;  a  current,  made 
to  pass  within  a  mass  of  iron  from  a  hotter  region  to  a  colder,  travels  against 

*  Prof.  O.  J.  Lodge  points  out  that  there  is  no  such  phenomenon  at  a  junction  of 
copper  and  zinc:  whence  he  concludes  that  there  is  at  such  a  junction  no  real  fall  of 
potential,  and  that  the  apparent  D.P.  of  copper-zinc  is  really  the  sum  of  a  copper-air 
and  a  zinc-air  contact-difference, 


XVI.] 


THOMSON'S  EFFECT. 


651 


progressively  rising  potentials  and  cools  the  iron  in  the  cooler  region ;  made 
to  pass  from  cold  to  hot  iron,  it  heats  the  iron  in  the  hotter  region.  It  thus 
tends  to  exaggerate  the  existing  differences  of  temperature.  These  effects 
are  reversed  in  copper  or  brass. 

The  convection  of  heat  by  a  current  of  electricity  in  unequally  heated 
iron  is  "  negative,"  that  is,  it  is  opposed  to  that  convection  of  heat  which 
would  be  brought  about  by  the  flow  of  water  through  an  unequally  heated 
tube.  In  copper,  on  the  other  hand,  the  electric  convection  of  heat  is 
"  positive." 

In  a  thermo-electric  circuit,  therefore,  the  current,  as  it  travels  in  the 
iron  from  hot  to  cold,  absorbs  heat  and  gains  energy ;  in  the  copper,  travel- 
ling from  cold  to  hot  it  again  absorbs  heat. 

The  Thermo-electric  Diagram  may  be  made  to  represent  the  Thomson 
and  Peltier  effects.  Let  Fig.  225  be  a  diagram  for  iron  and  copper  between 
the  temperatures  t,  and  tir  The  area  marked  "  Peltier  hot  junction  "  repre- 
sents the  amount  of  energy  absorbed  at  the  hot  junction  when  a  unit- 
current  passes;  the  area  marked  "Thomson-Fe"  represents  the  energy 


Fig.  225. 


THOMSON  - FE 


THOMSON  -  CU 


'? 


Abs. 


absorbed  from  the  iron  when  a  unit-current  passes  in  it,  from  hot  to  cold ; 
the  area  marked  "  Thomson-Cu  "  in  the  same  way  represents  the  Thomson 
effect  in  the  copper,  the  amount  of  energy  absorbed  from  the  copper  when 
a  unit-current  passes  in  it  from  cold  to  hot.  The  whole  shaded  area  thus 
represents  the  energy  absorbed,  by  the  cooling  of  the  hot  junction  and  of 
the  unequally  heated  iron  and  copper,  when  the  unit-current  runs  in  the 
direction  indicated  by  the  arrows.  Plainly,  if  the  hotter  junction  be  heated 
to  T°,  the  Xeutral  Point,  we  shall  have  the  two  Thomson  effects,  and,  at 
the  hot  junction,  no  Peltier  effect.  There  is  at  that  temperature  no  D.P. 
between  the  two  metals. 

Now  turn  to  the  energy  evolved.  This  takes  two  forms:  (1)  Heat 
liberated  at  the  colder  junction  (Peltier  effect)  ;  and  (2)  the  Energy  of 
Electric  Current.  The  latter,  when  the  current-intensity  is  unity,  is  equal 
to  the  E.M.D.P. ;  and  we  have  already  seen  that  this  E.M.D.P.  is  repre- 
sented by  the  area  between  two  metal-lines  and  the  ordinates  corresponding 
to  the  two  temperatures.  Hence  the  accompanying  diagram  (Fig.  226) 
needs  little  explanation.  If  the  colder  junction  be  at  a  temperature  of  T°, 


652 


ELECTRICITY  AND   MAGNETISM. 


[CHAP. 


the  Neutral  Point,  there  will,  at  that  junction,  be  no  Peltier  effect,  no  libera- 
tion of  energy  as  Heat. 

The  student  may  now  exercise  himself  in  showing  that  when  the  colder 
junction  is  at  temperature  T°  the  effect  is  the  reverse  of  that  obtained  when 
the  hotter  junction  is  at  T°;  that  when  one  junction  is  as  far  below  T°  as 


Fig.  226. 


the  other  is  above  T°,  the  area  representing  the  Current-Energy  vanishes  ; 
and  that  when  the  hotter  junction  is  at  a  temperature  farther  above  Tc 
.than  that  of  the  colder  is  below  it,  the  current  is  reversed. 

In  these  figures  the  energy  supplied  is  equal  to  the  energy  accounted 
for.  The  whole  arrangement  is  a  kind  of  thermic  engine,  in  which  Heat 
is  absorbed  from  a  Source,  partly  restored  to  a  Condenser  or  Sink,  and 
partly  converted  into  the  Energy  of  an  Electric  Current. 

The  Thomson  effects  are  themselves  reversed  in  iron  at  a  low  red  heat, 
and  probably  again  at  a  higher  temperature,  so  as  to  make  one  if  not  two 
new  neutral  points.  The  same  phenomena  occur  in  nickel  at  comparatively 
low  temperatures. 

When  a  circuit  is  composed  of  various  conductors  which 
successively  offer  different  resistances  to  the  current,  the  Heat 
produced  is  distributed  among  them,  to  each  according  to  its 
resistance,  or  its  total  potential-fall. 

Numerical  Example  :  —  A  circuit  consisting  of  one  cell  whose  E.M.D.P. 
is  1-8  Volts,  and  whose  internal  resistance  is  0-7313  Ohm,  and  of  an  external 
conductor  composed  of  6-170  metres  of  soft  copper-wire  4  sq.  mm.  in  cross- 
section,  in  which  is  interpolated  a  piece  of  platinum  wire  ^  mm.  in  diar. 
and  4  cm.  in  length,  will  have  a  total  resistance  amounting  to  —  Battery 
0-7333  Ohm,  copper  wire  ^  Ohm,  and  platinum  wire  (equivalent  to  a 

f  -  x 


,  , 

mercury  column   f  -  x  -  J  metres  long  and  -007854  sq.  mm.  in  section) 


•7417  Ohm;  or  on  the  whole  1-500  Ohms,  or 

900000,000000 

static  units.    We  assume  that  a  steady  current  can  be  set  up  and  maintained 
for  a  second  within  such  a  circuit,  and  further,  that  radiation  and  convec- 


xvi.]  HEAT  IN  CIRCUIT.  653 

rt  )-roqo 

tion  of  heat  may  be  set  aside.  Of  the  total  heat  produced,  is  developed 

1*500 

in  the  battery,      "J  in  the  copper  wire,  and  '— —  in  the  small  piece  of 

.1/ouU  JL*oUU 

platinum  wire.     The  total  heat  produced  in  a  second  is 

E2=  \  (l'SY+  1>5  I 

R       <  V300/    '  900000,000000  I 

C.G.S.  units  or  ergs;  this  is  21,600000  ergs  or  (21,600000-^-41,593000)  ca. 
The  heat  evolved  in  the  battery  —  4889  of  the  whole  —  would  be,  if  we  sup- 
pose the  battery  to  contain  1  kilogramme  of  material  of  a  mean  specific  heat 
of  0-8,  sufficient  to  raise  its  temperature  by  about  -000314°  C.  in  a  second ; 
that  evolved  in  the  copper  wire  (whose  weight  is  about  217  grammes  and  sp. 
heat  =  0-095)  by  about  -0004°  C. ;  while  that  liberated  in  the  platinum  wire 
(whose  weight  is  about  0-0276  grammes,  and  whose  sp.  heat  =  0-0325) 
would  be  competent  to  raise  it  in  a  second  to  the  temperature  of  289°  C. 

In  electric  welding,  the  two  pieces  of  metal  to  be  welded  together  are 
made  terminals  for  a  powerful  current.  They  are  brought  into  contact : 
the  current  runs  :  the  point  of  contact  offers  resistance  and  becomes  very 
hot :  and  the  hotter  it  is  the  worse  is  its  conductivity,  and  therefore  all  the 
greater  is  its  resistance. .  Heat  is  thus  locally  developed :  and  the  metal 
pieces  may,  by  this  means,  even  be  fused  together.  In  electric  blasting  and 
the  electric  cautery  the  current  is  made  to  flow  through  a  very  thin  piece  of 
platinum  wire,  which  locally  becomes  red-hot  or  white-hot ;  and  in  electric 
fuses  an  excessive  current  heats  a  specially  fusible  part  of  the  circuit  so  far 
that  it  melts,  and  thus  breaks  the  circuit  or  "  cuts-off  "  the  current. 

When  a  steady  current  is  divided  into  derived  currents 
(Fig.  221,  et  seq.),  the  division  is  such  as  to  correspond  to  the 
least  possible  value  of  2  (I2R)  in  the  branches ;  that  is,  to  the 
minimum  aggregate  production  of  Heat. 

Production  of  Light.  — When  one  part  of  a  circuit  presents 
a  relatively-great  resistance,  the  greater  part  of  the  heat  devel- 
oped within  the  circuit  is  concentrated  within  that  part.  When 
the  local  resistance  is  due  to  a  thin  platinum  wire  or  a  thin  fila- 
ment of  carbon  or  of  carbonised  paper  or  vegetable  fibre  or  paste, 
that  bad  conductor  is  so  far  heated  as  to  emit  a  considerable 
amount  of  light.  This  is  illustrated  by  the  various  forms  of 
incandescent  lamps  or  electric  "  glow-lamps." 

Those  in  which  the  carbon  filament  is  arranged  within  a  vacuum  give 
out,  according  to  the  type  of  lamp,  the  number  in  circuit,  and  the  intensity 
of  the  current  employed,  a  light  equal  to  that  of  from  1  to  1000  candles  each 
(usually  8  or  16).  The  ordinary  data  are :  — 2^  candle-power,  5  Volts  x  1-9 
Amperes  to  25  Volts  x  0-4  Amperes ;  8  c.-p.,  15  V  x  1-9  A  to  55  V  x  0-6  A  ; 
16  c.-p.,  30  V  x  1-85  A  to  105  V  x  0-58  A  ;  32  c.-p.,  55  V  x  2-0  A  to  105  V 
x  1-05  A  ;  100  c.-p.,  80  V  x  4-4  A  to  105  V  x  3-3  A  ;  200  c.-p.,  80  V  x 
8-5  A  to  105  V  x  6-5  A  ;  1000  c.-p.,  80  V  x  43-5  A  to  105  V  x  33  A.  The 
consumpt  of  energy  is,  per  candle-power,  from  3£  to  4  Ampere- Volts  per 
second,  or  from  0-0047  to  0-0054  horse-power,  absorbed  from' the  energy  of 
the  current. 


654  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

If  the  carbon  filament  be  so  constructed  as  to  present  the  form  of  a 
hollow  tube,  of  relatively- great  surface  and  small  actual  cross-section,  the 
luminous  efficiency  of  the  lamp  is  greatly  increased  (Bernstein,  Cruto). 

When  too  strong  a  current  is  driven  through  such  a  lamp, 
the  superficial  particles  of  the  heated  carbon  are  scattered 
throughout  the  vacuum ;  the  carbon  is  volatilised,  and  condenses 
on  the  wall  of  the  lamp ;  so  with  platinum  and  iridium  heated 
above  1700°  C. 

When  a  strong  electric  current  is  driven  through  a  carbon 
rod  to  a  thicker  piece  of  carbon,  the  thin  rod  becomes  heated , 
when  this  takes  place  in  air  the  carbon  burns  away  rapidly; 
but  if  the  rod  rest  loosely  by  one  end  upon  the  thicker  mass, 
the  contact  is  always  maintained,  and  the  light  is  fairly  steady 
so  long  as  any  carbon  remains. 

When  the  interposed  resistance  is  that  of  a  certain  thick- 
ness of  air,  the  current  will  not  pass  unless  the  interval  be  so 
small  or  the  difference  of  potential  on  its  two  sides  be  so  great 
that  a  spark  can  fly  across  it.  When  this  is  the  case  the  current 
is  established  across  the  interval.  If  the  poles  be  of  carbon, 
their  extremities  become  intensely  hot  and  wear  away  by  oxida- 
tion in  the  air.  The  intervening  air  is  so  good  a  conductor 
when  intensely  heated  that,  when  the  arc  has  once  been  estab- 
lished, the  poles  may  be  separated  to  a  distance  greater  than  the 
striking  distance  in  cold  air ;  still,  the  resistance  of  the  hot  air 
will  not  alone  explain  the  resistance  offered  by  the  voltaic  arc 
to  the  transmission  of  the  current.  This  is  due  to  a  kind  of 
thermo-electric  effect.  The  positive  pole  is  hotter  (4000°  C.) 
than  the  negative  (3000°  -  3500°  C.) ;  and  the  greater  part  of  the 
fall  of  potential  is  at  the  positive  carbon,  while  there  is  another 
potential-fall  at  the  negative  carbon.  These  sudden  falls  of 
potential  are  equivalent  to  a  reverse  E.M.D.P. ;  and  the  resist- 
ance of  the  arc  presents  two  terms,  the  one  constant,  the  other 
increasing  with  the  length  of  the  arc.  The  air  in  the  arc  is 
dark,  not  bright. 

The  temperature  attained  seems  to  be  that  of  the  volatilisation  of 
carbon.  In  small  arc-lamps,  this  temperature  is  attained  over  a  small  area ; 
in  large,  with  more  powerful  currents,  a  larger  area  attains  it;  but  the 
brightness  of  the  incandescent  carbon  per  unit  of  luminous  area  is  the 
same  in  both  large  and  small. 

The  positive  pole,  being  the  hotter,  wears  away  about  twice 
as  fast  as  the  negative,  and  becomes  hollowed.  The  problem  of 
electric  lighting  is  to  keep  the  arc  in  the  same  place,  the  carbons 


xvi.]  PRODUCTION  OF  LIGHT.  655 

being  allowed  to  approach  one  another  as  far  as,  and  only  as  far 
as,  is  necessary  in  order  to  make  up  for  their  wear. 

In  JablochkofPs  and  Jamin's  candles  the  two  carbons  were  rods,  parallel 
to  one  another  and  of  equal  length ;  the  arc  passed  between  their  apices. 
If  the  current  passed  in  one  direction  only,  one  carbon  would  wear  away 
faster  than  the  other ;  the  carbons  would  thus  cease  to  be  of  the  same  length. 
The  currents  used  must  rapidly  alternate  in  their  direction ;  both  carbons 
are  thus  equally  worn  away,  and  the  length  of  the  arc  is  constant.  The  usual 
fall  of  potential  in  Jablochkoff's  lamps  was  from  42  to  43  Volts  ;  the  inten- 
sity of  the  current  producing  the  light  from  8  to  9  Amperes,  and  the  candles 
per  horse-power  about  400. 

In  arc-lamps  two  carbon  points  are  placed  opposite  to  one 
another,  and  it  is  the  part  of  a  special  regulatory  mechanism  to 
keep  the  carbons  at  a  constant  distance  (3  to  4J  mm.)  apart. 
Such  regulating  mechanisms  depend  for  their  action  (1)  upon 
variations  of  intensity  of  the  current  traversing  the  lamp,  or 
(2)  upon  variations  in  the  differences  of  potential  between  the 
two  ends  of  the  arc,  or  (3)  upon  departures  from  a  predeter- 
mined relation  between  this  difference  of  potential  and  the 
intensity  of  the  current,  or  (4)  upon  variations  in  the  amount 
of  heat  developed  in  the  arc.  The  light  given  generally  varies 
very  much  according  to  the  angle  from  which  the  lamp  is 
viewed. 

The  resistance  of  the  voltaic  arc  is  4  to  10  Ohms ;  the  fall  of  potential  is 
from  32  to  58  Volts ;  a  12,000-candle  lamp  consumes  about  7  engine  horse- 
power, an  875-candle  lamp  about  1  engine  h.-p.,  or  f  h.-p.  electrical  (say  10 
Amperes  x  50  Volts). 

The  heat  developed  in  the  arc  has  been  utilised  by  Messrs.  Siemens  and 
Huntington,  who  produced  the  electric  arc  within  the  interior  of  a  crucible, 
and  by  its  means  fused  very  refractory  metals  with  considerable  expedition. 
M.  Moissan  has  recently  succeeded  in  effecting  many  extraordinary  chemi- 
cal reductions  by  the  temperature  of  an  arc  (450  A  and  70  V)  enclosed 
between  two  blocks  of  burned  lime.  At  the  temperatures  obtained,  he 
distilled  platinum  and  evaporated  silicium  and  carbon.  » 

When  the  electric  arc  is  produced  between  carbons  in  vacua 
a  beautiful  glow  is  obtained,  the  negative  pole  being  surrounded 
by  a  blue  aureole,  and  the  positive  by  a  stratified  pale-blue  light. 
The  carbon  evaporates,  the  vessel  becomes  filled  with  a  blue 
vapour  which  darkens  to  indigo,  and  this  condenses  and  renders 
the  whole  opaque. 

If  a  very  little  vapour  of  bisulphide  of  carbon  be  introduced  into  the 
vacuum,  the  light  becomes  insupportably  bright,  and  of  an  extremely  brill- 
iant green.  Its  spectrum  presents  channelled  regions  in  the  red,  yellow, 
green,  and  violet,  which  look  like  duplicates  of  one  another,  Reproduced  in 
different  colours  (Jamin). 


656  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

Geissler's  Vacuum  Tubes.  —  When  a  discharge  of  high 
D.P.,  as  from  an  electric  frictional-machine,  or  an  "induction 
coil,"  or  a  battery  of  400  Groves,  is  sent  through  a  mass  of 
rarefied  gas  (about  yoVo  atmos.)  contained  within  a  so-called 
vacuum  tube,  that  gas  glows  with  a  bright  light,  characteristic, 
as  regards  its  spectrum,  of  the  gas  exposed  to  this  operation. 
The  positive  pole  is  surrounded  by  a  bright  glow,  the  negative 
by  a  dark  space  and  a  set  of  striae,  and  in  this  case  the  negative 
pole  is  the  hotter. 

If  the  vacuum  be  very  good  and  the  tube  containing  the 
rarefied  gas  be  somewhat  narrow  at  its  middle,  the  glow  breaks 
up  into  striae,  which  flow  and  flicker  if  the  current  which  pro- 
duces them  be  in  the  slightest  degree  variable.  (See  p.  662.) 

The  discharge  through  a  vacuum  is  shown  to  be  disruptive 
by  the  fact  that  the  fall  of  potential  in  the  vacuum  tube  is  inde- 
pendent of  the  E.M.D.P.  of  the  circuit,  provided  that  this  be 
sufficient  to  produce  such  a  discharge  at  all;  and  there  is  a  large 
fall  of  potential  at  the  negative  pole. 

The  approach  of  external  conductors  repels  the  internal  glow  and  causes 
its  deflection ;  and  the  glow  is  deflected  by  a  magnet  in  the  same  way  as  a 
wire  bearing  a  current  would  be. 

In  very  high  vacua  the  discharges  from  the  two  poles  of  a  vacuum  tube 
appear  to  be  independent  of  one  another  :  each  pole  discharges  itself  with- 
out, as  it  were,  feeling  the  condition  of  the  opposite  pole  of  the  tube.  Even 
where  both  are  positive  they  may  discharge  towards  one  another  into  the 
same  space. 

Electrification  of  Radiant  Matter.  —  When  the  rarefac- 
tion of  a  gas  is  extreme  (one-millionth)  its  matter  becomes 
radiant.  The  movement  of  its  molecules  may  be  guided  and 
rendered  manifest  by  electrification.  In  a  Geissler's  tube,  the 
mattey  filling  which  is  radiant,  the  molecules  which  come  in 
contact  with  the  negative  pole  are  at  once  repelled  from  it  in 
lines  at  right  angles  to  its  surface.  Energy  is  imparted  to  these 
molecules  by  the  electrified  negative-pole,  and  where  these  mole- 
cules strike  each  other  or  other  molecules  they  produce  an 
internal  glow ;  where  they  strike  glass  (or  diamond  or  ruby) 
they  produce  light  and  cause  phosphorescence  ;  they  also  pro- 
duce heat,  so  that  when  they  are  directed  from  a  concave  nega- 
tive-pole upon  a  piece  of  platinum,  the  energy  conveyed  by 
them  brings  that  piece  of  metal  to  its  melting  point ;  and  when 
they  strike  a  movable  body  they  produce  obvious  mechanical 
effect  (Crookes). 


xvi.]  RADIANT  MATTER.  657 

Two  streams  of  molecules  proceeding  from  a  forked  negative-pole  repel 
one  another  like  two  similarly-electrified  gold  leaves,  and  the  negatively- 
electrified  particles  which  constitute  such  a  stream  are  attracted  and  deflected 
from  their  course  by  a  magnet. 

Chemical  Effects  of  a  Steady  Current— Electrolysis. — In 
most  cases,  if  a  liquid  permit  the  passage  of  a  steady  current 
through  it,  different  chemical  elements  or  groups  of  elements, 
contained  within  it,  travel  in  opposite  directions  along  the  lines 
of  force,  the  result  being  apparent  decomposition;  in  other 
words,  most  liquids  which  possess  conductivity  are  Electro- 
lytes. A  few  liquids,  such  as  alcohol  and  ether,  though  not 
absolutely  non-conductive,  are  not  decomposed  by  the  passage 
of  a  current.  As  a  rule,  substances  which  conduct  when  melted, 
but  insulate  when  solid  and  cold,  are  electrolytes,  e.g.,  glass. 

Let  us  take  as  an  example  the  effect  of  a  current  upon  a 
solution  of  hydrochloric  acid  in  water.  In  such  a  solution 
insert  two  platinum  plates, — the  one,  the  positive-electrode, 
connected  by  wire  with  the  copper  terminal  of  a  sufficient 
battery;  the  other,  the  negative-electrode,  with  the  zinc  or 
negative  terminal.  Hydrogen  is  liberated  on  the  surface  of  the 
negative-electrode ;  chlorine  is  liberated  upon  the  positive- 
electrode,  which  by  a  secondary  reaction  it  attacks  and  dissolves, 
with  the  formation  of  PtCl4. 

According  to  recent  researches,  the  mechanism  of  Electroly- 
sis seems  to  be  somewhat  the  following :  —  If  the  liquid  lying 
between  the  positive  and  negative  electrodes  were  pure  water, 
these  electrodes  would  be  brought  by  the  outside  battery  to  a 
definite  difference  of  potentials,  and  there  would  thereafter  be  no 
current,  but  a  condition  of  electrostatic  equilibrium  would  result, 
in  which  the  water  constituted  a  field  of  force ;  for  pure  water 
is  a  non-conductor.  If,  however,  the  water  contain,  say,  HC1  in 
solution,  then  some  hydrogen  and  some  chlorine  are  already 
dissociated  from  one  another,  and  exist  equally  disseminated 
throughout  the  solvent  as  separate  atoms  or  ions ;  the  hydrogen- 
atoms  are  positively  and  the  chlorine-atoms  negatively  charged 
with  definite  and  equal  quantities  of  electricity.  The  positive 
hydrogen-atoms  are  attracted  by  the  negative  electrode,  the 
negative  chlorine  by  the  positive.  Each  atom,  as  it  comes  up 
to  its  electrode,  discharges  its  electricity  into  the  general  cir- 
cuit ;  it  is  then  free  to  combine  with  another  similar  atom, 
similarly  discharged,  to  form  a  molecule  of  ordinary  free  non- 
electrified  chlorine,  or  of  hydrogen.  The  phenomena  of  Elec- 

2u 


658  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

trolysis  are,  therefore,  not  phenomena  of  decomposition,  but  of 
discharge  of  already-dissociated  ions  upon  the  electrodes ;  and 
the  number  of  free  charged  ions  which  can  come  up  to  the 
electrodes  in  a  given  time  limits  the  quantity  of  electricity 
which  they  can  bring  to  the  electrodes  in  that  time,  and  thus 
determines  the  Conductance  (or,  inversely,  the  Resistance)  of 
the  electrolyte.  Any  molecules  which  are  not  decomposed  by 
dissociation  appear  to  play  no  part  in  the  electrolytic  conduc- 
tion: and  the  electrolytic  conductivity  of  a  solution,  after  mak- 
ing due  allowance  for  dilution,  thus  measures  the  extent  to 
which  dissociation  has  taken  place  in  it. 

The  ions  cannot  travel  with  indefinite  velocity  through  the 
electrolyte  ;  each  kind  of  ion  has  its  own  specific  velocity  under 
the  electromotive  action  of  a  given  slope  of  potential. 

This  causes  differences  in  the  number  of  free  ions  which  reach  the 
electrodes,  and  corresponding  differences  in  the  quantities  of  electricity  dis- 
charged upon  the  electrodes  in  a  given  time ;  hence  the  total  Conductivity 
depends  upon  the  number  of  free  ions  of  each  kind  and  the  specific  velocity 
of  each :  and  it  increases  with  temperature  so  long  as  dissociation  remains 
incomplete,  but  no  further. 

In  the  electrolysis  of  a  solution  of  hydrochloric  acid  in  water,  since  the 
hydrogen  travels  five  times  as  fast  as  the  chlorine,  the  solution  becomes 
more  weakened  towards  the  positive  than  towards  the  negative  electrode : 
while  in  the  electrolysis  of  a  solution  of  potassium  chloride,  since  the 
velocities  of  the  ions  happen  to  be  nearly  equal,  the  solution  is  weakened 
almost  equally  at  the  two  electrodes. 

In  copper  sulphate,  CuSO4,  the  copper  plays  the  part  of  the 
hydrogen  of  the  previous  example,  while  the  part  of  the  atom  of 
chlorine  is  taken  by  the  atom-group  or  salt-radicle  SO4.  The 
copper  liberated  at  the  negative-electrode  forms  a  film  upon  that 
electrode ;  the  SO4  liberated  at  the  positive-electrode  reacts 
secondarily  upon  the  water  present;  SO4  +  H2O  =  H2SO4  +  O  ; 
the  positive-electrode  is  surrounded  by  sulphuric  acid,  and  oxy- 
gen is  liberated  on  its  surface. 

If  a  solution  of  sulphate  of  potash  be  electrolysed,  the  ions 
liberated  are  K,  K,  at  the  negative  and  SO4  at  the  positive  elec- 
trode ;  the  SO4  causes  the  evolution  of  oxygen  as  a  secondary 
product  at  the  positive-electrode ;  the  potassium,  by  the  reac- 
tion K  +  K  +  2H2O  =  H2  4-  2KHO,  causes  .the  evolution  of 
hydrogen  at  the  negative-electrode.  Here  the  water  seems  to 
have  been  decomposed;  the  apparent  decomposition  of  the 
water  is,  however,  a  secondary  result  of  the  liberation  of  the 
potassium  and  SO4  ions. 


xvi.]  ELECTROLYSIS.  659 

Alcohol,  oil,  and  bisulphide  of  carbon  are  practically  like  water,  non- 
conductors when  in  a  pure  state ;  but  if  metallic  salts  be  dissolved  in  them, 
the  solutions  are  electrolytes. 

The  secondary  reactions  met  with  in  electrolysis  depend 
upon  the  time  which  is  allowed  for  them,  and  are  therefore 
favoured  by  currents  of  small  intensity. 

If  copper  chloride  be  electrolysed  between  copper  elec- 
trodes, the  one,  the  negative-electrode,  is  thickened  by  a  deposit 
of  copper,  while  the  other  is  worn  away,  being  attacked  by  the 
chlorine ;  and  the  intervening  solution  of  copper  chloride  is,  if 
the  electrolysing  current  be  feeble,  maintained  in  its  state  of 
saturation;  but  if  the  electrolysis  be  very  rapid,  the  solution 
of  the  copper  electrode  does  not  keep  pace  with  the  evolution 
of  chlorine  upon  it,  and  the  solution  becomes  weaker  in  copper 
and  acid  in  its  reaction. 

The  secondary  reactions  observed  are  sometimes  very  peculiar.  When 
hydrochloric  acid  is  electrolysed,  the  chlorine  evolved  at  the  positive-electrode 
attacks  the  water  present  and  liberates  oxygen,  which  in  its  turn  attacks 
some  of  the  hydrochloric  acid  present  and  produces  chloric  and  perchloric 
acids.  When  Na2CO3  is  electrolysed,  Na  appears  at  the  negative  pole  (pro- 
ducing a  secondary  evolution  of  hydrogen)  and  CNaO3  at  the  positive  pole  ; 
this  last  group  reacts  upon  water  and  forms  oxygen  and  NaHCO3;  2CNaO3  + 
H0O  =  O  +  2  HNaCO3.  When  NaHCO3  is  electrolysed,  it  produces  Na  and 
CHO3,  and  then  2  CHO3  =  2  CO2  +  H2O  +  O.  When  formic  acid  (H.COOH) 
is  electrolysed  it  breaks  up  into  H  and  COOH ;  then  2  COOH  +  H2O  = 
2H.COOH  +  O,  or  formic  acid  and  oxygen;  but  the  oxygen  reacts  upon 
some  of  the  formic  acid  present,  and  then  H.COOH  +  O  =  H2O  +  CO2.  When 
fused  caustic-potash  is  electrolysed,  K  appears  at  the  negative,  HO  at  the 
positive ;  this  coalesces  into  H2O2,  and  breaks  up  into  H2O  and  O ;  but  if  the 
action  be  slow  the  K  acts  upon  some  of  the  water,  and  hydrogen  is  evolved. 
The  nascent  hydrogen  evolved  at  the  negative  pole  will  attack  aldehydes, 
forming  alcohols,  and  thus  certain  ill-tasted  rough  alcohols  may  be  greatly 
improved. 

Faraday's  Laws  of  Electrolysis. —  First  Law. —  The  quan- 
tity of  material  liberated  at  the  electrodes  from  a  given  elec- 
trolyte in  a  given  time  is  directly  proportional  to  the 
intensity  or  strength  of  the  actual  current;  or,  in  other 
words,  the  quantity  of  electricity  which  passes  in  a  given  time, 
in  a  circuit  of  which  a  given  electrolyte  forms  part,  is  propor- 
tional to  the  quantity  of  the  ions  actually  liberated  at  the  elec- 
trodes in  that  time. 

Faraday's  Second  Law.  —  This  Law  of  Electrochemical 
Equivalence  may  be  divided  into  the  following  propositions,  of 
which  the  fourth  may  be  regarded  as  a  paraphrase,  of  the  law 
itself : — 


660  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

1.  The  gramme-equivalent  of  a  metal  is  that  quantity 
which  will  chemically  replace  one  gramme  of  hydrogen.    For 
example  :  in  comparing  HC1   (H  =  1,  Cl  =  35-5)   with   AgCl 
(Ag  =  108,  Cl  =  35-5),  we  find  Ag  (=  108)  to  be  equivalent  to 
H(=  1)  ;  the  gramme-equivalent  of  silver  is  108  grammes.     In 
comparing  CuSO4  with  H2SO4  we  find  Cu  (=  6346)  equiva- 
lent to  H2(=2);   the  gramme-equivalent  of  copper  is  31-73 
grammes. 

2.  The  gramme-equivalent  of  a  salt-radicle  or  halogen  is  that 
quantity  which  will  combine  with  one  gramme  of  hydrogen. 
In  HC1,  35-5  grammes  of  chlorine  unite  with  1  gramme  of  hydro- 
gen ;  the  gramme-equivalent  of  chlorine  is  35'5  grammes.     In 
HNO3  (H  =  1,  NO3  =  62),  62  grammes  of  NO3  unite  with  1 
gramme  of  H  ;  the  gramme-equivalent  of  NO3  is  62.     In  H2SO4 
(H2  =  2,  SO4  =  96),  the  gramme-equivalent  of  SO4is48  grammes. 
In  H3PO4  (H3  =  3,  PO4  =  95)  the  gramme-equivalent  of  PO4  is 
31f. 

3.  The  gramme-equivalent  of  a  salt  or  acid  is  that  quantity 
which  contains  1  gramme-equivalent  of  the   halogen   or  salt- 
radicle. 

4.  When  a  current  whose  intensity,  after  the  current  has 
become  steady,  is  equal  to  A  Amperes  passes  in  a  solution  of  a 
salt,  0*000,  010352A  gramme-equivalents  of  the  salt  are  electro- 
lysed during  each  second  ;  0-000,010352J.  gramme-equivalents  of 
the  salt-radicle  or  halogen  being  liberated  at  the  positive  elec- 
trode and  a  corresponding  quantity  of  the  metal  at  the  negative. 

If  both  ions  be  monovalent,  as  in  Ag  |  N"O3,  the  electrolysis  of  the  solu- 
tion causes  the  liberation  or  deposition  of  0-000,010352/1  gramme-equiva- 
lents of  the  metal  as  well  as  of  the  negative  ion,  where  A  is  the  number  of 
Amperes.  A  current  of  1  Ampere  deposits  from  a  solution  of  pure  nitrate 
of  silver  0-001,118  gramme  of  silver  per  second,  or  4-025  grammes  per  hour; 
this  corresponds  to  0-000,010352  gramme  of  H  per  second. 

If  the  ions  differ  in  chemical  valency,  the  rule  holds  good  that  the 
amount  of  the  negative  ion,  the  halogen  or  the  salt-radicle,  liberated  at 
the  positive-electrode,  is  equal  to  0-000,010352^4  gramme-equivalents. 

Thus  let  a  current  pass  simultaneously,  in  "  series,"  through  a  solution 
of  cupric  chloride  and  a  solution  of  cuprous  chloride,  and  continue  to  pass 
through  both  solutions  for  five  minutes  ;  its  intensity  is  18  Amperes  ;  com- 
pare the  amounts  of  copper  deposited  on  the  negative-electrodes  in  the  two 
solutions.  In  each  the  amount  of  the  halogen  —  the  chlorine  —  liberated  is 
(-030010352  x  18  Amp.  x  300  sec.)  gramme-equivalents,'  or  (-000010352  x  18 
x  300  x  35-5)  grammes.  In  CuCl9  every  71  parts  of  chlorine  are  combined 
with  63-46  of  copper  ;  the  copper  deposited  from  the  CuCl2  is  therefore 


(-000010352  x  18  x  300  x  35-5)  x  grammes. 


xvi.]  ELECTROLYSIS.  661 

In  Cu2Cl2  every  71  parts  of  chlorine  are  combined  with  126-92  of  copper; 
the  copper  deposited  from  the  Cu2Cl2  solution  is  therefore 


(-000010352  x  18  x  300  x  35-5)  x  grammes,  double  the  quantity 

deposited  by  the  same  current  from  a  solution  of  cupric  chloride. 

When  acidulated  water  is  electrolysed,  0-0000103524  grm.-equivts.  of 
salt-radicle  are  set  free:  these  take  up  the  hydrogen  of  0-0000103524  grm.- 
equivts.  of  water,  and  set  free  0-0000103524  grm.-equivts.  of  oxygen,  that 
is,  (0-0000103524  x  8)  grammes.  Correspondingly,  0-0000103524  grammes 
of  hydrogen  are  set  free  at  the  negative-electrode.  Thus  it  is  said  that 
a  current  whose  intensity  is  4  Amperes  will  decompose  0-000,092,9614 
grammes  of  water  per  second  :  though  a  current  will  not  decompose  water 
at  all,  except  by  secondary  reactions  such  as  the  above.  A  Voltameter 
is  an  instrument  in  which  a  current  is  made  to  pass  through  acidulated 
water  between  platinum  electrodes  ;  the  hydrogen  and  oxygen  liberated  at 
the  electrodes  are  collected,  either  together  or  separately,  in  a  graduated 
tube  or  tubes,  and  measured  ;  a  simple  calculation  gives  the  strength  of  the 
current  actually  passing  through  the  voltameter.  Instead  of  sending  the 
whole  current  through  the  voltameter,  we  may  send  a  known  fractional 
part  of  it  by  arranging  the  instrument  in  a  Shunt. 

Since  each  Ampere  will  liberate  0-000010352  grm.-equivt.  of  salt-radicle 
or  halogen  per  second,  each  Coulomb  of  electricity  will  liberate  that  quan- 
tity independently  of  the  time;  and  each  C.G.S.  electrostatic  unit  will  lib- 
erate one  3000,000000th  part  of  this  quantity,  i.e.,  0-000000,000000,0034506 
gramme-equivalents.  This  last  quantity  is  otherwise  known  as  the  electro- 
static Electrochemical  Equivalent  of  the  salt-radicle  or  halogen  liberated 
or  salt  electrolysed. 

The  Energy  of  a  current  passing  a  quantity  Q  down  a  constant  poten- 
tial-fall E  is  equal  to  EQ;  when  Q  =  1  this  energy  is,  numerically,  E  ergs; 
but  it  is  also,  if  no  energy  be  lost  or  gained  collaterally,  equal  to  the  work 
done  by  the  unit  quantity  in  electrolysing  one  electrochemical  equivalent  of 
a  salt.  Hence,  conversely,  the  chemical  energy  liberated  by  the  production 
of  one  electrochemical  equivalent  of  a  salt  may  be  measured  in  terms  of  a 
potential-fall  or  an  electromotive  difference  of  potential. 

The  E.M.D.P.  in  a  galvanic  cell  may  thus  be  computed  in  terms  of  the 
chemical  energy  set  free  in  it.  Let  us  enquire  what  is  the  value  of  the 
E.M.D.P.  of  a  Daniell's  cell.  Here  we  have  a  chemical  action  going  on 
which  liberates  electric  energy  :  this  energy,  if  not  utilised  as  the  energy  of 
a  current  of  electricity,  may  wholly  appear  as  heat  ;  this  heat  has  been 
measured  in  various  ways,  and  the  mean  result  of  several  observations  is 
that  (between  extreme  values  714  and  805)  the  amount  of  heat  liberated 
when  one  gramme  of  zinc  is  dissolved  in  the  cell  amounts  to  760  ca  or 
31,610,680000  ergs.  The  gramme-equivalent  of  zinc  is  32-645  grammes 
(Marignac);  the  electrostatic  electrochemical  equivalent  of  zinc  is 
(0-000000,000000,0034506  x  32-645)  =  0-000000,000000,112645  grammes.  The 
amount  of  energy  liberated  on  the  solution  of  one  electrostatic  electrochemi- 
cal equivalent  of  zinc  is  therefore  (0-000000,000000,112645  x  31,610,680000) 
ergs  or  0-00356  ergs.  This  being  equal  to  the  amount  of  energy  liberated 
during  the  solution  of  one  electrostatic  electrochemical  equivalent  of  zinc 
in  a  Daniell's  cell  and  the  production  of  a  current  of  one  electrostatic  unit- 
intensity  for  one  second,  is  necessarily  equal  to  the  energy  which  would  be 
spent  by  a  current  of  the  same  intensity,  and  enduring  for  the  same  time, 


662  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

on  reversing  the  whole  chemical  process  which  results  in  the  production  of 
an  electrochemical  equivalent  of  zinc  sulphate,  that  is,  on  electrolysing  one 
electrochemical  equivalent  of  that  salt.  This  we  have  seen  to  be  numerically 
equal  to  E,  the  difference  of  potential  under  which  the  unit  current  passes. 
The  E.M.D.P.  of  a  Daniell's  cell  is  therefore  E  =  0-00356  electrostatic  units 
of  difference  of  potential.  This  is  equal  to  1-068  Volts;  a  theoretical  result 
which  does  not  depart  widely  from  the  experimental  values,  which  range 
between  1  Volt  and  1-124:  Volt.  This  mode  of  computation  is  due  to  Lord 
Kelvin. 

Electrolysis  in  Gases.  —  Cold  gas  will  not  conduct,  if  dustless :  hot 
gas  will :  it  seems  that  some  amount  of  dissociation  is  necessary  for  con- 
duction. Those  gases  which  are  most  readily  dissociated  are  the  best  con- 
ductors. Mass  for  mass,  highly  rarefied  gases  have  high  conductivities : 
and  dissociated  molecules  appear  to  be  concerned  in  conduction.  These 
molecules  appear  to  be  disposed  in  chains  here  and  there  along  the  path  of 
the  current,  those  of  each  kind  acting  together,  and  the  average  length  of 
the  chains  being  the  same  as  the  distance  between  the  strice.  When  a  cur- 
rent goes  through  rarefied  steam,  oxygen  and  hydrogen  are  liberated,  in 
approximately  the  same  amount  as  in  a  voltameter  traversed  by  the  same 
current  (Perrot  and  J.  J.  Thomson).  A  current  once  started  in  a  gas  can 
be  kept  up  with  comparative  ease ;  the  molecules  have  been  sufficiently  dis- 
sociated to  act  as  carriers.  When  oxygen  is  sent  through  a  space  across 
which  a  silent  discharge  is  passing,  it  becomes  in  part  converted  into  ozone, 
especially  if  there  be  rapid  variations  in  the  potential-difference.  Prof. 
Schuster  has  shown  reason  for  believing  that  in  highly  rarefied  gases  there 
is  considerable  dissociation  of  molecules  into  atoms  around  the  negative 
electrode.  In  mercury-vapour,  which  is  monatomic,  the  phenomena  of  glow 
are  the  same  round  both  terminals. 

In  Electrolysis,  Work  is  done  by  the  current ;  its  Energy  is 
spent  in  separation  of  the  charged  ions  of  the  electrolyte.  The 
energy  absorbed  in  the  electrolytic  production  of,  say,  9  grammes 
of  oxy-hydrogen  gas  from  acidulated  water,  is  approximately 
equal  to  the  Heat  liberated  by  chemical  action  and  change  of 
physical  state  when  1  gramme  of  hydrogen  and  8  of  oxygen  are 
exploded  together  and  condensed  into  water. 

Work  done  on  electrolysis,  if  such  work  be  other  than  chemical,  causes 
divergences  from  Faraday's  second  law.  Such  work  is  done  when  the  pro- 
ducts of  electrolysis  of  a  liquid  become  gaseous,  or  when  the  metal  of  an 
electrolyte  is  lifted  up  towards  the  negative-electrode ;  while  the  energy  of 
the  current  is  increased  when  these  circumstances  are  reversed. 

The  relation  of  the  energy  liberated  by  the  chemical  action 
of  the  battery  to  the  heat  produced  in  the  battery  and  the 
energy  spent  in  doing  electrolytic  work  is  represented  by 
the  equation  — 

Battery-energy  =  Heat  evolved  in  battery  +  Electrolytic  work  done. 
The  last  term  cannot   be  greater  than   the   first;    if   it   were, 
the  heat  developed  in  the  battery  would  be  a  negative  quan- 


xvi.]  ELECTROLYSIS.  663 

tity,  and  the  battery  would  cool  itself  as  well  as  all  surrounding 
objects. 

That  this  should  go  on  indefinitely  is  impossible :  and  in  such  cases  the 
E.M.D.P.  of  the  battery  falls  as  the  temperature  falls :  and,  further,  heat- 
ing the  battery  causes  a  rise  of  E.M.D.P.  Similarly,  if  the  battery  become 
heated  in  action,  heating  it  causes  a  fall  in  its  E.M.D.P.  If  the  battery 
become  neither  heated  nor  cooled,  as  in  the  case  of  Daniell's  cell,  heating  or 
cooling  it  from  without  causes  no  change  in  its  E.M.D.P. ;  and  it  is  only  in 
such  cases  that  the  E.M.D.P.  can  be  directly  calculated  from  the  chemical 
energy  alone. 

In  a  Daniell  cell  the  replacement  of  one  grm.-equivt.  of  copper  by  one 
grm.-equivt.  of  zinc  is  attended  with  the  evolution  of  24,200  ca  of  heat ;  in 
a  Grove  the  energy  evolved  is  47,000  ca  for  every  grm.-equivt.  of  Zn  dis- 
solved. When  1  grm.  H  and  8  grm.  O  unite  to  form  one  grm.-equivt.  of 
water,  34,462  ca  of  heat  are  evolved.  A  single  Grove  cell  can  therefore 
electrolyse  acidulated  water;  a  cell  with  a  potential-difference  equal  to 
f  If  H  times  that  of  a  Daniell  would  just  be  able  to  do  so;  a  single  Daniell 
cannot.  Two  such  cells  are,  however,  able  to  effect  electrolysis  of  water ; 
the  energy  supplied  by  two  cells  arranged  in  series  is  double  that  supplied 
by  one,  even  though  the  resistances  be  so  adjusted  that  the  current  pro- 
duced is  of  the  same  intensity :  for  Energy  per  second  =  El,  and  if  E  be 
doubled  while  I  remains  the  same,  the  energy  is  doubled. 

A  single  cell  will,  however,  electrolyse  water  if  the  positive-electrode  be 
of  a  substance  such  as  copper,  which  will  combine  with  oxygen,  and  thereby 
liberate  energy  sufficient  to  make  up  the  deficiency  in  that  supplied  by  the 
cell. 

The  process  of  electrolysis  is  turned  to  practical  account  in  the  arts  of 
electroplating,  etc. ;  the  article  to  be  covered  with  a  metallic  film  is  made 
the  negative-electrode  in  a  suitable  solution  of  the  metal.  The  positive- 
electrode  is  often  itself  made  of  the  metal  to  be  deposited  from  the  solution  : 
as  the  metal  is  deposited  from  the  solution  upon  the  negative-electrode,  the 
positive-electrode  is  attacked  and  its  substance  dissolved  in  the  solution, 
which  is  thus  kept  saturated  if  the  action  be  not  too  rapid. 

In  the  electrolysis  of  mixtures,  the  reaction  which  requires  the  least 
energy  is  the  first  to  be  completed;  and  thus,  by  regulating  the  potential- 
difference,  the  different  metals  may  be  successively  deposited  from  a  mixed 
solution. 

When  a  mixed  solution  of  acetates  of  lead  and  copper  (Nobili),  or  a 
solution  of  litharge  in  caustic  potash  (Becquerel),  is  electrolysed  between 
two  electrodes,  of  which  the  negative  is  a  sharp-pointed  platinum  wire  or 
steel  needle,  while  the  positive  is  a  large  plate  of  silver,  german-silver,  sil- 
vered copper  (spangle  metal),  or  even  thin  sheet-iron,  the  current  being  one 
of  relatively-great  E.M.D.P.  (15  to  20  very  small  Bunsen  cells  mounted  in 
file),  there  is  formed  on  the  positive-electrode  a  series  of  rings  concentric 
with  the  point  of  the  negative-electrode,  —  Nobili's  rings,  produced  by 
deposition  of  PbO9.  If  the  positive-electrode  be  complex  and  consist  of 
two  or  more  points,  or  if  currents  be  made  to  run  through  some  of  these 
points  towards  the  plate,  while  in  others  the  direction  is  reversed,  the  rings 
are  modified  into  representations,  in  iridescent  hues,  of  complete  systems  of 
equipotential  surfaces  (Guebhard).  , 


664  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

Polarisation  of  Electrodes.  —  When  platinum  electrodes 
have  been  used  in  the  electrolysis  of  water,  the  negative  one 
is  found  to  contain  a  certain  quantity  of  hydrogen  not  only 
adherent  to  its  surface,  but  also  occluded  within  its  substance ; 
while  the  positive  one  carries  similarly  a  certain  quantity  of 
oxygen.  These  oxygen  and  hydrogen  films  tend  to  produce  a 
reverse  current,  which  weakens  the  main  current.  When  the 
main  current  ceases,  the  reverse  current  continues  for  some  time 
and  dies  away  gradually. 

The  smallest  potential-difference  between  the  electrodes  is  sufficient  to 
direct  the  ions  so  as  to  set  up  this  state  of  polarisation ;  and  the  reverse  cur- 
rent tending  to  be  produced  by  this  is  equal  and  opposite  to  the  main  current, 
so  that  between  the  two  there  is  no  effect,  until  the  latter  reaches  a  certain 
limit.  When  it  exceeds  this  limit,  a  current  passes;  but  it  is  only  the 
excess  of  the  main  current  above  the  virtual  reverse  current  which  can 
pass  through  the  electrolyte. 

The  occluded  hydrogen  is  very  gradually  reduced  by  the  oxygen  brought 
to  it  by  a  reverse  current,  and  a  corresponding  quantity  of  hydrogen  is 
liberated  on  the  positive  pole,  where  it  is  occluded  by  the  electrode  or  oxi- 
dised by  the  occluded  oxygen  or  dissolved  by  the  water.  Thus,  though  a 
reverse  current  passes  when  the  main  current  ceases,  no  products  visibly 
appear. 

In  Grove's  gas-battery  a  number  of  such  electrodes  are  arranged  so 
as  to  give  a  current,  or  —  which  amounts  to  the  same  thing  —  a  circuit 
is  arranged  in  which  a  current  passes  through:  (1)  water;  (2)  a  plate  of 
platinum  standing  partly  in  water,  partly  in  hydrogen  gas ;  (3)  conducting 
wire ;  (4)  a  plate  of  platinum  standing  partly  in  an  atmosphere  of  ozonif- 
erous  or  electrolytic  oxygen,  partly  in  the  water. 

If  electrodes  of  palladium  be  used  in  the  decomposition  of  water,  the 
negative  one  absorbs  as  much  as  936  vols.  of  hydrogen,  with  which  it  forms 
an  alloy,  greater  in  size  than  the  original  electrode. 

The  electrodes  in  such  a  case  are  said  to  be  polarised,  and 
their  polarisation  within  a  given  substance  is  (if  the  resistance 
of  that  substance  be  not  so  great  as  to  prevent  any  perceptible 
current  from  passing  under  any  potential-difference)  the  best 
test  as  to  whether  that  substance  is  really  an  electrolyte.  Warm 
glass  is  thus  found  to  be  actually  an  electrolyte. 

The  capacity  of  electrodes  so  polarised  is  very  great.  Two  square  inches 
of  platinum  electrode  immersed  in  dilute  sulphuric  acid  have  (Varley)  when 
the  E.M.D.P.  is  one-fiftieth  that  of  a  Daniell's  cell,  a  capacity  equal  to  that 
of  an  electrostatic  condenser  whose  plates  have  an  area  of  80,000000  square 
inches  separated  by  |  inch  of  air ;  i.e.,  a  capacity  of  175  microfarads :  while 
as  the  E.M.D.P.  increases,  the  capacity  increases  also. 

Electrodes  of  amalgamated  zinc  are  not  at  all  polarised  when  used  to 
transmit  a  current  through  a  perfectly  neutral  solution  of  ZnSO4  (Beetz). 

When  a  powerful  current  is  sent  through  a  metal  immersed  in  dilute 
sulphuric  acid,  hydrogen  evolved  on  its  surface  offers  local  resistance,  and 


xvi.]  POLARISATION  OF  ELECTRODES.  665 

heat  is  there  evolved;    the   metal    may  rapidly  become  even  white-hot 
(Lagrange  and  Hoho's  process  for  welding). 

Secondary  Cells  and  Batteries.  —  When  acidulated  water 
is  electrolysed  between  electrodes  of  lead,  the  negative  electrode 
remains  bright,  but  the  positive  becomes  covered  with  a  film  of 
PbO2.  When  the  current  ceases,  if  the  positive  and  negative 
electrodes  be  connected  by  a  conducting  wire,  a  reverse  current 
—  a  polarisation-current,  or  Secondary  Current  — 
passes ;  the  film  of  PbO2  is  electro-negative  like  the  copper  in 
an  ordinary  cell,  the  lead  electro-positive,  like  the  zinc ;  and  the 
current  passes  in  the  conducting  wire  from  PbO2  to  Pb;  for 
which  reason  the  PbO2  pole  of  the  secondary  cell  or  battery  is 
called  the  positive  pole. 

The  reverse  action  appears  to  be  (Gladstone  and  Tribe)  in  the  main  the 
following :  — 

Pb02    H2S04  I  H2S04  I  Pb  =  PbO    H20    H2S04  I  PbSO4 

which  becomes 

PbSO4  |  H2O  |  H2O  |  PbSO4. 

On  passing  in  a  charging  current,  the  reaction  is,  for  thin  layers, 
PbSO4,  H2O,  O;  H2,  SO4Pb  =  PbO2,  SO4H2;  H2SO4,  Pb. 

An  arrangement  of  this  kind,  into  which  a  current  of  elec- 
tricity can  be  passed  and  a  reverse  or  secondary  current  obtained 
at  will,  is  a  Secondary  Cell;  and  secondary  cells  may  be 
grouped  into  Secondary  Batteries.  Planters  original  form  of 
secondary  cell  consisted  of  two  large  lead  electrodes,  separated 
by  a  sheet  of  felt,  rolled  up  into  a  spiral  and  immersed  in  10% 
dilute  sulphuric  acid.  Faure  improved  this  by  covering  both 
electrodes  with  a  layer  of  red-lead  of  about  10  kilogrs.  per  sq. 
metre,  held  on  by  layers  of  felt  and  parchment  between  the 
opposed  plates.  When  a  current  has  been  passed  into  a 
so-called  '  Faure-accumulator '  for  some  time,  the  red-lead  on 
the  surface  of  the  negative-electrode  is  converted  into  spongy 
lead,  while  that  on  the  positive-electrode  is  oxidised  to  PbO2. 
This  improvement  greatly  abridged  the  tedious  process  pre- 
viously necessary  for  Charging  Planters  4  accumulators.' 

In  the  Faure-Sellon-Volckmar  « accumulators '  there  is  no  felt ;  the  plates 
of  lead  are  pierced  or  cast  with  holes,  into  which  there  is  compressed  a  quan- 
tity of  red-lead,  of  reduced  lead,  or  of  a  salt  of  lead,  or  a  mixture  of  PbO, 
Pb8O4,  and  PbSO4.  One  of  these,  weighing  140  kgr.,  and  composed  of  43 
plates,  gave  (Hospitalier)  a  current  of  120  Amp.  for  6  hours. 

The  efficiency  of  these  batteries  is  in  part  due  to  the  fact  that  their 
porous  metal  or  oxide  is  in  close  contact  with  the  lead  plate,  and  is,  on 


666  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

account  of  its  porosity,  able. to  retain  large  quantities  of  the  oxygen  or 
hydrogen  which  is  evolved  when  an  external  current  is  passed  through  the 
accumulator. 

When  the  secondary  current  has  passed  for  some  time,  both 
films  become  mainly  converted  into  sulphate  of  lead,  and  the 
apparatus  is  ready  for  a  renewed  charge. 

When  a  secondary  battery  is  charged  by  two  or  three  Grove 
cells  and  disconnected  from  them,  it  will  be  rapidly  discharged  if 
connection  be  established  between  its  poles  by  means  of  a  thick 
wire.  From  sixty  to  seventy  per  cent,  or,  in  the  newer  forms,  a 
still  greater  proportion  of  the  energy  actually  sunk  in  it  can  be 
recovered  in  the  form  of  the  energy  of  the  secondary  current, 
especially  if  the  battery  be  not  allowed  to  run  down  too  far. 
When  the  cells  of  a  secondary  battery,  charged  side  by  side,  are 
disconnected  from  the  source,  and  then  connected  in  series  and 
discharged,  the  electric  current  produced  is  one  of  "high  ten- 
sion" or  great  fall  of  potential.  The  D.P.  of  a  single  cell  is 
about  2-25  Volts,  and  hence  a  hundred  such  cells,  first  arranged 
in  surface  and  charged  by  prolonged  connection  with  a  few 
cells,  can  be  made,  if  arranged  in  series,  to  pass  for  a  short  time 
a  current  of  E.M.D.P.  =  225  Volts;  such  a  current  produces  a 
high  temperature,  together  with  vibration  and  crumpling  of  the 
conducting  wire.  The  internal  resistance  of  these  cells  is  very 
small,  being  only  0-006  Ohm  in  a  single  cell  whose  surface  is 
about  300  sq.  cm. 

A  secondary  cell  of  about  18  Ibs.  weight,  the  plates  in  which  are  each 
about  2  sq.  feet  in  area,  will,  when  charged  by  three  Leclanches,  keep  at  a 
white  heat  during  5  to  10  minutes  a  platinum  wire  of  ^  inch  diar.  and  8 
inches  long.  A  pile  of  this  weight,  kept  constantly  connected  with  three  or 
four  Leclanche  elements  (which  require  little  attention  beyond  that  of  keep- 
ing them  moist),  is  a  very  convenient  means  of  heating  such  a  thing  as  a 
galvano-cautery  wire,  which  must  be  raised  to  a  high  temperature  for  a 
short  time.  The  rate  of  discharge  should  not  exceed  1  Ampere  per  sq. 
decimetre  of  plates. 

Electrical  Storage  of  Energy. —  When  energy  is  stored  up 
in  bent  steel  springs,  about  3924  megergs  per  kilogramme  can 
be  stored  up  —  i.e.,  40  kilogrs.  can  be  lifted  through  1  metre  by 
the  elasticity  of  a  spring  weighing  one  kilogramme. 

When  it  is  stored  up  in  compressed  air,  1.  kilogr.  of  air  com- 
pressed to  one-sixth  contains  2,250,000  megergs,  of  which  about 
450,000  can  be  recovered  in  the  form  of  work. 

When  it  is  stored  in  secondary  batteries,  about  500,000 
megergs  are  stored  up  per  kilogr.  of  secondary  battery.  Of  these, 


xvi.]  ELECTRICAL   STORAGE   OF  ENERGY.  667 

from  250,000  to  330,000  may  be  recovered  if  the  batteries  be 
used  within  a  day  or  two  after  charging. 

The  great  fault  of  these  "  accumulators  "  in  their  present 
form  is  their  want  of  durability. 

Equalisation  of  a  Current.  —  A  current  passed  through  one 
plate  of  an  Electrostatic  Condenser  will  be  apparently  absorbed 
when  the  current  is  increased,  and  will  be  given  out  equably 
when  the  current  falls  off ;  such  a  condenser  is  therefore  com- 
petent to  play  the  part  taken  by  a  flywheel  in  the  mechanical 
transmission  of  power. 

Similarly,  Secondary  Cells  may  be  made  to  serve  as  regu- 
lators of  a  current  if  they  be  fitted  up  in  the  course  of  the 
conductor  in  the  manner  indicated  in  Fig.  227.  The  direction  of 

Fig.  227. 


Z- 


the  secondary  current  is  indicated  by  the  dotted  lines  and  arrows 
connected  with  the  secondary  cell ;  it  opposes  the  main  current 
in  CZnCuD,  but  aids  it  in  DEC. 

THE  DYNAMICAL  PROPERTIES  OF  A  STEADY  CURRENT. 

If  a  straight  conducting  wire,  forming  part  of  a  wide 
circuit,  bearing  a  steady  current,  be  passed  vertically  through  a 
hole  in  a  piece  of  card  or  of  silver-paper  adjusted  to  a  horizontal 
position,  and  if  iron  filings  be  then  sprinkled  upon  the  card,  and 
if  the  card  be  gently  tapped  downwards  so  that  the  filings  may 
leap  into  positions  spontaneously  assumed  by  them,  they  will  be 
found  to  range  themselves  in  concentric  circles  round  the  current, 
while  each  filing  becomes,  for  the  time  being,  a  little  magnet. 

The  space  round  the  current  is  therefore  an  Electromag- 
netic Field  of  Force,  permeated  by  concentric  circular  Lines 
of  Magnetic  Force  or  of  Magnetic  Induction  and  by 
Magnetic  Equipotential  Surfaces,  which  ,are  at  right 
angles  to  these.  The  magnetic  equipotential  surfaces  all  have 


'U-II7IRSIT7' 
At*  « 


668  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

the  line  of  the  current  for  their  common  edge  or  boundary.  If 
the  current  be  straight,  these  equipotential  surfaces  are  planes ; 
and  if  they  were  visible,  and  if  the  current  could  be  looked  at 
end-on,  so  as  to  appear  a  mere  point,  these  surfaces  would  seem 
to  radiate  from  it  like  equidistant  radii  from  a  centre. 

The  lines  of  Magnetic  Force  mark  the  direction  in  which  an 
ordinary  magnet,  such  as  a  small  compass-needle,  when  placed 
within  the  field,  tends  to  place  itself.  The  one  end  of  the 
magnet  is  driven  in  one  direction,  the  other  end  equally  in  the 
opposite  direction,  along  these  lines  of  force:  the  magnet  is 
acted  upon  by  a  couple,  which  acts  upon  the  two  extremities  or 
Poles  like  the  hands  on  the  handles  of  a  copying  press  —  one 
pole  being  pushed  or  repelled,  the  other  being  pulled  or  attracted, 
until  the  magnet  lies  along  a  line  of  force.  The  moment  of  the 
couple  gradually  diminishes  as  this  position  is  being  assumed, 
and  the  couple  ultimately  ceases  to  produce  farther  rotation; 
and  further,  since  one  pole  is  attracted  as  much  as  the  other  is 
repelled,  the  magnet  as  a  whole  undergoes  in  such  a  field  no 
movement  of  translation. 

The  direction  in  which  a  current  tends  to  throw  the 
positive  or  north-seeking  pole  of  a  magnet  placed  in  its 
neighbourhood  is  shown  by  Fig.  228.  This  direction  is  called 
the  Positive  Direction  of  the  lines  of  magnetic 
Fig.228.  force.  The  Current  in  the  figure  passes  verti- 
cally  Upwards ;  the  Positive  pole  is  thrown  to  the 
Left  Hand  of  the  Current.  This  expression, 
left  hand  of  the  current,  is  obtained  by  suppos- 
ing the  current  to  be  replaced  by  a  person 
whose  head  is  at  B  and  feet  at  A,  and  who  turns 
so  as  constantly  to  keep  the  magnet-pole  in  full 
view.  The  relation  between  the  direction  of  the 
current  and  the  positive  direction  of  the  lines  of  magnetic  force  is 
always  the  same  as  that  between  the  propulsion  of  the  point  and 
the  twist  of  the  hand  in  the  ordinary  use  of  a  European 
corkscrew.  Conversely,  action  and  reaction  being  opposite, 
stationary  positive  magnet-poles  tend  to  throw  movable  conduct- 
ors, bearing  upward  currents,  to  their  left.  For  negative  mag- 
net-poles the  directions  given  are  reversed. 

Proposed  Mnemonic  Rule.  —  If  a  pen  be  held  in  the  right 
hand  in  the  usual  way,  the  penholder  may  represent  the  wire, 
and  the  direction  of  flow  of  ink  (towards  the  point)  the  direc- 
tion of  the  current ;  if  then  the  thumb  be  stretched  across  the 


xvi.]  FIELD   SURROUNDING  A  CURRENT. 

penholder  it  will  represent  the  magnet,  and  the  thumb-nail  its 
marked  or  north-seeking  pole.  The  same  relation  may  be  still 
more  simply  brought  to  mind  by  laying  the  thumb  across  the 
forefinger  of  the  right  hand ;  either  of  these  will  then  represent 
the  current  (flowing  towards  the  finger-tip  or  the  thumb-tip,  as 
the  case  may  be),  the  other  the  magnet. 

A  magnet-pole  may  be  made  to  rotate  round  a  current  by  keeping  the 
other  pole  in  the  axis  of  rotation.  In  general,  magnet-poles  tend  to  rotate 
round  currents  so  long  as  these  are  maintained ;  and,  conversely,  currents 
tend  to  rotate  round  magnet-poles:  and  the  deflection  of  a  magnet  by  a 
current  is  only  a  particular  case  of  this.  The  action  between  a  current  and 
a  magnet-pole  is  thus  at  right  angles  to  the  line  joining  them. 

There  is  a  magnetic  field  even  within  the  wire ;  if  a  magnet  be  made 
itself  a  conductor,  the  steady  current  being  led  from  its  midpoint  to  one 
end,  it  will  spin. 

Further,  Linear  Currents  act  upon  other  Linear  Cur- 
rents ;  but  they  do  not  throw  them  to  the  right  or  left ;  they 
attract  or  repel  them  directly. 

In  Fig.  229,  AB  is  a  steady  current ;  round  it  there  are  lines 
of  magnetic  force,  the  number  of  which  per  sq.  cm.,  at  any  point, 
varies  inversely  as  the  mean  distance  from  the  axis  of  the  wire. 
Let  another  current,  parallel  and  in  the  same  direction,  be 
brought  to  CD :  then,  be-  Fig.229 

tween  AB  and  CD  the  lines  B  Fj 

of  force  of  the  two  currents  ,^"~" 

are    opposed,    but    beyond  ^'.^.--si—"--^--.-----.  , 

c* —  —-"-•*- — 


CD    they    concur   in    their 
direction.       The    result   is 
that   the    medium    beyond  t..~- 
CD  is  in  a  state  of  greater  v*-. 
constraint     than    that    be- 
tween AB  and  CD ;   so  is 
that  to  the  left  side  of  AB ; 
and  the  two  conductors  are 

impelled  towards  one  another.  If  the  current  in  AB  be  opposed 
to  that  in  CD,  the  directions  of  the  lines  of  force  coincide 
between  AB  and  CD,  but  are  opposed  beyond  AB  or  CD,  and 
the  stress  is  such  as  to  drive  AB  and  CD  asunder.  If  the 
currents  in  AB  and  CD  be  at  right  angles,  approaching  one 
another  (Fig.  230),  the  current  in  AB  has  lines  of  force  whose 
direction  is  upward  on  the  left  side  of  AB,  downward  on  its 
right.  The  current  CD  has  lines  which  would  depress  a  posi- 
tive pole  placed  in  the  upper,  and  raise  a  positive  pole  placed 


670 


ELECTRICITY   AND    MAGNETISM. 


[CHAP. 


in  the  lower  part  of  the  figure. 

Fig.230. 


Up 


Up 


Down 


The  concurrence  of  lines  of 
force  is  in  the  upper  part  of 
I      the    diagram  (Fig.  230):    if 
the  current  AB  be  fixed,  the 
•£  conductor  CD  is  repelled  in 
^  a  direction  contrary  to  that 
|  of  the  course  of  the  current 
in    AB.      If   the    current   in 
CD  pass  from  D  towards  C, 
the   conductor  CD  tends  to 
move  in  the  same  direction  as  the  current  in  AB. 


Down 


Fig.  231. 


1 

ft          "A 

1 
(  / 

/ 

'      1 

These   results   were   first   summarised  in   Ampere's   formula.      The 
mechanical  force  of  Attraction  between  very  small  elements  of  two  linear 

circuits  is  equal,  in  dynes,  to  ii'  •  II' 
(2  sin  0  sin  0'  cos  w  — cos  0  cos  Of) 
/d2,  where  i  and  i'  are,  in  mag- 
netic measurement  (p.  707),  the  in- 
tensities of  the  currents  passing  in 
the  two  wires ;  I  and  /'  the  lengths 
of  the  elements ;  d  the  distance 
between  their  midpoints  ;  and  0, 
0',  w  are  the  angles  which  deter- 
mine their  relative  direction  as 
follows  :  —  Each  element  makes  an  angle  6  or  0'  with  the  line  AB ;  but  fur- 
ther, these  elements  are  situated  in  planes  which  make  an  angle  co  with  one 
another. 

We  may  take  some  particular  cases. 

Let  both   currents  lie  in  the  plane   of  the   paper ;    the  angle    <o  =  0 ; 
cos  <o  =  1 ;  F  =  ii'  -  IV  -  (2  sin  0  sin  0'  -  cos  0  cos  0')/d2. 

1.  Let  both  currents  be  parallel  to  AB ;   0  =  0,  &  =  0 ;  sin  0  =  sin  & 
=  0  ;  cos  6  —  cos  0'  =  1 ;  F  =  —  ii'  •  IV /d2.     Two  elements  of  current,  end-on 
to  one  another,  and  running  in  the  same  direction,  repel  one  another. 

2.  Let  both  be  at  right  angles  to  AB  ;  0  =  0'  =  90° ;  cos  0  =  0 ;  sin  0  = 
1 ;  F  =  *2  ii'  •  ll'/d*.      Two  elements  of  current,  parallel  and  abreast  of  one 
another,  running  in  the  same  direction,  attract  one  another.     If  they  be 
opposed  in  direction,  the  product  ii'  is  negative ;  then  F  is  negative,  and  the 
mutual  action  is  one  of  repulsion. 

3.  Let  one  be  at  right  angles  to  AB,  the  other  parallel  to  it ;  0  =  90° ; 
0'  =  0.     Cos  0  =  0  =  sin  0' ;  cos  0'  =  1  =  sin  6.     F  =  0.     There  is  no  appre- 
ciable action  between  two  extremely  small  elements  one  of  which  points 
end-on  and  at  right  angles  to  the  centre  of  the  other. 

4.  Let  one  be  at  right  angles  to  AB,  the  other  at  any  other  angle,  say 
45°.     B  =  90°,  &  =  45°.     F  =  V2  •  ii'  .  tt'/d*.     Two  currents  diverging  from 


*  If  in  this  case  I  =  V  =  d  =  1  cm.,  and  if  1=1'  =  I  electromagnetic  unit,  F  =  2 
dynes ;  but  by  using  smaller  intensities,  each  equal  to  (1  E.-M.  unit  -*•  \/2),  F  becomes 
equal  to  one  dyne.  The  reduced  intensity  so  found  is  the  electrodynamic  unit 
of  current-intensity,  the  basis  of  a  system  of  measurement  now  disused,  but  still 
occasionally  referred  to,  especially  in  French  works. 


XVI.] 


AMPERE'S   FORMULA. 


671 


a  point  attract  one  another;  if  one  of  the  currents  be  reversed,  they  repel 
one  another ;  if  both  be  reversed,  so  that  both  converge  upon  an  angle,  they 
attract  one  another. 

5.  Two  elements,  B  and  A,  parallel  in  their  direction ;  the  resultant 
force  is  one  of  attraction  between  B  and  A.  A  similar  element  at  A'  attracts 
B  towards  A'.  Of  these  two  attractions  the  resultant  is  towards  O.  Pairs 
of  such  elements,  symmetrically  ranged  to  an  infinite  distance  on  either  side 


232. 


of  O,  make  up  an  infinite  conductor  whose  attraction  for  B  is,  in  the  aggre- 
gate, F  =  2  (u'/OB)  x  length  of  element  B.  The  attraction  of  an  infinite 
current  for  an  element  of  current  running  parallel  to  it  is  directed  along 
a  line  at  right  angles  to  both  currents,  and  is  inversely  proportional  to  the 
distance  between  them. 

6.  The  element  B  flows  towards  O ;  A  and  A'  are  equal  elements,  sym- 
metrically arranged,  of  an  infinite  current  along  AA' :   A  attracts  B :  A' 


Resultant 


Pig.  233. 


equally  repels  B ;  the  resultant  is  in  a  line  parallel  to  AA',  and  tends  to 
drive  the  element  B  in  the  direction  A' A. 

The  formulae  of  Grassmann,  Clausius,  von  Helmholtz,  and  others,  we 
here  pass  over. 

In  all  the  above  examples,  it  is  assumed  that  the  medium  between  the 
two  wires  is  air ;  if  it  be  not  air,  all  the  above  expressions  for  the  value  of  F 
will  have  to  be  multiplied  by  a  factor  /u,,  special  to  each  medium,  and  known 
as  the  Magnetic  Permeability,  p.  685. 

The  difference  between  an  Electrostatic  and  a  Current  Field  is,  that  in 
the  former  the  Lines  of  Electric  Force  are  steady,  while  in  the  latter  they 
are  in  motion.  In  the  former  case  there  is  no  magnetic  induction ;  in  the 
latter  there  is.  The  relation  between  the  lines  of  Electric  Force  and  those 
of  Magnetic  Induction  may  be  expressed  by  a  simple  illustration.  Con- 
ceive a  guide-post,  which  of  course  points  upwards,  bearing  two  pointers, 
pointing  respectively  K  and  E. ;  let  the  guide-post  itself  represent  a  Line 
of  Electric  Force,  directed  upwards ;  let  the  one  pointer,  the  one  pointing 
Northwards,  be  marked  "  This  way  the  Direction  of  Motion  of  the  Line  of 
Electric  Force,  of  Momentum,  of  Transmission  of  Energy,  and  of  Repulsion 
of  a  Current  in  the  Electromagnetic  Field ;  "  and  the  oth^r,  the  one  point- 
ing Eastwards,  be  marked  "  This  way  the  Magnetic  Force  and  Induction :  " 


672 


ELECTRICITY  AND   MAGNETISM. 


[CHAP. 


and  further,  conceive  the  pointers  to  shrink  to  nothing  when  the  guide-post 
is  at  rest,  but  to  grow  simultaneously,  proportional  to  one  another,  when  the 
post  is  moved.  Then  if  a  model  of  this  be  made,  and  turned  into  any  posi- 
tion, and  moved  by  translation,  but  always  so  as  to  follow  the  direction  of 
the  Motion-pointer,  it  will  denote  the  respective  directions  of  the  lines  of 
Electric  Force,  of  the  lines  of  Magnetic  Force  and  Induction,  and  of  their 
Motion  in  the  Field.  A  surface  joining  the  two  rectangular  pointers  would 
be  at  right  angles  to  the  post  and  would  represent  an  Electric  Equipotential 
Surface  in  the  field ;  one  joining  the  post  with  the  Motion-pointer  would 
be  at  right  angles  to  the  magnetic  force,  and  would  represent  a  Magnetic 
Equipotential  Surface  in  the  field. 

If  a  wire  bearing  a  straight  steady  current  be  bent  into  a 
closed  circuit  or  loop,  its  equipotential  surfaces  are  modified 
into  a  series  of  bowl-shaped  surfaces  which  still  have  the  wire, 
the  contour  of  the  circuit,  for  their  common  edge  or  boundary. 
A  circular  current  would  have  equipotential  surfaces  whose 
general  form,  looked  at  in  section,  is  indicated  by  the  undotted 
curves  of  Fig.  234,  in  which  the  lines  of  force  are  shown  at  right 

Fig.234. 


angles  to  the  equipotential  surfaces.  To  one  side  of  the  circuit 
the  magnetic  potential  is  positive,  to  the  other-  it  is  negative. 
The  positive  side  of  the  circuit  is  such  that  an  observer  stand- 
ing girdled  by  the  current,  with  his  head  in  the  positive  region, 
would  see  the  current  pass  round  him  from  his  right  towards  his 
left  hand. 


XVI.] 


EQUIPOTENTIAL  SURFACES. 


673 


If  another  circular  circuit  be  brought  near,  and  if  the  direc- 
tion of  the  current  within  it  be  the  same  as  that  within  the 
former,  the  lines  of  force  or  of  induction  of  the  two  circuits 
coalesce,  and  the  two  circuits  attract  one  another.  The  result- 
ant system  of  surfaces  and  lines  takes  the  form  indicated  in  Fig. 
235.  The  field  of  force  between  the  two  circuits  is  approxi- 
mately uniform.  The  lines  of  force  are  all  closed  curves,  but 
some  of  them,  those  which  pass  up  the  centre  of  the  region 
between  the  circuits,  take  a  relatively-wide  sweep  into  space, 
and  seem  to  radiate  from  or  converge  upon  the  external  face  of 
either  circuit. 

A  large  number  of  such  circular  circuits  arranged  so  as  to 
have  a  common  axis,  and  thus  to  form,  as  it  were,  the  outline  of 

Fig.  235. 


EquJpotentlal  Surfaces  only.  Equipotential  Surfaces  &  Lines  of  Force. 

a  cylinder,  would  form  a  so-called  Solenoidal  system.  Such  a 
system  would  have  lines  of  force  or  of  induction  radiating  from 
each  extremity,  taking  a  more  or  less  ample  sweep  into  space, 
returning  into  the  opposite  extremity  and  passing  up  the  axial 
region  of  the  cylinder  from  the  negative  to  the  positive  region ; 
each  line  of  force  or  induction  being  thus  a  closed  curve. 
The  external  electro-magnetic  field  of  such  a  solenoid  system 
would  be  identical  with  that  produced  by  a  system  consisting  of 
an  attracting  disc  at  the  one  and  a  repelling  disc  at  the  other 

2x 


674  ELECTRICITY   AND    MAGNETISM.  [CHAP. 

extremity  of  the  solenoid ;  and  such  a  solenoid  would  by  one  of 
its  extremities  attract  the  north  and  repel  the  south  pole  of  a 
Fig  236    compass-needle ;  while  by  the  other  it  would  attract 
the  south  and  repel  the  north  pole.     Such  a  solenoid 
would,  so  far  as  its  external  action  is  concerned, 
act  like  a  bar-magnet;  and  Ampere's  theory  of 
Magnetism  is,  that  magnets  and  solenoid  systems  of 
currents  are  fundamentally  identical. 

A  solenoid  may  be  roughly  realised  by  winding  a  wire  into 
a  narrow  spiral  and  bringing  the  two  extremities  back  to  the 
same  point.  The  error  introduced  in  each  turn  of  the  spiral 
by  its  departure  from  a  perfect  ring-form  is  roughly  compen- 
sated by  the  return  of  the  wire  (Fig.  236). 

This  identity  of  action  of  Magnets  and  of  Sole- 
noidal  Steady  Current-systems  being  premised,  we  now  proceed 
to  give  a  rapid  summary  of  the  main  phenomena  of  Magnetism. 

MAGNETISM. 

Some  bodies  —  a  piece  of  loadstone,  a  compass-needle,  a 
wire  spiral  through  which  a  current  is  passing  —  tend,  when 
suspended  by  their  centre  of  gravity,  to  lay  themselves  in  a 
definite  direction,  and  to  place  a  definite  line  within  them, 
their  Magnetic  Axis,  in  a  definite  direction,  which,  roughly 
speaking,  lies  north  and  south.  The  same  bodies  have  the 
power  of  attracting  iron.  Such  bodies  are  called  Magnets. 

Curiously  enough,  this  directive  power  is,  according  to  Gore,  shared  by 
crystals  of  Cyanite,  an  anhydrous  monosilicate  of  alumina. 

Magnets  may  be  divided  into  Permanent  (loadstone,  hard 
steel  magnets)  or  Temporary  (a  solenoid  current,  or  an  Electro- 
magnet, i.e.,  a  bar  of  soft  iron,  whose  magnetic  properties  are 
induced  by  the  presence  of  an  electric  current  circulating 
round  it,  but  endure,  in  soft  iron,  no  longer  than  the  persist- 
ence of  that  current);  or  again,  into  Natural  (loadstone)  and 
Artificial. 

The  constituent  particles  of  a  magnet  are  themselves  mag- 
nets. A  permanent  magnet  may  be  cut  into  a  very  large  num- 
ber of  minute  fragments,  each  of  which  will  be  a  little  magnet, 
the  original  magnetic  axis  in  which  will  continue  to  point  to 
the  magnetic  north  and  south.  When  a  steel  bar  is  converted 
into  a  permanent,  or  a  soft-iron  bar  into  a  temporary  magnet, 
some  operation  must  be  effected,  not  upon  the  mass  as  a  whole, 


xvi.]  MAGNETS.  675 

but  upon  its  constituent  molecules,  or  groups  of  molecules. 
The  magnetic  axis  of  a  bar-magnet  or  compass-needle  coincides 
more  or  less  closely,  but  hardly  ever  with  perfect  accuracy, 
with  its  geometrical  axis  of  figure.  The  magnetic  axis  joins 
the  two  Poles  of  the  magnet. 

One  mode  of  expressing  the  mechanical  action  of  magnets  is  to  feign  a 
distribution  of  imaginary  magnetic  matter  at  the  Poles ;  positive  at  the 
one  pole,  equal  and  negative  at  the  other ;  the  attractions  and  repulsions 
observed  are  exercised  mainly  to  and  from  these  poles. 

Another  method  is  to  feign  a  distribution  of  magnetic  matter  partly 
over  the  surface,  partly  within  the  substance  of  the  magnet  (Poisson)  or 
over  the  surface  only  (Gauss  and  Green). 

Positive  and  negative  magnetic  distribution  may  be  feigned  to  be  either 
heaping-up  of  positive  matter  towards  or  at  positive  poles,  and  of  negative 
matter  towards  or  at  negative ;  or  else  to  be  distribution  in  excess  and 
defect  respectively  (or  inversely)  of  one  and  the  same  all-pervading  imagi- 
nary "  magnetic  fluid." 

These  modes  of  representation  are  convenient  for  calculation  and  expo- 
sition merely ;  and  indeed  the  only  case  in  which  it  can  be  said  that  there 
is  in  a  magnet  a  real  Pole,  a  point  at  which  the  imaginary  mass  may  be 
considered  as  concentrated,  is  that  of  an  ideally-thin,  infinitely-long,  uni- 
formly-magnetised wire.  In  every  other  case  the  distribution  of  forces  in 
the  field  surrounding  the  magnet  is  more  complex ;  but  it  is  convenient  to 
assume,  to  begin  with,  that  every  Magnet  has  two  Poles,  each  of  which  is  a 
Point. 

A  long  thin  bar  so  magnetised  that  all  its  molecules  would, 
considered  as  magnets,  be  absolutely  equal,  would  have  its 
poles  at  its  ends.  Such  a  theoretical  bar-magnet  is  called  a 
Solenoidal  Magnet.  In  practice  the  action  of  bar-magnets  is 
the  same  as  that  of  a  theoretical  Solenoid  whose  Poles  are  at 
a  somewhat  less  distance  from  another  than  the  extremities  of 
the  bar :  for  which  reason  the  Poles  of  a  bar-magnet  are  often 
said  to  be  within  its  substance,  at  a  short  distance  from  its  ends. 

The  North-seeking  pole  of  a  magnet  is  called  its  Positive 
pole ;  the  other,  its  south  pole,  is  called  its  negative  pole. 

In  different  magnets,  unlike  poles  attract  one  another ; 
like  poles  repel  one  another. 

The  force  F  of  repulsion  or  attraction  between  the  poles 
varies  inversely  as  the  square  of  the  distance  between 
them.  It  also  varies  directly  with,  and  furnishes  the  basis  for 
the  measurement  of,  the  strength  m  of  each  pole,  or  the  quan- 
tity m  of  imaginary  magnetic  matter  conceived  to  be  concen- 
trated at  each.  It  is  therefore,  in  air,  F  =  mm'/d2,  and  is  repul- 
sive when  the  poles  are  similar,  both  positive  or Jboth  negative ; 
attractive  when  they  are  dissimilar. 


676  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

If  any  other  medium  than  air  intervene  between  the  mutually 
repelling  or  attracting  magnet-poles,  F  =  mm'/^2,  where  fj,  is  the 
Magnetic  Permeability  of  the  medium. 

Two  poles  are  said  to  be  of  Equal  Strength  when  they  can 
replace  one  another  in  their  action  upon  external  magnetic  poles. 

Two  poles  are  said  to  be  each  of  Unit  Strength  if  they  be 
equal  and  have  between  them  a  repulsion-force  equal  to  one 
dyne  when  their  mutual  distance  is  one  centimetre  through  air. 

A  unit  pole  placed  at  1  cm.  in  air  from  a  similar  pole  of 
m  units  will  be  repelled  with  a  force  of  m  dynes. 

The  opposite  poles  of  a  magnet  are  of  strengths  opposite  but 
numerically  equal;  these  are  +  m  and  —  m;  their  sum  is  zero; 
the  sum  of  the  magnetisms  of  every  magnet  is  always  zero. 

It  is  not  possible  to  isolate  a  single  magnetic  pole  or  to 
produce  any  numerical  difference  of  strength  between  the  two 
poles  of  a  magnet. 

Round  a  magnet  there  is  a  Magnetic  Field  of  Force,  per- 
meated by  Magnetic  Lines  of  Force  or  Induction  and 
Magnetic  Equipotential  Surfaces.  An  isolated  positive- 
pole  (if  such  a  thing  were  possible)  placed  in  the  neighbour- 
hood of  the  positive  extremity  of  a  bar-magnet  would  be 
repelled  and  would  travel  to  the  negative  extremity,  not  by  the 
shortest  path,  but  by  following  the  wide  sweep  of  any  line  of 
force  on  which  it  might  happen  to  lie. 

The  Direction  of  a  magnetic  Line  of  Force  is  the  direction 
in  which  a  positive  pole  is  driven,  or,  conversely,  a  negative 
pole  pulled  upon. 

A  magnet  within  a  magnetic  field  is  acted  upon  by  a 
Couple :  its  positive  pole  is  driven,  its  negative  pole  drawn,  in 
the  direction  of  the  Lines  of  Force  passing  through  them :  the 
magnetic  axis  of  the  magnet  tends  to  coincide  as  nearly  as 
possible  with  a  line  of  force  passing  through  its  centre.  If  this 
line  of  force  be  curved,  the  axis  of  the  magnet  is  set  tangentially 
to  it. 

If  there  be  magnetic  attraction  between  any  two  magnetised  surfaces, 
the  Lines  of  Magnetic  Force  run  across  from  the  one  surface  to  the  other. 
If  Magnetic  Lines  of  Force,  proceeding  from  the  respective  surfaces,  present 
their  sides  to  one  another,  the  medium  between  the  two  surfaces  tends  to 
expand  so  that  these  lines  of  force  come  to  lie  farther  apart ;  and  the  two 
surfaces  accordingly  appear  to  repel  one  another. 

The  Condition  of  a  Magnetic  Field  at  a  point  is  determined 
(1)  by  the  Direction  of  the  Line  of  Force  passing  through  the 
point,  and  (2)  by  the  local  Intensity  or  Strength  h  of  the 


xvi.]  MAGNETIC   FIELD.  677 

field  —  i.e.,  by  the  amount  of   mechanical  force  with  which  a 
unit-pole  there  situated  is  repelled  or  attracted. 

The  unit  intensity  of  field  would  be  that  produced  by  a  unit  of  magnetic 
quantity  at  1  cm.  distance  through  air.  This  unit  of  intensity  is  called  a 
Gauss. 

When  the  local  intensity  is  h,  a  magnet  whose  length  is  I, 
and  whose  poles  have  the  respective  values  m  and  —  m,  is  acted 
on  by  a  couple :  the  force  acting  on  the  positive  pole  is  F  =  mh, 
and  its  moment  round  the 
midpoint  of  the  magnet  is 
(mh  x  ^7)  ;  the  moment  of 
that  acting  on  the  negative 
pole  is  ( —  m  x  —  h  x  ji) 
=  |-  mhl :  the  moment  of  the  couple  is  thus  mh  •  I.  At  a  spot  where 
the  intensity  h  =  1  —  that  is,  within  a  unit  magnetic  field 
—  the  moment  is  equal  to  mZ,  the  numerical  Strength  of  either 
pole  multiplied  by  the  Length  of  the  magnet.  This  moment  ml 
is  called  the  Magnetic  Moment,  &(,  of  the  magnet. 

If  a  magnet  of  length  I  be  broken  into  fragments,  each  of  length  l/n, 
the  magnetic  moment  of  each  fragment  is  m  x  l/n ;  the  sum  of  the  magnetic 
moments  of  all  the  n  fragments  is  n  x  m  •  l/n  =  ml=ffil,  the  magnetic  moment 
of  the  original  magnet,  and  every  fragment  possesses  poles  of  strength  m 
and  —  m,  equal  to  those  of  the  original  magnet. 

When  two  equal  magnets  are  arranged  thus — N— S,  N-S, 
the  extreme  poles  are  effective,  the  intermediate  ones  mask  one 
another ;  when  work  is  done  upon  them  in  separating  them,  the 
original  condition  is  restored  and  all  the  poles  are  again  mani- 
fest. When  n  such  magnets  are  connected  in  this  way,  all  the 
poles  except  the  extremes  mask  one  another.  A  uniform  bar- 
magnet  I  cm.  long  and  o  sq.  cm.  in  cross-section,  and  therefore 
having  a  volume  of  lo  cub.  cm.,  may  be  considered  as  a  collec- 
tion of  'lo  magnets,  each  1  cm.  in  length  and  1  sq.  cm.  in  section, 
and  therefore  each  of  unit  volume.  The  magnetic  moment  of 
the  whole  is  equal  to  that  of  lo  such  magnets :  the  magnetic 
moment  of  each  of  these  unit-volume  magnets  is  that  of  the 
entire  magnet  iB  =  m?,  divided  by  lo,  the  volume,  and  is  there- 
fore equal  to  tn/0.  This  is  3E,  the  so-called  Intensity  of  Mag- 
netisation of  the  bar-magnet. 

If  the  intensity  of  magnetisation  of  a  bar-magnet  were  equal 
throughout,  its  poles  would  be  situated  exactly  at  its  extremities. 
We  generally  find,  however,  that  magnets  present  abnormalities 
in  this  respect,  and  that  they  may  even  have  secondary  poles, 


678  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

produced  by  local  inequalities  in  this  intensity  and  by  consequent 
deficient  compensation  of  the  internal  poles. 

Such  complex  distributions  can,  within  a  bar,  generally  be  represented 
by  the  superposition  of  a  number  of  solenoids  of  different  lengths. 

A  steel  sphere  will  be  magnetised  uniformly  if  it  be  placed 
for  some  time  within  a  uniform  magnetic  field.     It  then  has  a 
F.   23g  moment  equal  to  that  of  a  small  axial  mag- 

net NS,  Fig.  238;  and  it  tends  to  lay  its 
axis  NS  along  that  line  of  force  which 
passes  through  its  centre. 

The  Earth  considered  as  a  magnet  is 
not  uniformly  magnetised;  its  intensity  of 
magnetisation  is  not  equal  throughout;  it 
does  not  act  upon  bar-magnets  placed  near 
its  surface  exactly  as  a  distant  bar-magnet 
would  do,  for  the  law  of  its  action  is  not  even  approximately 
expressible  by  any  formula  less  complicated  than  one  which 
contains  at  least  twenty-four  coefficients  (Gauss). 

Terrestrial  Magnetism.  —  The  neighbourhood  of  the  surface  of  the 
Earth  is  a  great  Magnetic  Field,  nearly  uniform  within  such  small  spaces  as 
the  interiors  of  rooms.  The  lines  of  force  point  in  the  northern  hemisphere 
downwards  and  northwards  :  in  the  southern  hemisphere  upwards  and  north- 
wards. A  compass-needle  thus  tends  to  place  itself,  in  the  northern  hemis- 
phere, so  that  its  magnetic  axis  points  downwards  and  to  the  Magnetic  North, 
which  is  inclined  to  the  west  of  the  true  or  Geographical  North  by  a  so-called 
Declination  or,  as  sailors  call  it,  a  'variation'  of  17°  6'  (at  Greenwich, 
1894) ;  20°  13'  at  Edinburgh,  22°  11'54"  at  Valentia ;  4°  5'  48"  W.  at  Wash- 
ington, D.C.,  1893;  14°  22'  54"  E.  at  Los  Angeles,  1893.  This  declination 
towards  the  west  is  at  present  decreasing  at  Greenwich  by  6'  per  annum ;  at 
Edinburgh  by  7'-8.  If  the  needle  cannot  move  except  round  a  vertical  axis, 
its  axis  cannot  point  downwards  or  upwards :  it  therefore  tends  to  point  to 
the  Magnetic  North,  lying,  as  it  does  so,  with  its  magnetic  axis  in  a  line 
situated  in  the  same  vertical  plane  with  the  true  line  of  force.  This  plane 
is  the  Magnetic  Meridian;  and  the  magnetic  axis  of  a  given  magnet 
may  be  found  if  the  magnetic  meridian  passing  through  the  place  of  obser- 
vation be  known. 

To  find  the  magnetic  meridian,  a  single  observation  is  not  sufficient.  The 
axis  of  figure  of  a  needle  may  not  coincide  with  its  magnetic  axis,  and  the 
needle  (which  is  provided  with  an  agate  cup  on  each  of  its  flat  faces)  is  there- 
fore observed  when  it  lies  poised  on  one  side,  and  again  when  it  lies  on  the 
other.  The  mean  of  the  two  positions  gives  the  position  of  the  magnetic 
axis  of  the  needle,  and  therefore  indicates  the  magnetic  meridian.  The 
downward  or  upward  direction  of  the  lines  of  force,  their  departure  from  the 
horizontal  line,  is  the  Inclination  or  Dip  of  the  needle.  This  is  down- 
wards in  the  northern  hemisphere,  upwards  in  the  southern.  A  needle 
suspended  on  a  horizontal  axle  will,  by  the  mean  of  two  readings,  give  the 
inclination  of  its  magnetic  axis.  Friction  prevents  the  attainment  of  very 


xvi.]  TERRESTRIAL  MAGNETISM.  679 

great  accuracy  in  this  measurement;  but  it  can  be  greatly  diminished 
by  slinging  the  axle  of  the  needle  upon  silk  threads.  The  inclination  is 
at  Greenwich,  1894,  07°  16'  30",  diminishing  by  l\'  per  annum ;  at  Edin- 
burgh, 70°  30' ;  at  Valentia,  68°  45' .  7,  1893 ;  in  Washington,  71°  4'  30", 
1893 ;  at  Los  Angeles,  59°  29',  1893. 

If  the  inclination  be  found,  and  if  the  horizontal  component  fj  of  the  at- 
tractive force  of  the  earth's  magnetism,  acting  upon  a  unit  pole,  be  known, 
we  have  the  data  required  for  determining  the  whole  intensity  of  the  earth's 
magnetic  field  in  the  direction  of  the  lines  of  force  at  any  point.  The  hori- 
zontal component  ij  at  Greenwich  in  1894  is  0-1832  dynes,  increasing  by  -0002 
per  annum ;  Washington,  0-19860,  1893 ;  Los  Angeles,  0-2725,  1893.  The 
vertical  component  at  Greenwich  is  0-4374  dynes,  scarcely  changing,  since 
1889,  from  year  to  year;  Washington,  0-57928,  1893;  Los  Angeles,  0-4630, 
1893. 

The  line  along  which  the  lines  of  force  are  horizontal,  and  at  which  the 
Inclination  or  Dip  is  equal  to  zero,  is  the  Magnetic  Equator,  which  does 
not  coincide  with  the  geographical  equator,  and  is  not  a  great  circle  of  the 
earth.  The  lines,  roughly  parallel  to  the  magnetic  equator,  along  which  the 
Dip  is  equal,  are  the  MagneticParallels:  these  are  lines  along  which  equi- 
potential  surfaces  cut  the  surface  of  the  earth.  The  intensity  of  the  earth's 
magnetic  force  may  be  indicated  by  the  distance  between  these  parallels, 
just  as  those  maps,  which  give  contour-lines  indicating  equal  levels,  may 
show  by  the  crowding  together  or  separation  of  these  lines  the  tendency  of 
water  to  rapid  or  to  slow  flow  over  the  face  of  a  country.  The  magnetic 
parallels  are  not  great  circles  of  the  earth ;  they  are  not  even  parallel  to  one 
another  ;  in  circurnpolar  regions  they  are  irregularly  elliptical,  and  the  needle 
points  to  their  centres  of  curvature.  A  Magnetic  Pole  is  a  spot  where 
the  equipotential  surfaces  of  the  magnetic  field  graze  the  earth's  surface  ;  the 
needle  there  stands  vertical,  the  dip  being  90°.  There  are  two  true  poles, 
one  Arctic  (negative),  the  other  Antarctic  (positive),  together  with  other 
points  towards  which  surrounding  magnetic  needles  seem  to  converge,  but 
which  are  only  the  centres  of  curvature  of  the  irregularly-shaped  magnetic 
parallels.  The  line  joining  the  Magnetic  Poles  does  not  coincide  with  any- 
thing which  may  be  termed  the  Magnetic  Axis  of  the  earth. 

The  terrestrial  magnetic  field  undergoes  remarkable  Variations.  The 
direction  of  the  lines  of  force,  and  therefore  the  dip,  the  declination  and  the 
position  of  the  magnetic  north,  as  also  the  intensity,  undergo  secular  changes ; 
and  there  are  other  changes,  some  of  which  depend,  like  the  period  of  sun- 
spots,  upon  a  cyclical  period  of  about  eleven  years,  others  upon  the  rotation 
of  the  sun,  upon  the  position  of  the  moon,  upon  the  time  of  the  year  and  the 
hour  of  the  day ;  while  other  disturbances,  productive  of  electrical  currents 
in  the  crust  of  the  earth,  so  powerful  and  so  irregular  as  sometimes  to  render 
telegraphic  signalling  perfectly  unintelligible  —  disturbances  known  as  Mag- 
netic Storms,  and  possibly  due  to  long  waves  in  the  Ether  — are  observed  to 
occur  with  special  frequency  in  sympathy  with  outbreaks  of  sunspots  and  of 
solar  storms  and  appearances  of  the  Aurora  Borealis.  The  nature  of  the 
undoubted  connection  between  the  Sun  and  the  magnetism  of  the  earth  is 
in  the  highest  degree  obscure ;  it  is  clear,  however,  that  the  sun  and  moon 
cannot  exercise  any  important  direct  effect  as  magnets,  although  when  one 
side  of  the  sun  is  turned  towards  us  the  terrestrial  magnetic  intensity  is 
greater  than  when  that  side  is  turned  away. 

The  diurnal  variations  have  been  traced  by  Schuster  to  causes  above 


680  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

the  earth's  surface ;  probably  those  electric  currents  in  the  upper  regions  of 
the  atmosphere  which  are  (Balfour  Stewart)  produced  by  the  action  of  the 
Sun,  and  which  give  rise  to  the  Aurora  Borealis,  and  are  best  marked 
during  periods  of  maximum  sunspots. 

To  find  the  Magnetic  Moment  of  a  Magnet.  —  We  must  combine 
the  magnetic  moment  JH  of  the  magnet  with  fj,  the  horizontal  component 
of  the  intensity  of  the  Earth's  Magnetic  Field  at  the  place  of  observation. 
By  one  process  we  can  find  the  value  of  iftflfj ;  by  another  we  can  find  that 
of  i!H/fy ;  from  these  data  we  can  find  not  only  iftfl,  the  magnetic  moment 
(for  «J  x  fH/fj  =  ffl2),  but  also  ij,  for  flfty  +  (fft/ij)  -  t2- 

1.  To  find  iiftfj  ("Method  of  Vibrations"):  —  Suspend  a  very 
long  magnet  by  one  thread  attached  to  its  centre ;  load  it  so  that  it  may 
swing  horizontally  round  a  vertical  axis :  observe  the  time  of  its  oscillation 
under  the  earth's  magnetic  attraction  of  one  pole  and  repulsion  of  the  other. 
The  time  of  a  complete  oscillation  is  T  =  2*-VrN/55l,  where  N"  is  the 
moment  of  inertia  (page  162).  In  the  simple-pendulum  formula  (page 
212)  T  =  27rVN/Gl.  Here  G,  the  weight  of  the  pendulum,  is  replaced  by 
ijm,  the  attraction  of  the  one,  and  —  fym  the  repulsion  of  the  other  pole,  or 
together  by  2fjm.  Also,  the  magnet  of  length  I  swings  suspended  on  its 
midpoint  like  a  couple  of  simultaneously-oscillating  pendulums,  each  of 
length  \  =  %l.  Whence  T,  the  period  of  a  complete  oscillation  to-and-fro, 
T  =  27r\/N/tjm/  =  27rVX/|Hfj.  T  can  be  found;  N"  can  be  ascertained 
for  any  given  needle  ;  whence  ilHfj  may  be  calculated. 

This  operation  is  difficult ;  for  |5l,  the  magnetic  moment,  and  therefore 
the  rate  of  oscillation,  varies  with  every  slight  vibration  or  change  of  state 
or  of  the  temperature  of  the  suspended  magnet. 

If  we  take  the  torsion  of  the  suspending  thread  into  account  we  find 
that  a  restitution-pressure  has  been  developed,  proportional  to  the  displac- 
ing force  F  =  Jjm  and  also  to  I ;  it  is  therefore  proportional  to  fym  •  I  and  con- 
sequently to  fyJH ;  call  it  p  -  fjfH.  The  couple  tending  to  restore  the  needle  to 
its  mean  position  is  not  fjfH  but  fyfH  +  p  •  fyiftfl ;  and  the  divisor  in  the  value 
of  T  is  not  Vptt,  but  VfrfaT7§jjI. 

To  find  the  value  of  N,  the  Moment  of  Inertia,  we  must  attach  to  the 
oscillating  needle  a  mass  whose  moment  of  inertia  N",  is  known  from  geo- 
metrical considerations.  Let  this  be,  for  instance,  a  ring  of  rectangular 
cross-section  whose  mass  is  m  and  whose  radii  are  r,  and  rlt ;  the  moment 
of  inertia  is  (No.  6,  p.  163)  \m  (ry/2  -f  r/2).  When  this  ring  is  fixed  to  the 
needle  in  such  a  way  that  the  horizontal  oscillations  of  the  needle  cause 
the  heavy  mass  to  rotate  horizontally  round  its  own  centre,  the  time  of  a 
complete  oscillation  is  increased  to  Ty,  which  is  equal  to  27r\/(N  -f  N^/fjfH. 
These  data  are  sufficient  to  give  the  value  of  N",  the  moment  of  inertia  of 
the  needle. 

If  the  needle  be  a  straight  wire  of  length  /,  N"  =  ml2/ 12  ;  whence  fjfH 
=  7r2'ra/2/3T2,  where  m  is  the  mass  of  the  needle  in  grammes  (Equation  a). 

TofindfH/ij  ("Deflection-Methods"):  —  We  may  use  the  same 
needle  to  produce  a  deflection  in  a  compass-needle  free  .to  swing  round  its 
midpoint ;  by  observing  the  deflection  from  the  Magnetic  Meridian  (p.  679) 
when  the  compass-needle  has  come  to  rest  under  the  influence  of  the  two 
couples,  we  find  the  ratio  of  these  couples  and  thus  learn  the  value  of  fH/ij. 
There  are  two  main  methods ;  the  End-on  deflection-method  and  the  Broad- 
side deflection-method. 


XVI.] 


MAGNETOMETKY. 


681 


In  the  former,  the  End-on,  Fig.  239,  DE  is  the  deflected  magnet  or 
compass-needle;    AB   the    magnet   whose 
moment  we  are  investigating ;  0  the  deflec- 
tion ;  d  the  distance  between  C  and  the  Fig.239. 
midpoint  of  AB,  a  distance  very  consider- 
able as  compared  with  the  dimensions  of  A  B 
DE  ;  I  the  length  of  AB  ;  then 
£H           2d 

a  formula  which,  when  the  length  I  of  the 

magnet  AB  is  small  in  comparison  with  the  distance  d,  becomes  tan  0  = 


I......L 

«       /E 


In  the  latter,  the  Broadside  method,  Fig.  240,  AB  is  fixed  so  that  its 
midpoint  is  in  a  line  with  the  mag-  Fig.  240. 

netic    meridian    passing  through   C,  [ 

and  d  being,  as  before,  the  distance  I  ^-^ 

between  the  centres  of  the  magnets,  !        --•-' 

the  deflection  0  is  such  that  -*    Q^^E 


which,  if  I  be  relatively  insignificant, 
becomes  tan  0  =  iftfl/fjrf3 ;  the  twisting 
couple  in  this  case  is  therefore  half 
that  due  to  AB  when  the  end-on 
method  is  applied. 

By  blending  equations  (a)  and  (b) 
we  find  that  the  data  of  the  end-on 
measurement  give 
£fl  =  (7rZ/2T)(d2-i/2)V2tan  0  •  m/'dd, 

and 

fj  =  (7rl/T(d*  -  i/2))  V2mrf/3  tan  0, 
expressions  which  involve  only  meas- 
urable terms,  and  which  give  numeri- 
cal  values  for   Jj    and   jjH    in   proper 
C.G.S.   units.      Similarly    by    blend- 


*  DE  is  supposed  so  small  that  all  forces  acting  on  it  act  along  the  line  BC,  and 
that  deflections  do  not  modify  the  forces  upon  it.  This  distance  BC  is  (d  —  5^)  ;  the 
strength  of  B  is  fR/l;  the  strength  of  D  is  m  ;  the  force  between  B  and  D  is 
kl)2-  Similarly  the  force  between  A  and  D  is  opposite  in  sign,  and  equal 


to  mffl/l(d+kl)2.    The  force  upon  D  is  thus 


2d/  (d*  — 


An  equal  force  acts  upon  E.  The  couple  acting  on  DE  is  thus  DE  x  mJH  •  2d/ 
(tf»—  I/*)2  =  f&.&l  •  2d/(d2  —  J/2)2,  where  fH,  is  the  magnetic  moment  of  DE.  When 
DE  is  deflected  through  an  angle  6  this  couple  becomes  2£fl£H,cZcos0./(d2  —  kl'2)2- 
When  this  couple  is  in  equilibrium  with  the  terrestrial  horizontal  couple  (Jj  •  m  •  DE  •  sin  6) 
or  (f)  •  m,  sin  6),  _ 

d-cos  6./(d?  —  W)2  =  fr  •  HJ(  .  sin  0,  whence 

--  (d2- 


f  It  is  supposed  that  all  parts  of  DE  are  appreciably  at  the  same  distances  from 


682  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

ing  equations  (a)  and  (c)  we  may  interpret  the  data  of  the  broadside 
method. 

If  we  have  any  doubt  as  to  the  true  value  of  I,  the  distance  between  the 
"  poles  "  of  AB,  we  can  find  it  by  repeating  at  a  different  distance  d  the 
observations  which  lead  to,  say,  equation  (b).  We  now  have  a  different  0, 
a  different  d,  but  still  the  same  /.  From  two  such  equations  we  can  obtain 
the  numerical  value  of  /. 

Magnetic  Potential.  —  A  magnetic  pole,  if  isolated,  would 
be  surrounded  by  concentrically-spherical  equipotential  surfaces 
traversed  by  radial  lines  of  force.  But  a  magnet  has  two  poles 
of  opposite  kind,  and  the  field  of  force  around  it  presents  a  char- 
acter approximately  represented  by  Fig.  234,  if  the  lines  there 
marked  Equipotential  Surfaces  be  held  to  represent  Lines  of 
Magnetic  Force,  and  vice  versd. 

The  potential  at  any  point  due  to  the  positive  pole  m  at  distance  d, 
through  air,  is  m/d ;  that  due  to  negative  pole,  —  m,  at  distance  d',  is  —  m/d' ; 
due  to  both  together  the  potential  is  O  =  {m/d  —  m/d'}  =  m(\/d  —  !/</'), 
which  has  the  same  value  for  every  point  on  one  and  the  same  equipotential 
surface. 

In  any  other  medium  than  air,  the  permeability  being  ti,  the  force 
between  two  magnetic  poles  m  and  tn'  is  F  =  m  •  m'//^2 1  then,  just  as  in  the 
corresponding  electrostatic  case  where  F  =  Q  •  Qf  /Kd2,  the  Magnetic  Poten- 
tial O  at  a  point  varies  inversely  as  /x;  the  field-intensity  h  also  varies 
inversely  as  /A ;  and  the  magnetic  induction  per  sq.  cm.,  b,  which  is  always 
equal  to  /xh,  is  independent  of  //,. 

Magnetic  Shell.  —  We  may  arrange  a  number  of  extremely 
short  magnetised  bars  side  by  side,  so  that  their  similar  poles  all 
point  in  the  same  direction ;  a  metal  sheet  is  thus  built  up,  of  which 
the  one  face  is  negatively,  the  other  positively  magnetised. 
Such  a  sheet  is  called  a  Magnetic  Shell.  The  Magnetic  Moment 
of  such  a  shell  is  the  sum  of  the  magnetic  moments  of  all  its 
portions.  Let  it  be  supposed  to  be  first  a  continuous  shell,  and 
then  to  be  divided  into  portions  each  1  sq.  cm.  in  area.  Each 
such  portion  will  have  a  magnetic  moment  $E.  The  magnetic 
moment  of  each  such  unit-area  portion,  if  this  be  invariable  over 
the  whole  shell,  is  called  the  Strength,  ?>,  of  the  Shell ;  it  is 
equal  to  the  magnetic  quantity  per  unit  of  area  x  the  thickness 
of  the  shell. 

AB  as  the  central  point  C  is;  i.e.,  at  distance  d  from  the  midpoint  AB,  at  distance 
VcF2  -\-  ?/2  from  either  A  or  B.  The  pole  D  is  attracted  by  the  one  end  of  AB  and 
repelled  by  the  other;  in  each  case  with  a  force  m-ffi/l-  AU2,  or  mffi/1-  (d2  +  ^)2- 
The  resultant  force  on  D  is  parallel  to  AB,  and  is  therefore  equal  to  the  whole  force 
acting  X  i '/  \A?2  +  i/2 ;  i.e.,  it  is  equal  to  mfH /  (eZ2+i/2)f.  The  resultant  on  E  is  equal 
and  opposed  in  direction.  The  couple  on  DE  is  therefore  mfH/  (d2  +  i/2)i  X  DE  X  cos  6 
=  ffl,  •fflL-cosfl./Ccff  +  i/2)^.  This  is  equal  to  the  terrestrial  couple  t-Jft-sintf; 
whence  tan  0 


xvi.]  MAGNETIC   SHELL.  683 

The  quantity  of  magnetism  per  unit  of  area  is  the  Mag- 
netic Superficial  Density,  9,  =  <p/d,  where  d  is  the  thick- 
ness of  the  shell. 

The  Potential  of  a  Magnetic  Shell  upon  a  Unit  Positive-Pole 
placed  at  any  point  facing  the  Positive  Aspect  of  the  Shell  will, 
in  air,  be  the  product  (see  p.  200)  of  the  Strength  of  the  Shell 
into  the  apparent  Surface  of  the  shell,  as  seen  from  the  unit- 
pole  —  this  apparent  surface  being  measured  by  the  projection 
of  the  shell  upon  an  ideal  sphere  whose  centre  is  occupied  by 
the  unit-pole,  and  whose  radius  is  1  cm.,  or,  in  other  words,  by 
the  value  of  the  Solid  Angle  co  subtended  by  the  Shell  at  the 
point.  [Magnetic  Potential  fl  =  &>?'//*•] 

If  the  shell  be  fixed,  a  positive  pole  would  tend  to  move 
away  to  regions  of  less  potential,  and  thus  to  travel  round  to  the 
negative  side  of  the  shell.  If  the  unit  pole  be  fixed,  a  shell 
would  tend  to  move  in  such  a  way  as  to  diminish  its  apparent 
area,  and  even  to  present  what  is  equivalent  to  a  negative 
area,  namely,  its  negative  side,  to  the  positive  pole ;  it  would 
therefore  tend  to  rotate. 

In  the  immediate  neighbourhood  of  a  magnetic  shell  the  angle  subtended 
by  it  is  2?r;  the  potential  near  the  positive  surface  is  therefore  27rcp,  where  £> 
is  the  strength  of  the  shell ;  near  the  negative  surface  it  is  —  2Tr<f ;  hence, 
when  a  unit  positive-pole  moved,  in  air,  from  the  -f  to  the  —  surface,  4?r^ 
units  of  work  would  be  done  by  it. 

Equivalence  of  Magnetic  Shell  and  Electric  Circuit.  — 
The  Equipotential  Surfaces  in  the  neighbourhood  of  a  magnetic 
shell  are  such  that  from  every  point  on  any  one  of  them  the  area 
of  the  shell  will  for  that  surface  appear  invariable.  But  equi- 
potential  surfaces  as  determined  by  this  criterion  are  identical 
in  form  with  those  bowl-shaped  equipotential  surfaces  which  sur- 
round a  closed  circuit  bearing  a  steady  current  of  electricity 
(Fig.  234) ;  provided  that  the  contour  of  the  shell  and  that  of 
the  circuit  be  the  same.  A  magnetic  Shell  and  a  Closed  Current 
of  electricity  may  therefore  have  in  their  vicinity  an  identical 
Magnetic  or  Electromagnetic  Field;  and  an  electro- 
magnetic field  is  a  magnetic  field  produced  by  a  current. 

Magnetic  Shells  and  equivalent  Currents  of  the  same  con- 
tour can  thus  replace  one  another,  the  difference  being  that  the 
shell  is  impervious,  while  the  circuit  is  not.  Hence,  when  a 
closed  current-bearing  circuit  is  placed  with  its  positive  face 
facing  a  positive  magnet-pole,  there  is  mutual  repulsion.  The 
positive  pole  is  repelled  along  the  lines  of  magnetic  force,  which 


(584  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

trend  positively  from  the  positive  face  of  the  circuit ;  such  a 
pole  would  tend  to  travel  repeatedly  through  the  circuit  along 
the  closed  lines  of  force.  The  potential  energy  of  the  system, 
which  always  tends  to  become  a  minimum,  is  thus  found  to  have 
no  fixed  value,  but  to  depend  on  the  number  of  times  the  pole 
has  passed  through  the  circuit.  This  anomalous  result  would  be 
due  to  the  continuous  supply  of  energy  by  the  current  itself. 

As  regards  the  Lines  of  Force  and  of  Induction  in  the  elec- 
tromagnetic field  surrounding  a  current-bearing  Circuit,  these 
are,  for  one  particular  intensity  or  strength  of  current,  exactly 
the  same  as  those  in  the  Magnetic  Field  surrounding  a  magnetic 
Shell  of  the  same  contour  and  of  a  given  strength  ;  and  by  adopt- 
ing a  system  in  which  that  particular  Intensity  of  the  Current  is 
said  to  be  the  same  numerically,  in  air,  as  the  Strength  of 
the  equivalent  magnetic  Shell,  we  are  able  to  state  all  electrical 
and  magnetic  quantities  in  terms  of  Magnetic  or  Electromag- 
netic Measurement. 

In  other  words,  i  =  <p,  where  ^  is  the  strength  of  the  equivalent  shell ; 
and  if  the  area  of  shell  or  circuit  be  A,  At  =  A<f>  =  the  Magnetic  Moment  of 
the  equivalent  shell ;  but  if  the  circuit  be  coiled,  so  as  not  to  be  a  single 
loop,  but  to  surround  its  own  axis  n  times,  the  magnetic  moment  of  the 
circuit  =  Am;  and  if  the  medium  constituting  the  field  be  any  other  than 
air,  the  magnetic  moment  Ay  of  the  equivalent  shell  =  /x  •  An  -  i,  or  for  a 
single  loop,  <p  —  /w,  •  i. 

Magnetic  Induction.  —  Soft  iron  filings  are  attracted  by  a 
magnet,  and  themselves  become  temporary  magnets.  This  they 
do  even  though  they  be  not  in  contact  with  a  magnet,  but  merely 
exposed  to  such  forces  as  can  act  upon  them  within  a  magnetic 
field.  Soft  iron  completely  loses  its  magnetic  properties  when 
removed  from  the  neighbourhood  of  a  magnet ;  but  a  steel  or 
hard  iron  bar,  which  is  with  greater  difficulty  induced  to  become 
a  magnet,  will  not,  when  removed  from  the  field,  entirely  lose 
its  magnetic  state,  but  preserves  a  certain  Residual  Mag- 
netisation. The  property  of  steel  or  hard  iron,  in  virtue  of 
which  it  slowly  takes  up  and  slowly  parts  with  a  magnetic  con- 
dition, is  traditionally  named  its  Coercitive  Force.  Any  vibra- 
tion or  jar  which  facilitates  relative  movement  of  particles  of 
the  iron  will  enable  its  molecules  to  yield  to  the  inducing 
forces,  and  will  facilitate  the  magnetisation  of  the  iron :  and 
after  its  removal  from  the  field,  such  a  jar  will  facilitate  its  loss 
of  magnetic  condition. 

A  poker  suspended  near  the  earth's  surface  and  repeatedly  struck  will 
become  feebly  magnetic ;  so  does  an  iron  ship  which  is  exposed  to  much 
hammering  during  construction ;  and  all  working  machinery  is  magnetic. 


xvi.]  MAGNETIC  INDUCTION.  685 

The  effects  of  the  inducing  forces  within  a  magnetic  field  differ  from 
those  within  an  electric  field  of  force  in  the  following  respects: —  (1)  The 
action  is  one  which  affects  the  state  of  each  molecule ;  (2)  There  is  no  repul- 
sion of  a  mass  of  iron  or  steel  which  conies  in  contact  with  a  magnet ;  and 
(3)  The  power  of  taking  up  a  magnetic  condition  in  any  marked  degree  is 
limited  to  a  very  small  number  of  bodies,  though  to  a  slight  extent  it  is  pos- 
sessed by  all. 

The  strength  of  the  poles  of  an  induced  magnet  depends  on 
the  nature  of  the  magnetic  field,  and  therefore  on  the  strength, 
the  distance,  the  direction,  the  form,  of  the  inducing  magnet ; 
and  also  upon  the  nature  of  the  body  acted  upon,  its  form,  its 
direction,  its  temperature,  and  its  size. 

In  some  substances  the  magnetisation  induced  is  such  that 
the  north  pole  of  the  induced  magnet  lies  as  far  as  possible 
along  the  lines  of  force,  —  as  far  as  possible  away  from  the 
north  pole  of  the  inducing  magnet.  Such  substances  —  iron, 
nickel,  cobalt,  manganese,  chromium,  oxygen,  etc.  —  are  Para- 
magnetic or  Ferromagnetic. 

In  other  substances  the  direction  of  the  induced  magnetisa- 
tion is  the  reverse  of  this.  Such  substances  —  bismuth,  anti- 
mony, silver,  copper,  hydrogen,  nitrogen,  etc. — are  Diamagnetic. 

Intermediate  between  these  are  such  substances  as  air,  which 
do  not  become  magnetic,  or  but  very  slightly  so. 

In  still  other  substances  the  induced  magnetisation  is  not 
parallel  to  the  lines  of  force,  but  is  along  certain  Lines  of  Induc- 
tion within  the  body,  whose  direction  depends  upon  the  molec- 
ular agglomeration  or  the  crystalline  constitution  of  the  body. 

The  Lines  of  Induction  within  an  induced  magnet 
must  therefore  be  distinguished  from  the  Lines  of  Force, 
with  which  they  do  not  in  all  cases  coincide ;  within  a  mag- 
netic field  in  air,  on  the  other  hand,  they  are  coincident  in  all 
respects. 

The  magnetisation  induced  in  a  magnetisable  body  exposed 
to  induction  is  proportional  to  the  local  strength  h  of  the  field  ; 
and  the  number  of  lines  of  magnetic  induction  per  sq.  cm.,  or 
the  Magnetic  Induction  or  Flux  per  sq.  cm.,  within  that  body, 
is  b  =  yu-h,  where  ^  is  a  coefficient,  the  Coefficient  of  Mag- 
netic Induction,  or  the  Permeability,  or  Inductivity,  of 
the  substance.  The  Total  Magnetic  Induction  or  Flux,  or  the 
Total  Number  of  Lines  of  Magnetic  Induction  across  a  given 
area  A  is  B,  =  Ab  if  b  be  uniform  over  that  area. 

The  number  of  Lines  of  Magnetic  Induction  round  a  pole  ttt  is  always 
B  =  47rm;  •*•  b  =  B/4:7rr2  =  m/r2  per  sq.  cm.  at  distance  r:  the  number  of 


686  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

Lines  of  Force,  within  a  medium  of  permeability  /A,  is  47rtn//x  all  round  the 
pole  :  .•.  h  =  m/r2/jL  per  sq.  cm.,  at  distance  r. 

The  number  of  lines  of  induction  from  1  sq.  cm.  of  either  face  of  a  flat 
magnetic  shell,  of  superficial  density  s  =  (b/4?r),  is  47rs=b  lines  per  sq.  cm.  ; 
for  the  lines  do  not  diverge,  but  all  pass  in  one  direction,  4?r  lines  from  each 
unit  of  magnetic  quantity.  Conversely,  if  there  be  b  lines  of  induction 
crossing  each  sq.  cm.  of  a  given  area,  these  will  be  associated  with  a  dis- 
placement or  separation  of  magnetism,  or  Magnetic  Flux,  along  the  line 
of  their  direction,  to  the  amount  of  m'  =  s  =  b/47r  units  of  magnetic  quantity 
per  sq.  cm.  This  magnetic  displacement,  b/47r  =  /xh/47r  per  sq.  cm.,  multi- 
plied by  half  the  inducing  force  h  per  sq.  cm.,  gives  the  local  magnetic 
energy  of  the  magnetic  field,  i.e.,  /xh'2/87r  ergs  per  cub.  cm.  ;  provided  that 
that  energy  be  wholly  induced  and  be  stored  up  conservatively  in  the  field, 
so  that  there  is  no  tendency  to  retention  of  magnetisation  when  the  induc- 
ing cause  ceases  to  act. 

If  we  consider  the  air  in  the  neighbourhood  of  a  magnet, 
we  find  a  certain  number  of  lines  of  force  or  of  induction  pass- 
ing through  a  given  bulk  of  it.  If  we  replace  the  given  bulk 
of  air  by  an  equal  bulk  of  iron,  we  find  the  lines  of  induction 
passing  through  the  iron  to  be  more  numerous  than  those  pre- 
viously within  the  undisturbed  magnetic  field.  The  equipoten- 
tial  surfaces  are  also  farther  apart  within  the  iron,  so  that  iron 
may  be  said,  on  the  analogy  of  Electric  Conduction,  to  transmit 
inductive  effect  better  than  air  or  a  vacuum. 

Faraday  called  the  permeability  //,  the  Conductivity  for  Lines  of  Mag- 
netic Force. 

The  number  of  lines  developed  in  a  body  of  permeability  /x,  exposed  to 
the  influence  of  induction  within  a  field  of  magnetic  force  h,  is  such  as  to 
meet  two  requirements  ;  those  lines  shall  remain  which  would  have  been 
present  had  there  been  no  magnetisable  body  brought  into  the  field  ;  that 
is,  there  shall  be  h  lines  per  sq.  cm.  ;  and  secondly,  there  shall  be  lines 
developed  in  consequence  of  the  presence  of  the  magnetisable  body,  the 
number  of  which  lines  is  proportional  to  the  magnetic  separation  or  dis- 
placement induced,  per  sq.  cm.  cross-section  of  the  magnetised  body.  This 
displacement,  being  proportional  to  the  strength  h  of  the  field,  is  equal  to, 
say,  xh  units  of  magnetic  quantity  per  sq.  cm.  ;  K  is  the  Coefficient  of  Induced 
Magnetisation  for  the  substance  acted  upon,  or  its  Magnetic  Suscepti- 
bility :  and  the  number  of  induced  lines  running  through  the  magnetised  sub- 
stance, in  the  direction  along  which  the  displacement  has  occurred,  is  47r-/ch 
per  sq.  cm.  Together,  the  lines  per  sq.  cm.  =  b  =  h(l  +  4?™)  ;  whence 


For  air,  earth,  and  almost  all  unmagnetisable  substances,  K  is  approxi- 
mately =  0,  and  //.  =  1  ;  for  bismuth  K  =  —  0-000,0025,  and  //,  -  0-999,968,584. 
Bismuth  is  the  most  strongly  diamagnetic  substance  known,  and  has  the 
least  known  permeability.  For  iron,  K  varies  according  to  the  inducing 
force  h  applied,  and  tends  to  fall  to  0  for  successive  increments  of  h  ;  the 
value  of  (b  —  h)  therefore  tends  to  a  limit.  When  h  is  about  50  in  good 
soft  iron,  K  is  about  25-39  and  //,  about  320;  but  K  falls  to  nearly  0  and 
p.  to  1,  for  increments  of  h,  when  h  is  about  24,000,  and  /xh  about  45,400. 


xvi.]  MAGNETIC   PERMEABILITY.  687 

The  value  of  JJL  also  varies  to  some  extent  with  vibration,  the  mechanical 
condition,  and  the  temperature. 

The  permeability  /u  of  a  substance  can  be  measured  by  comparing  the 
deflection  produced  in  a  distant  magnet  by  a  magnetising  coil,  first  alone, 
and  then  provided  with  a  core,  consisting  of  the  substance  to  be  examined ; 
or  by  arranging  an  exploring  coil  of  wire  so  as  to  embrace  all  the  lines  of 
induction  developed  through  a  given  cross-section  of  that  substance  in  a 
field  of  known  magnetic  intensity,  and  connecting  this  exploring  coil 
with  a  Ballistic  Galvanometer  (p.  713) ;  then,  when  the  substance  is  mag- 
netised or  acted  upon  by  the  inducing  current,  a  secondary  current  (p. 
700)  is  induced  in  the  galvanometer  circuit,  and  the  throw  of  the  needle 
affords  the  means  of  comparison  with  an  earth-inductor  (p.  718)  of  known 
power. 

In  medical  magneto-electric  machines,  an  adjustable  piece  of  soft  iron 
is  used,  in  order  to  weaken  the  field  of  the  permanent  magnet  when  it  is 
brought  near  the  magnet-poles,  by  drawing  off,  through  its  substance,  some 
of  the  lines  of  that  field. 

A  diamagnetic  substance  has  the  reverse  property;  fewer 
lines  of  induction  pass  through  it. 

The  result  oi  this  distortion  of  the  lines  in  the  magnetic 
field  is  to  set  up  stresses,  which  tend  to  cause  an  iron  bar  to 
assume  a  position  parallel  with  the  lines  of  force ;  while  a  bar 
of  bismuth  tends,  in  a  non-uniform  field,  to  move  into  a  position 
of  least  negative  magnetisation,  and  to  lie  across  these  lines,  at 
right  angles  to  them  if  possible.  Substances  which  in  the  form 
of  bars  take  up  this  cross-position  in  a  non-uniform  field  — 
Diamagnets  —  comprise  the  great  majority  of  the  substances 
found  in  nature. 

In  a  uniform  field,  a  magnetically  isotropic  substance  (i.e.,  one  in 
which  all  directions  are  magnetically  similar),  if  its  form  be  spherical, 
becomes  simply  magnetised  and  remains  at  rest.  If  its  form  be  ellip- 
soidal or  otherwise  elongated,  it  tends  to  rotate  until  its  greatest  length 
lies  parallel  to  the  lines  of  force,  and  this  whether  it  be  paramagnetic  or 
diamagnetic. 

If  the  substance  be  aelotropic  (i.e.,  having  different  susceptibilities,  K, 
in  different  directions)  and  spherical,  the  sphere  tends  to  rotate  until  it 
brings  its  direction  of  greatest  susceptibility  parallel  to  the  lines  of  force. 
If  it  be  elongated  we  have  two  cases :  (1)  in  the  case  of  very  small  sus- 
ceptibility the  form  has  no  effect,  and  the  axis  of  greatest  susceptibility 
comes  to  lie  along  the  lines  of  force ;  (2)  in  the  case  of  great  susceptibility 
the  longest  axis  lies  parallel  to  the  lines  of  force. 

In  a  non-uniform  field,  an  isotropic  sphere  tends  to  move  along 
the  Lines  of  Slope  (p.  200)  into  regions  of  greater  force  if  the  substance 
be  paramagnetic,  into  regions  of  less  force  if  it  be  diamagnetic.  An  elon- 
gated body,  if  it  be  paramagnetic,  lies  along  the  lines  of  force,  a  position  in 
which  it  lies  as  far  as  possible  in  the  strongest  part  of  the  field ;  if  it  be 
diamagnetic  it  rotates  so  as  to  lie  across  the  lines  of  force,  a  position  in 
which  it  lies  on  the  whole  in  the  weakest  possible  part  of  the  field. 


688  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

If  the  substance  be  aelotropic,  the  force  tending  to  produce  motion 
along  the  Lines  of  Slope  is  greatest  when  the  axis  of  greatest  paramag- 
netic susceptibility  is  parallel,  or  that  of  least  diamagnetic  is  at  right 
angles  to  the  lines  of  force :  and  in  the  case  of  crystals,  the  susceptibility 
being  small,  there  will  be  rotation  into  this  position  whatever  the  form  of 
the  crystal. 

A  diamagnetic  substance  has  therefore  two  obvious  characteristics : 
(1)  a  bar  of  it  places  itself  across  the  lines  of  force  in  a  non-uniform  field, 
and  (2)  a  sphere  of  it  in  a  non-uniform  field,  such  as  the  neighbourhood  of 
either  pole  of  a  magnet  or  of  an  electric  current,  is  repelled  into  regions  of 
weaker  force. 

A  paramagnetic  substance  in  bar-form  places  itself  along  the  lines  of 
force,  and  a  sphere  of  it  is  attracted  by  either  pole  of  a  magnet. 

With  these  we  may  compare  a  sphere  of  an  already  magnetised  sub- 
stance, which  is  attracted  by  the  one  pole  of  a  magnet  and  may  be  repelled 
by  the  other,  if  the  disturbance  of  its  magnetic  condition  due  to  induction 
do  not  overpower  the  already  existing  magnetic  distribution. 

Any  substance  in  which  /x  has  a  less  value  than  it  has  in  the  surrounding 
medium  will  behave  as  if  it  were  diamagnetic :  for  example,  a  tube  filled 
with  a  weaker  solution  of  ferric  chloride  and  immersed  in  a  stronger  solu- 
tion of  the  same  salt  will  take  up  a  cross-position  in  a  strong  magnetic  field, 
though  both  solutions  are  ferromagnetic. 

Limit  of  Magnetisation.  —  The  intensity  of  induced  mag- 
netism increases  with  the  intensity  of  the  magnetic  field,  and 
varies  with  that  intensity ;  but,  in  the  case  of  iron,  within 
certain  limits  only.  As  the  intensity  of  the  magnetic  field 
rises,  the  induced  magnetism  of  soft  iron  verges  towards  a 
limit.  This  would  appear  to  favour  that  theory  of  induction 
which  regards  the  magnetisation  of  induced  iron  not  as  created, 
but  as  directed  by  the  forces  within  the  field.  According  to 
this  theory  (Weber's),  the  molecules  of  iron  are  already  little 
magnets,  but  their  directions  are  promiscuously  discrepant. 
When  they  come  into  a  magnetic  field  they  are  directed  so  as 
to  lie  with  their  axes  parallel  to  lines  of  force,  and  the  whole 
mass  of  iron  thereupon  becomes  obviously  magnetic.  The  lines 
of  force  of  the  directed  magnetic  molecules  are  added  to  those 
of  the  electromagnetic  field ;  and  this  operation  has  obviously  a 
limit,  beyond  which  any  increase  in  the  number  of  lines  can  be 
due  to  the  electromagnetic  field  only.  The  value  of  p,  therefore 
falls  off  as  h  increases.  In  diamagnets  there  is  some  reason  for 
believing  that  the  particles  are  magnetised  de  novo  in  the  mag- 
netic field ;  but  the  particle-magnets  thus  produced  are  feeble, 
and  their  strength  does  not  tend  to  a  limit. 

When  an  iron  bar  is  magnetised  there  seems  to  be  an  actual  twist  set 
up  in  it. 


XVI.] 


MAGNETS  AND  ELECTROMAGNETS. 


689 


Fig.241. 


Magnets  used  to  be  made  by  exposing  steel  for  some  time 
to  the  influence  of  an  existing  magnetic  field ;  as  by  rubbing 
bent  or  straight  bars  from  centre  to  ends  with  the  opposite  poles 
of  bar-magnets,  or  by  leaving  them  in  contact  by  their  extremi- 
ties with  the  opposite  poles  of  a  strong  horse-shoe  magnet  or 
electromagnet. 

Magnets  are  now  produced  by  electric  currents.  A  simple 
bar-magnet,  as  we  have  seen,  tends  to  lie  across  a  current,  its 
positive  pole  to  the  left,  its  negative  to  the  right  of  the  current. 
If  a  bar  of  soft  iron  be  placed  along  the  lines  of  force  within 
an  Electromagnetic  Field,  it 
becomes  a  temporary  magnet, 
or  Electromagnet;  there  is  a 
kind  of  separation  of  magnet- 
isms; the  left-hand  end  of 
the  bar  becomes  magnetically 
positive,  the  right-hand  end 
negative.  If  the  current  be 
wound  round  the  bar,  so  that 
every  part  of  the  current  exerts 
a  similar  action  upon  the  bar, 
the  bar  becomes  strongly  mag- 
netic (Fig.  241). 

If  it  be  of  very  soft  iron  it 
loses  this  property  the  instant 
the  current  ceases ;  but  if  it  be  of  steel  (or  nickel,  or  cobalt), 
and  if  the  current  be  powerful  and  continued  for  some  time,  it 
becomes  a  permanent  magnet. 

Cobalt,  iron,  and  steel  become  more  susceptible  to  magnetic  induction 
when  they  are  slightly  warmed.  At  about  785°  C.,  a  soft  iron  or  steel  mag- 
net suddenly  loses  all  its  magnetism,  with  evolution  of  energy  as  Heat; 
nickel  does  so  at  a  lower  (635°  C.)  and  cobalt  at  a  higher  temperature  (that 
of  melting  copper).  If  exposed  to  magnetic  induction  while  above  785°  C., 
iron  manifests  no  susceptivity ;  then  shut  off  the  inducing  current  and  allow 
the  bar  to  cool ;  the  bar  becomes  magnetic  when  it  reaches  that  temperature. 
Steel  presents  similar  phenomena  at  690C-880°,  according  to  its  composition. 
An  alloy,  Fe  75,  Ni  25,  cannot  be  magnetised  unless  below  —  20°  C. ;  if  then 
magnetised,  it  remains  magnetic  up  to  580°  C. ;  it  then  loses  its  magnetism 
and  does  not  recover  it  unless  and  until  cooled  to  —  20°  C.  There  is  some 
molecular  change  at  these  critical  temperatures;  at  about  the  same  tem- 
peratures, the  electrical  resistivity  and  the  thermo-electric  properties  also 
change,  and  there  is  a  sudden  evolution  of  latent  heat  on  cooling  ( J.  Hop- 
kinson). 

Thermo-magnetic  Motors.  —  A  soft-iron  disc  in  a  strong  field  may 
become  magnetised  along  a  particular  diameter;  but  if  one  end  of  that 

2Y 


690  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

diameter  be  heated,  the  susceptibility  of  the  iron  is  diminished  along  that 
diameter,  and  a  cooler  diameter  is  pulled  round  to  take  its  place.  Continu- 
ous rotation  may  thus  be  set  up,  with  transformation  of  Heat  into  Work 
by  magnetic  means  (Thomson  and  Houston).  In  Edison's  pyro-rnagnetic 
motor,  a  drum  is  made  up  of  soft-iron  tubes :  some  of  these  are  heated  by 
hot  air,  others  cooled  by  cold  air  blown  down  them  :  the  cool  tubes  are  con- 
tinuously drawn  into  the  strongest  part  of  the  field,  but  no  sooner  do  they 
reach  it  than  they  are  heated,  and  have  to  make  way  for  their  cooler  suc- 
cessors. A  motor  of  this  kind  gave  3  horse-power,  at  120  revolutions  per 
minute. 

When  an  iron  or  cobalt  bar  is  magnetised  it  becomes  longer  and  some- 
what more  slender,  but  does  not  appreciably  alter  in  volume ;  it  also  emits 
a  slight  sound,  a  "magnetic  tick."  A  nickel  or  a  steel  bar  shortens  and 
thickens. 

Magnetisation,  induced  or  residual,  in  an  iron  or  steel  wire  is  diminished 
on  stretching,  provided  that  the  magnetisation  correspond  to  an  inducing 
force  above  a  certain  critical  value  known  asVillari's  Critical  Value; 
this  being  (Lord  Kelvin)  about  24  times  the  terrestrial  intensity.  Below 
that  critical  value,  tension  increases  the  magnetisation  of  the  magnetised 
wire.  In  nickel  the  magnetisation  is  always  diminished.  The  effects  of 
transverse  expansive  stress  are  opposed  to  those  of  longitudinal  stretching. 

Energy  is  absorbed  during  magnetisation,  and  if  an  electro- 
magnet be  made  and  unmade  in  frequent  quick  succession,  it 
becomes  hot ;  the  energy  is  derived  from  that  of  the  intermit- 
tent inducing  current. 

Hysteresis.  —  When  a  soft  iron  core  of  permeability  fi  is  put  into  a 
solenoid  field,  of  intensity  h,  the  number  of  lines  which  traverse  the  core  is, 
per  sq.  cm.,  b  =  /xh,  where  /x  itself  depends  upon  the  value  of  h.  When 
h  =  0,  b  =  0,  to  begin  with ;  but  if  we  induce  magnetism  in  ordinary  iron, 
it  does  not  lose  all  its  magnetism  when  we  withdraw  the  inducing  force  h. 
To  take  an  example,  of  Prof.  Ewing's :  h  =  0,  b  =  0,  to  begin  with :  h  was 
raised  to  90,  and  b  was  then  14,000 ;  when  h  came  back  to  0,  the  value  of  b 
in  the  iron  had  only  sunk  to  10,500  :  to  make  b  sink  to  0,  it  was  necessary  to 
make  h  =  —  24 :  when  h  was  —  90,  b  was  —  14,000 ;  when  h  again  became 
0,  b  was  —  10,500 ;  h  had  to  be  increased  to  +  24  before  b  became  0 ;  and 
when  h  was  90,  b  was  again  14,000.  On  successive  oscillations  of  the  value 
of  h  between  +  90  and  —  90,  the  same  cycle  was  repeated ;  and  on  plotting 
out  these  results  in  a  diagram  it  will  be  seen  that  an  area  is  described,  after 
the  fashion  of  the  Indicator  Diagram,  showing  products  of  b  into  h,  and 
such  that  the  area  of  the  figure  is  equal  to  STT  x  the  number  of  ergs  per 
cub.  cm.  wasted  at  each  half-alternation  by  being  transformed  into  Heat ; 
for  the  Energy  of  Field,  within  the  magnet  in  this  case,  is  h-b/87r  ergs  per 
cub.  cm.,  in  any  medium.  Vibration  or  a  high  temperature  reduces  this 
effect.  The  form  of  the  area  described  is  somewhat  like  a  f  ,  varying  in 
central  thickness  from  one  substance  or  condition  to  ano'ther. 

If  the  induced  magnetism  of  iron  be  due  to  the  directive  action  of  the 
magnetic  field,  the  residual  magnetism  of  steel  may  perhaps  be  due  to  a  sort 
of  imperfect  elasticity  of  the  medium  surrounding  the  particles ;  the  par- 
ticles are  wrenched  into  definite  directions,  and  retain  these  as  a  permanent 
set,  or  very  slowly  reassume  their  discrepancy  of  direction  (Maxwell). 


xvi.]  MAGNETISATION. 

If  we  dissolve  away  the  outer  skin  of  a  steel  magnet  by  means  of  acid, 
we  find  (Jamin)  that  the  remainder  has  a  very  small  intensity  of  magneti- 
sation. Perhaps  the  outer  shell  is  the  hardest  part  of  the  magnet  and  has 
the  greatest  amount  of  the  so-called  coercitive  force,  the  least  amount  of 
elasticity  of  the  medium. 

Astatic  Arrangements.  — A  needle  tends  to  point  to  the  magnetic 
north  ;  but  it  is  often  desirable  to  mask  the  action  of  the  earth's  magnetism, 
in  order  to  increase  the  ratio  of  the  torque  due  to  any  deflecting  magnet  to 
that  due  to  the  terrestrial  intensity,  and  thereby  to  increase  the  sensitiveness 
of  galvanometers.  This  may,  roughly,  be  done  by  bringing  another  magnet  of 
opposite  effect  into  the  neighbourhood,  so  as  nearly  to  neutralise  the  earth's 
directive  force ;  or  again  by  coupling  together  on  the  same  suspending  thread 
two  equal  magnets  with  their  poles  opposed.  In  the  latter  case  the  earth 
tends  to  direct  the  two  magnets  in  opposite  senses,  and  if  the  two  magnets 
were  equal  and  their,  axes  parallel,  the  joint  system  would  be  practically 
unaffected  by  the  earth's  directive  action. 

It  is  better  to  enclose  a  single  needle  in  a  shell  of  very  soft  iron,  as  in 
Lord  Kelvin's  marine  galvanometer.  This  shell,  within  the  earth's  mag- 
netic field,  becomes  magnetic.  The  Fig  242 
needle  is  now  under  the  inducing  action 
of  two  magnets,  the  earth  and  the  in- 
duced shell.  The  actions  of  these  are 
opposed,  and  if  the  shell  be  thick 
enough  are  approximately  equal :  the 
earth's  magnetic  field  is  thus  nearly 
destroyed  within  the  shell,  and  the 
magnet  is  free  to  obey  the  directive 
impulse  of  any  current  which  may  be 
sent  round  it.  Such  a  shell  acts  as  a 
Magnetic  Screen;  and  such  a  screen, 
efficient  as  a  protection  from  the  influence  of  an  external  magnet,  may  be  a 
sphere,  an  infinite  or  a  very  large  plane,  or  an  equipotential  surface  of  any 
form. 

Magnetic  Circuit.  —  The  actual  phenomena  of  a  Magnet  are  in  some 
respects  better  correlated  on  considering  the  Lines  of  Induction  in  an  Electro- 
magnet than  they  are  on  considering  the  Poles  of  a  permanent  Magnet.  Sup- 
pose a  solenoid,  of  n  turns  and  of  length  I  cm.,  and  bearing  a  current  of 
intensity  i ;  the  magnetic  force  h  at  any  point  inside  the  solenoid,  in  any 
medium,  is  4?rt  •  n/l.  The  induction  per  sq.  cm.,  the  number  of  lines  of  induc- 
tion per  sq.  cm.,  is  b  =  /xh  =  ^irni  •  p/l.  The  Total  Induction  along  the  elec- 
tromagnet is  B  =  cross-sectional  area  x  b  =  kirni  •  (J.A./L  Write  this  B  = 
(47rni)  -4-  (Z//xA)  ;  then  the  Magnetic  Induction  or  Flux  B  is  said  to  be 
equal  to  the  "Magnetomotive  Force"  ±irni  (=h-Z),  divided  by  the 
"Reluctance"  (J//*A).  The  lines  of  induction,  each  of  which  is  closed 
upon  itself,  form  together  a  "  closed  circuit,"  like  an  electric  current ;  and 
the  above  expression  is  analogous  to  Ohm's  Law,  I  —  E/R.  The  ideal 
Magnetic  Circuit  is  a  magnetised  ring,  magnetised  so  that  the  lines  of 
induction  pass  continuously  and  equably  round  the  ring,  each  of  them 
being  circular.  Such  a  ring  has  no  "  poles,"  for  there  is  no  place  at  which 
the  lines  of  force  escape  into  the  outer  air.  But  if  we  cut  such  a  ring,  so 
as  to  produce  an  air-gap  in  it,  we  then  have  two  poles,  or^  rather  polar 
regions ;  the  lines  are,  as  it  were,  unwilling  to  bridge  the  air-gap,  and  thus, 


692  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

in  order  to  produce  a  given  b,  it  is  then  necessary  to  apply  a  greater  h ;  the 
lines  tend  to  become  fewer  in  number,  while  of  those  which  remain,  some 
find  their  way  back  by  shorter  return-paths  through  the  surrounding  air, 
there  then  being  "  leakage  "  of  these  lines.  The  air-gap  has  acted  as  if  a 
considerable  length  of  iron,  //,  times,  or  say  400  times  its  own  length,  had 
been  inserted  in  the  magnetic  circuit,  and  the  Reluctance  of  that  circuit 
had  been  correspondingly  increased.  But  the  formula,  in  the  form  B  = 
(47rm)  -=-  S(///u.A),  is  general;  there  is  always  some  kind  of  a  return-path, 
the  reluctance  of  which  is  ascertainable.  In  the  ordinary  horse-shoe  mag- 
net, the  circuit  for  the  lines  is  up  and  down  the  U,  and  through  the  cross- 
bar or  armature ;  many  lines  escape,  however,  at  the  air-joints,  and  do  not 
travel  through  the  armature.  In  an  ordinary  bar-magnet,  a  large  propor- 
tion of  the  lines  of  induction  escape  laterally;  and  at  the  terminal  faces 
they  are  crowded  towards  the  edge  of  that  face,  so  that  the  field  is  stronger 
towards  that  edge.  In  any  dynamo,  the  air-gaps  should  be  reduced  as  far 
as  possible,  so  as  to  diminish  the  reluctance  of  the  circuit :  and  in  every 
magnetic  circuit  in  which  electromagnets  are  employed,  the  soft  iron  parts 
must  be  so  designed  that,  with  the  desired  value  of  B,  b  shall  not  exceed 
a  limit  which  suits  the  iron  made  use  of,  say  16,000  per  sq.  cm.,  e.g.,  h  =  50, 
H  then  =  320 ;  for  if  the  iron  be  too  slender,  and  if  the  current  be  forced 
up  so  as  to  produce  the  required  value  of  B,  b  tends  to  grow  to  values  at 
which  fj.  falls  off. 

The  lines  which  are  enabled,  by  the  presence  of  air-gaps,  to  return 
through  the  air  tend  to  shorten  and  shrink;  hence  a  permanent  magnet 
tends  to  become  demagnetised.  Keeping  its  armature  upon  a  permanent 
magnet  tends  to  favour  the  retention  of  the  energy  of  magnetisation,  for 
a  good  path  is  thus  provided  for  the  lines. 

The  general  problem  of  magnetic  induction  is  a  problem  of  potential 
and  lines  of  force,  in  which  the  body  acted  upon  consists  of  perfectly-con- 
ducting molecules  scattered  through  an  absolutely  non-conducting  medium. 
This  kind  of  problem  involves  difficulties  of  calculation,  but  is  of  the  same 
nature  as  that  of  electrostatic  induction  through  a  heterogeneous  dielectric, 
or  that  of  conduction  of  heat  or  of  electricity  through  a  heterogeneous  con- 
ductor, or  that  of  the  flow  of  a  frictionless  incompressible  fluid  through  a 
heterogeneous  porous  material.  A  given  amount  of  force  within  the  mag- 
netic field  produces  a  certain  amount  of  separation  of  magnetisms  and  a 
corresponding  density  of  magnetic  distribution ;  this  may  be  regarded  as 
an  arrested  flow,  an  accumulation,  which  is  proportional  to  the  continuous 
flow  which  is  dealt  with  in  problems  of  conduction  :  and  the  nature  of  the 
substance  acted  upon  brings  into  the  calculation  a  term,  the  Coefficient 
of  Magnetic  Induction  or  the  Permeability,  /x,,  which  resembles  the  perme- 
ability of  bodies  to  fluids,  or  the  specific  inductive  capacity  of  electrostatic 
dielectrics. 

As  to  the  nature  of  magnetism,  Ampere's  theory  is  that 
every  molecule  of  a  magnetic  substance  is  the  seat  of  a  separate 
current,  circulating  round  it  in  a  plane  at  right  angles  to  the 
magnetic  axis.  This  explanation  meets  most  of  the  facts  with 
great  readiness  ;  but  in  view  of  the  doctrine  of  the  Conservation 
of  Energy  we  must  postulate  the  entire  absence  of  resistance  to 
these  molecular  ciirrents  —  a  circumstance  of  which  it  is  some- 
what difficult  to  form  a  clear  conception. 


xvi.]  MATURE   OF   MAGNETISM. 

When  all  the  molecules  of  a  substance  have  their  currents 
running  in  the  same  direction,  and  when  all  these  currents  are 
equal,  the  substance  is  uniformly  magnet- 
ised, and  in  the  interior  any  two  contigu-        

ous  molecules  (Fig.  243)  have  currents  in  f  j\ 

opposed  directions  whose  effect  on  exterior  f  j )( 

particles  is  nil.     The  result  of  the  whole  V  /) 

is  equivalent  to  a  superficial  sheet  of  elec-     •**• 

trie  current,  the  action  of  which  may  be  approximately  reduced  by 
a  kind  of  centre-of-gravity  problem  to  the  action  of  two  Poles. 

As  to  the  direction  of  the  currents  within  a  magnet :  a 
person  standing  on  the  Arctic  Pole  of  the  earth  would,  if  those 
currents  to  which  the  earth's  magnetism  is  supposed  to  be  due 
were  visible  to  him,  see  them,  or  rather  their  resultant,  the  cur- 
rent-sheet, travelling  over  the  surface,  circulating  round  him 
from  east  to  west ;  those  in  front  of  him  would  therefore  travel 
towards  his  right  hand.  The  observer  there  situated  would  be 
at  the  negative  end  of  the  earth;  the  Positive  pole  is  its 
Southern  pole  —  that  pole,  namely,  from  which  the  positive  or 
north-seeking  end  of  the  compass-needle  is  driven.  An  observer 
stationed  at  the  positive  or  southern  pole  of  the  earth,  the 
Antarctic  Pole,  would  therefore  see  these  currents  pass  round 
him,  still  from  east  to  west,  but  apparently  towards  his  left. 
These  currents  within  a  magnet  are  known  as  Amp£ re-cur- 
rents. 

Dimensions  of  Magnetic  Measures,  in  Air.  —  Quantity  of  mag- 
netism, m :  force  =  mm'  -=-  distance2 ;  whence,  j  ust  as  in  the  case  of  electric 
quantity,  p.  603,  the  dimensions  of  magnetic  quantity  are  [m]  =  [M^U/T]. 

Magnetic  Force,  or  Strength  or  Intensity  of  Field,  h: 
mechanical  force  acting  on  unit  quantity  of  magnetism ;  its  Dimensions  are 
Mechanical  Force  [ML/T2]  -  Magnetic  Quantity  [M*L*/T]  =  [M4/L*T]. 

Magnetic  Moment,  ffil  =  ml :  a  magnetic  quantity  x  a  length ;  [iffil]  = 
[MiLt/T]. 

Intensity  of  Magnetisation,  I:  magnetic  moment  per  unit  of  vol- 
ume ;  [I]  -  [MiLi/T]  +  [L«]  -  [Mi/L*T]. 

Magnetic  Potential,  O:  work  done  in  moving  unit  quantity  of  mag- 
netism ;  its  dimensions  are  those  of  (Work  done)  -4-  (Magnetic  Quantity  m 
moved);  [Q]  =  [Work/m]  =  [ML2/T2]  *  [M*L*/T]  =  [MiL*/T] ;  the 
same  dimensions  as  those  of  electric  potential,  electrostatically  measured. 

Magnetic  Surface-density,  s:  quantity  of  magnetism  per  unit  of 
area  ;  [?]  -  [m/ Area]  =  [MW/T]  -4-  [L2]  =  [Mi/HT]. 

Strength  of  Shell,  <]P :  Surface-density  x  thickness ;  [<jp]  =  [s  x  thick- 
ness] =  [Mi/LiT]  X  [L]  =  [MiLi/T]. 

Magnetic  Induction,  b  per  sq.  cm. ;  Number  of  Lines  of  Induction 
from  a  given  Pole  across    a  given   Area;    [47rm/area]  = 
=  [M*/L4T]. 


694  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

Coefficients  of  Induced  Magnetisation,  /c(=Intensity  of  Mag- 
netisation -r-  Magnetic  Force),  and  of  Magnetic  Induction,  p  (  =  Magnetic 
Induction  -4-  Magnetic  Force):  Numbers  simply;  no  Dimensions. 

Magnetic  Dimensions  in  medium  of  permeability  p.  —  Magnetic 
Quantity,  [^M^U/T]  :  Magnetic  Field-Intensity,  [Ms/LiT/xi]  :  Magnetic 
Moment,  ^MiL*/TTj  Intensity  of  Magnetisation,  [/x^M*  /  L*T]  :  Mag- 
netic Potential,  [M*L*  /  T/x*]  :  Magnetic  Surface-Density,  [/AM*  /  L*T]  : 
Strength  of  Shell,  [^M*L»  /  T]  :  Magnetic  Induction  per  sq.  cm.,  b, 
/cand/x,  |>]. 


Magnetic  Rotatory  Polarisation  of  Light.  —  If  a  plane- 
polarised  beam  of  light  or  of  radiant  heat  be  sent  through  a 
magnetic  field  occupied  by  a  transparent  medium,  its  plane 
will,  by  the  retardation  or  acceleration  in  phase  of  one  of  its 
circular  components,  be  rotated.  The  sense  of  this  rotation 
depends  upon  the  direction  of  the  lines  of  force  and  upon  the 
nature  and  chemical  constitution  of  the  medium  ;  its  amount 
upon  the  thickness  and  the  nature  and  physical  state  of  the 
medium  and  upon  the  intensity  of  the  magnetic  field,  resolved 
in  the  direction  of  the  ray  ;  but  it  does  not  occur  in  free  Ether. 

This  has,  in  the  hands  of  Becquerel  and  Lord  Rayleigh,  been  made  the 
basis  of  a  method  of  measurement  of  the  intensity  of  a  current  ;  the  current, 
when  passed  through  a  solenoid,  produces  within  this  an  electromagnetic 
field,  the  intensity  of  which  is  proportional  to  the  strength  of  the  current  ; 
a  plane-polarised  beam  sent  through  glass  along  the  axis  of  the  solenoid  is 
rotated  to  an  extent  proportionate  to  the  current-strength. 

The  direction  of  rotation  of  the  plane  of  polarisation  is  in 
most  cases,  including  flint  glass  and  thin  films  of  iron,  cobalt, 
Fig.244.  and  nickel,  positive,  being  that  shown  by  Fig. 
244,  in  which  AB  represents  a  line  of  magnetic 
force,  and  the  arrowed  circle  represents  the 
direction  of  rotation  of  the  plane.  Whether 
the  ray  travel  in  the  direction  AB  or  in  the 
direction  BA,  the  absolute  rotation  imparted  to 
it  on  its  transmission  through  the  magnetic 
field  remains  the  same  ;  whence,  if  it  be  re- 
A  fleeted  from  a  mirror  and  sent  back  through 
the  field,  the  rotation  of  its  plane  will  be  doubled. 

In  uniaxial  crystals,  the  rotation  is  most  marked  along  the  axis.  In  a 
concentrated  solution  of  perchloride  of  iron  it  is  negatiye. 

Hall's  Experiment.  —  A  film  of  metal,  in  the  form  of  a  cross,  laid 
upon  glass.  A  current  from  a  battery  passes  through  two  opposite  arms  of 
the  cross,  and  does  not  affect  a  galvanometer  connected  with  the  other  two 
arms.  When  the  cross  is  made  to  face  the  lines  of  force  of  a  strong  mag- 
netic field,  a  small  constant  current  is  indicated  by  the  galvanometer  (Fig. 


XVI.] 


MAGNETIC   ROTATORY  POLARISATION. 


695 


Fig. 245. 


245).     The  strength  of  this  current  depends  upon  the  intensity  of  the  mag- 
netic field  and  the  strength  of  the  primary  current ;  and  also  upon  the  kind 
of  metal  of  which  the  film  consists. 
Its  direction  depends  upon  the  direc- 
tion of  the  field  and  that   of   the 
primary  current,  and  on  the  metal 
of  the  film. 

Kerr's  Experiment.  —  Polar- 
ised light  reflected  from  the  polished 
face  of  a  magnet  undergoes  rotation 
of  the  plane  of  its  polarisation;  when 
reflected  from  the  north-seeking  pole, 
the  rotation  is  negative.  When  reflected  from  the  sides  of  a  magnet,  it  also 
undergoes  rotation,  the  sense  of  which  varies  with  the  plane  of  polarisation 
and  the  angle  of  incidence.  Kundt  has  also  found  a  variety  of  phenomena 
of  rotation  of  the  plane  of  polarisation  of  light  transmitted  through  thin 
magnetised  metallic  films. 

Thus  far  we  have  dealt  with  Steady  Currents  and  Steady 
Fields,  electrostatic,  magnetic,  or  electromagnetic.  We  have 
now  to  consider  the  properties  of  Varying  Currents  and  Fields. 


Fig.  246. 


THE  VARIABLE  PERIOD. 

When  an  open  circuit  is  abruptly  closed  for  an  instant,  and 
an  instantaneous  current  is  produced  in  the  wire,  this  current  is 
not  felt  simultaneously 
over  the  whole  circuit. 
In  the  case  (a)  of  Fig. 
246  the  galvanometers 
B  and  Bf  twitch  first 
when  the  interrupted 
circuit  is  momentarily 
completed ;  in  case  (5)  of  that  figure,  under  similar  circum- 
stances, the  galvanometers  A  and  A'  twitch  first.  The  dis- 
tance between  A  and  B  must  be  great  in  order  to  show  this 
effect. 

By  the  time  a  state  has  been  arrived  at,  in  which  a  steady 
current  passes,  a  certain  amount  of  energy  has  been  elastically 
accumulated  in  the  dielectric  ;  but  between  the  instant  at  which 
the  current  begins  to  flow  and  that  at  which  it  has  assumed 
its  steady  state,  there  is  a  period  of  adjustment,  the  variable 
period. 

During  this  period  the  field  has  Kinetic  Energy,  and  there^is  a  Displace- 
ment- or  Polarisation-Current. 


696 


ELECTKICITY  AND   MAGNETISM. 


[CHAP. 


When,  as  in  Fig.  247,  a  battery  of  which  one  pole  is  con- 
nected to  earth  has  its  other  pole  suddenly  brought  into  commu- 
Fig.247.  nication  with  a  long 

wire  whose  other 
extremity  is  con- 
nected with  the 
earth,  the  time 
which  elapses  be- 
fore the  current 
through  the  wire 
becomes  steady  is 
found  to  vary  as  the  square  of  the  length  of  that  wire. 

In  long  lines,  the  time  spent  in  acquiring  at  any  point  of  the  wire  a 
certain  definite  intensity  of  current  is  approximately  proportional  to  RC/2/E, 
where  E  is  the  D.P.  employed,  R  the  resistance,  and  C  the  electrostatic 
capacity  of  the  wire  per  cm.,  and  I  its  length.  The  time  which  elapses 
before  a  certain  proportion  of  the  ultimate  intensity  is  attained  varies 
approximately  as  RC/2;  it  is  also,  in  practice,  not  independent  of  the  effec- 
tive D.P.  set  up  by  the  galvanic  cell  employed.  The  product  RC/2,  which 
measures  the  Electrostatic  Retardation,  is  thus  of  great  importance 
in  long  lines ;  but  in  short  lines,  the  effects  of  electrostatic  retardation  are 
masked  by  those  of  self-induction  and  the  induction  of  other  circuits. 

A  lightning  discharge  through  a  lightning  conductor  is  so  brief  that  the 
laws  of  steady  flow  do  not  hold  good  :  it  is  of  advantage,  in  order  to  diminish 
the  risk  of  lateral  divergence,  to  render  the  current  more  uniform,  or,  in 
other  words,  to  retard  it ;  for  this  purpose  the  capacity  of  the  conductor 
should  be  increased,  and  therefore  its  surface;  and  lightning  conductors 
should  be  broad  flat  plates  of  metal  rather  than  compact  rods.  The  effect 
of  self-induction  of  the  current  also  aids  in  bringing  about  this  result ;  cur- 
rents running  parallel  and  in  the  same  direction  retard  one  another.  The 
circumstance  that  the  flow  is  too  brief  to  affect  the  interior  of  the  wire  to 
any  considerable  extent  also  aids  in  making  it  more  important  to  increase 
the  relative  surface  of  the  conductor  than  to  increase  its  cross-section ;  for 
the  phenomena  of  so  abrupt  a  discharge  are  practically  restricted  to  the  field 
of  force,  the  dielectric,  surrounding  the  wire. 

In  a  uniform  wire,  OL,  between  whose  extremities  a  dif- 
ference of   potential  is  maintained   equal  to   OP   (Fig.   248), 

the   ultimate  Line  of 
Potentials  is  PL ;  and 
when  such  a  distribu- 
tion of  potentials  has 
once    been    produced 
along   the   conductor, 
Ohm's  law  is  obeyed; 
L   but  at  various  instants 
during  the  preliminary  variable  period,  the  distribution  of  poten- 


Fig.248. 


XVI.] 


THE   VARIABLE   PERIOD. 


697 


Fig.249. 


tials  along  the  wire  is  such  as  is  indicated  by  the  curved  lines, 
1,  2,  3,  etc.,  sketched  in  Fig.  248. 

The  momentary  and  local  intensity  is  always  the  momentary  and  local 
E/R(=  Potential-Slope  -4-  Resistance  per  linear  cm.),  but  during  the  vari- 
able period  it  varies  from  point  to  point  and  from  instant  to  instant. 

When  the  extremity  of  a  long  wire  is  momentarily  charged 
by  contact  with  a  charged  conductor  or  with  one  pole  of  a  bat- 
tery, its  home  end  suddenly  acquires  a  high  potential,  which  is 
immediately  thereupon  reduced  by  communication  of  the  charge 
acquired  by  the  ex- 
tremity of  the  wire  to 
the  rest  of  the  wire. 
In  Fig.  249  the  end  O 
of  the  conductor  OL  is 
suddenly  raised  to  the 
potential  OP.  A  point 
such  as  A  is  found,  as 
it  were,  to  leap  up  to  a 
high  potential  and  then  to  descend.  A  wave  of  sudden  increase 
of  potential  thus  travels  along  the  conductor,  but  falls  off  pro- 
gressively, both  in  abruptness  and  in  height,  the  farther  it  travels. 

At  the  distant  end,  for  a  short  interval  after  the  circuit  has 
been  actually  completed,  no  effect  is  perceived;  the  current  then 
begins  to  become  sensible :  and  it  then,  if  the  contact  be  kept  up 
at  the  home  end,  appears  to  increase  in  intensity  after  the  man- 
ner indicated  by  the  Arrival-Curve  represented  in  Fig.  250. 
A  current,  even  though  Fig.  250. 

it   be    constantly   main- 
tained  at  the  home  end, 
would  take    an   infinite 
time  to  acquire  its  maxi- 
mum value  at  the  distant     5  /" 
end  of  such  a  conductor     °-         / 
as  an  Atlantic  cable,  if             / 

that  conductor  had,  when  

the  current  commenced  Time 

to  traverse  it,  been  uncharged ;  it  would,  however,  require  only 
about  108  seconds  to  attain  -f-$  of  its  maximum  value,  and  about 
the  fifth  part  of  a  second  to  attain  T^  of  its  maximum  value. 
The  apparent  velocity  of  transmission  of  signals  in  a  given  con- 
ductor is  thus  seen  to  be  mainly  an  affair  of  the  dejicacy  of  the 
instruments  which  detect  the  current  on  its  arrival  at  the  distant 


698  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

end,  and  is  perfectly  distinct  from  the  velocity  of  propagation  of 
an  electromagnetic  disturbance  ;  and  it  depends  on  the  capacity 
of  the  conductor,  for  the  transmission  is  greatly  delayed  in  con- 
ductors whose  capacity  is  great,  such  as  submarine  cables,  appre- 
ciably so  in  long  air-lines,  inappreciably  so  in  short  air-lines. 

The  attainment  of  the  steady  state  is  greatly  facilitated,  though  the 
currents  produced  are  weakened,  by  leakage.  The  signals  produced  are 
thus  rendered  clear  (Oliver  Heaviside). 

When  the  current  suddenly  stops  after  having  acquired  a 
steady  flow,  its  cessation  at  the  distant  end  presents  a  similar 
deliberation. 

When  a  wire  is  momentarily  connected  with  a  charged  body  and  then 
connected  with  the  earth,  or  "  put  to  earth,"  the  arrival-curve  at  its  distant 
end  is  a  curve  due  to  the  superposition  of  two  arrival-curves ;  the  first  of 
these  is  the  arrival-curve,  resembling  that  of  Fig.  250,  due  to  the  contact 
with  the  charged  body ;  the  second  is  curved  in  the  opposite  sense,  and  is 
due  to  the  sudden  discharge  of  the  conductor.  The  dotted  curve  of  Fig. 
250  is  the  result  of  the  superposition  of  two  such  opposed  arrival-curves. 
This  curve  indicates  that  there  is  an  abrupt  and  brief  variation  of  potential 
at  the  end  of  the  wire  distant  from  the  galvanic  cell. 

More  effective  still  than  this  in  producing  an  abrupt  and  brief  current 
is  the  process  of  following  up  each  positive  charge,  immediately  after  putting 
to  earth,  with  a  negative  one,  after  which  the  wire  is  again  put  to  earth. 
The  disadvantage  of  this  is  that  the  potential,  while  it  abruptly  ceases  to  be 

positive,  sinks  at  once  to  a 
negative  condition,  as  in  Fig. 

Fig.25l.  251;   for  which  reason  it   is 

X..  customary  so  to  arrange  the 

mechanism  at  the  signalling 
station  that  each  apparently 
simple  making  of  contact  is  in 
reality  a  complex  operation,  in 
which  an  odd  number  of  cur- 
rents  of  opposite  kinds  are  sent 
in  rapid  succession  into  the 

wire,  the  wire  being,  after  each,  put  to  earth ;  each  of  these  currents  being 
briefer  than  its  predecessor,  and  correcting  it.  The  arrival-curve  for  such 
a  combination  indicates  an  abrupt  rise  of  potential,  an  abrupt  fall,  and  then 
a  slightly-wavy  line,  which  at  no  point  diverges  to  any  material  extent  from 
the  base-line. 

Even  these  methods  are  increased  in  effect,  the  arrival-curve  being  ren- 
dered still  more  abrupt,  by  the  use  of  Condensers.  Each  condenser  is  com- 
posed of  a  large  number  of  plates  of  tinfoil  separated  by  waxed  paper  and 
paraffin :  the  alternate  plates  are  in  metallic  communication  with  one 
another.  One  series  of  alternate  plates  in  each  condenser  is  in  communica- 
tion with  the  cable ;  the  other  set  is  in  communication  with  the  galvanic 
battery  or  with  the  galvanometer  G  (Fig.  252). 

Any  sudden  variation  in  the  potential  of  the  landward  plates  of  the  home 
condenser  is  immediately  followed  by  an  equally-sudden  flow  of  electricity 


XVI.] 


THE   VARIABLE   PERIOD. 


699 


into  or  from  the  cableward  plates  of  that  condenser :  this  flow  takes  place 
either  from  or  into  the  cable  itself ;  this  disturbance  is  propagated  along 
the  cable ;  the  potential  of  the  cableward  plates  of  the  condenser  at  the 
receiving  station  is  affected ;  by  induction  the  distribution  of  electricity  in 
the  landward  plates  of  that  condenser  is  affected,  and  a  current  passes 
through  the  galvanometer,  either  from  the  condenser  to  the  earth  or  in  the 

Fig.  252. 


reverse  direction.  On  connection  of  the  home  condenser  with  the  positive 
pole  of  the  battery  employed,  a  positive  current  runs  through  the  distant 
galvanometer  Gr  to  the  earth ;  and  on  putting  the  home  condenser  to  earth 
a  reverse  current  passes  through  G,  which  may  be  corrected  as  before. 

Even  if  the  key  be  kept  permanently  pressed  down  at  the  transmitting 
station  the  current  passing  through  G  is  but  momentary,  for  both  condensers 
quickly  assume  a  condition  of  electrostatic  equilibrium. 

During  the  Variable  Period,  the  Lines  of  Force  are  slipping  along  the 
wires  with  the  velocity  of  Light.  They  travel  with  their  ends  on  the  wires, 
and  approximately  lie  at  right  angles  to  these,  until  the  condition  approaches 
that  of  the  steady  state.  During  the  variable  state,  they  accumulate  (or 
thin  away)  in  the  field ;  when  the  steady  state  has  been  attained,  there  is 
no  accumulation  of  them  in  the  field,  but  only  transit,  while  their  direction 
becomes  approximately  parallel  to  that  of  the  wires.  At  the  same  time, 
during  the  variable  period,  the  Lines  of  Transmission  of  Energy  through 
the  field  are  themselves  in  motion ;  and  the  axis  of  the  wire  is  the  last 
thing  to  be  affected.  During  this  period  there  may  be  production  of 
induced  currents  in  neighbouring  conductors. 


ELECTROMAGNETIC  CURRENT-INDUCTION. 

If  there  be  a  closed  current-bearing  circuit,  with  its  positive 
face  facing  a  positive  magnetic  pole,  there  will  be  mutual  repul- 
sion between  that  circuit  and  pole ;  and  if  the  current  and  the 
magnetic  pole  be  brought  nearer  one  another,  then,  since  work 
must  be  done  in  order  to  bring  about  this  approach  in  the  face 
of  mutual  repulsion,  the  potential  energy  of  the  system  is 
increased  by  a  fixed  amount:  a  portion  of  this  energy  takes 
the  form  of  a  temporary  increase  of  the  current  in  the  closed 
circuit;  while  the  remainder  may,  by  induction,  produce  an 
increased  magnetic  condition  in  the  magnet. 

Now  replace  the  magnet  by  an  equivalent  closed  circuit 


700 


ELECTRICITY   AND   MAGNETISM. 


[CHAP. 


("  circuit  B  "),  the  positive  aspect  of  which  faces  the  positive 
aspect  of  the  original  closed  circuit  ("circuit  A").  These  two 
circuits,  again,  repel  one  another :  and  if  work  be  done  in  forc- 
ing them  together,  the  energy  appears  in  the  form  of  a  tempo- 
rary increase  in  the  intensities  of  both  currents,  and  is  presently 
converted  into  heat  in  the  circuits. 

Conversely,  when  the  magnet  or  the  equivalent  circuit  is 
withdrawn,  there  is  a  corresponding  temporary  diminution  in 
the  corresponding  current-intensities,  and  possibly  in  the  corre- 
sponding magnet-strength. 

These  increases  and  diminutions  in  current-intensities  are 
equivalent  to  the  Induction  of  New  Currents.  The  duration  of 
these  new  induced  currents  is  limited  to  the  Variable  Period,  the 
time  spent  in  changing  the  relative  positions  of  the  mutually- 
inducing  magnets  or  currents. 

The  lines  of  electric  force  accumulate,  or  else  fall  off,  upon  and 
parallel  to  the  circuit-wire.  The  steady  state  is  thus  interfered  with,  and 
an  electrical  effect  is  produced,  analogous  to  mechanical  acceleration. 

If  a  circuit  bearing  a  current  be  brought  towards  a  circuit 
capable  of  bearing  a  current,  and  if  the  former,  the  inducing  cur- 
rent, have  its  positive  face  turned  towards  the  circuit  approached, 
there  will  be  two  effects  produced:  (1)  an  increase  in  the  inten- 
sity of  the  inducing  current,  and  (2)  a  new  current  developed 
by  induction  in  the  circuit  approached,  which  had  previously 
appeared  to  bear  no  current.  This  current  has  its  positive  face 
turned  towards  the  approaching  positive  face  of  the  inducing 
current,  and  is  therefore  opposed  to  it  in  its  direction. 

If  two  wires  be  laid  alongside  one  another  (Fig.  253),  and 
if  one  of  these  wires  be  connected  with  the  two  poles  of  a  battery, 

and  thus  form  part  of  a  Primary  or 
Battery  Circuit;  while  the  other 
wire  is  merely  a  part  of  a  complete 
metallic  circuit,  a  so-called  Secon- 
dary Circuit ;  then,  when  contact  is 
suddenly  made  in  the  primary  cir- 
cuit, a  current  of  brief  duration  — 
a  duration  not  exceeding  in  time 
the  variable  state  -of  the  primary 
current  —  is  produced  in  the  secon- 
dary circuit  and  is  known  as  the 
Secondary  Current.  The  primary  current  and  the  secondary 
current  are,  in  the  wires  laid  alongside  one  another,  opposed  in 
their  direction. 


Fig. 253. 


xvi.]  ELECTROMAGNETIC   CURRENT-INDUCTION.  7Q1 

So  long  as  the  intensity  of  the  primary  current  remains 
constant,  the  secondary  circuit  has  in  it  no  current ;  but  any 
increase  is  accompanied  by  a  brief  opposed  secondary  current. 

When  the  primary  current  is  diminished,  the  primary  cir- 
cuit again  presents  a  variable  state ;  and  so  long  as  that  variable 
state  lasts,  there  is  again  a  current  in  the  secondary  circuit,  which 
is  on  this  occasion  in  the  same  direction  as  the  waning  primary 
current.  When  the  primary  current  stops  abruptly,  there  is  a 
very  abrupt  secondary  current,  parallel  to  the  ceasing  current. 

These  secondary  currents  represent  a  definite  amount  of 
energy  subtracted  from  the  energy  of  the  primary  current,  —  an 
amount  which  depends  only  on  the  initial  and  final  states  or 
intensities  of  that  current.  Being  of  extremely  short  duration, 
they  are  of  correspondingly  great  intensity  and  high  potential. 
The  secondary  current  produced  on  breaking  the  primary  cur- 
rent is  briefer,  and  therefore  more  intense  than  that  produced 
on  making  it. 

When  a  magnet  is  thrust  into  the  axis  of  a  bobbin  which 
forms  part  of  a  closed  circuit,  there  is  a  current  produced  in 
that  circuit.  The  current  is  opposed  in  direction  to  the  mag- 
netic molecular-currents,  the  Ampere-currents,  of  the  pole  which 
is  introduced  first.  If  a  long  magnet  be  drawn  wholly  through 
such  a  coil,  there  is  at  first  a  current  in  one  direction  as  the  one 
pole  approaches ;  then,  as  its  midpoint  passes  the  midpoint  of 
the  coil,  the  current  is  ra7,  but  is  reversed  as  the  opposite  pole 
passes  out.  The  current  is  at  first  opposed  to  the  Ampere- 
currents  of  the  approaching  pole;  and  as  all  parts  of  a  bar- 
magnet,  looked  at  end-on,  have  their  currents  in  the  same 
direction  in  space,  the  induced  current  changes  in  its  direction 
as  the  magnet  passes  through. 

These  statements  may  be  generalised  by  saying  that  wher- 
ever a  closed  circuit,  capable  of  bearing  an  electric  current,  lies 
wholly  or  in  part  in  a  Magnetic  or  Electromagnetic  Field  of 
Force,  any  disturbance  in  the  Intensity  of  the  Field  of  Force 
will  induce  a  Current  in  the  circuit ;  and  the  direction  of  the 
induced  current  is  determined  by  the  rule  (Lenz's  Law)  that 
the  new  current  will  increase  the  already-existing  resistances, 
or  develope  new  resistance  to  that  disturbance  of  the  field  which 
is  the  cause  of  induction. 

A  telephone  circuit  passing  through  a  disturbed  field  of  force  will  pick 
up  signals :  for  example,  at  every  lightning  flash  the  instrument  is  heard  to 
roar;  and  in  order  to  prevent  such  effects  of  induction,  no  part  of  the  current 


702  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

is  entrusted  to  earth,  but  the  double  wire  necessary  is  coiled  round  itself  so 
as  to  form  a  strand  composed  of  two  insulated  wires.  The  effect  of  induc- 
tion on  one  wire  is  then  equal  to  the  opposite  effect  of  induction  on  the  other 
wire.  During  thunderstorms  military  mine-fuses  have  been  known  to  explode 
through  induction  in  the  wires  controlling  them. 

We  have  seen  that  a  closed  current,  A,  whose  positive  aspect  faces  a 
positive  magnetic  pole  or  face  of  a  magnetic  shell  or  equivalent  electric  cur- 
rent, B,  sends  towards  the  latter,  from  within  its  own  contour,  a  number  of 
positive  lines  of  force  or  of  induction,  which  radiate  from  its  positive  face. 
If  we  change  our  standpoint,  and  regard  .the  current  first  mentioned  —  a 
current  borne  by  circuit  A  —  as  placed  within  the  magnetic  field  of  the  shell 
B  or  the  electromagnetic  field  of  the  equivalent  circuit,  then  the  positive  lines 
proceeding  from  the  latter  are,  as  regards  the  circuit  A,  negative,  for  they 
trend  not  from  but  towards  its  positive  aspect. 

Circuit  A,  as  we  have  seen,  tends  to  move  by  translation  to  a  greater  dis- 
tance from  circuit  B.  It  will  also  tend  to  rotate  until  its  negative  aspect 
faces  the  positive  side  of  B ;  it  is  then  attracted  towards  B. 

In  the  former  case,  as  A  moves  away,  the  number  of  negative  lines 
which  pass  towards  its  area  diminishes.  In  the  latter  case  —  that  of  rota- 
tion —  A  tends,  as  it  turns,  first  to  set  itself  edge-on  to  B's  lines  of  induction, 
and  then  so  to  place  itself  (its  negative  face  opposite  to  B's  positive  face) 
that  the  lines  of  induction  which  emerge  from  B  positively  also  emerge  from 
A's  positive  face  positively,  and  are  positive  with  respect  to  A. 

In  either  of  these  cases,  translation  or  rotation,  the  number  of  negative 
lines  met  by  the  area  of  A  is  diminished  as  far  as  possible,  or  the  number  of 
positive  lines  embraced  by  its  contour  attains  a  maximum. 

A  little  movable  circuit  may  be  made  —  De  la  Rive's  floating  battery  — 
by  thrusting  a  strip  of  copper  and  a  strip  of  zinc  through  a  cork,  and  con- 
necting them  by  an  arch  of  copper  wire :  when  the  whole  is  floated  in  water, 
the  arch  tends  to  lay  itself  at  right  angles  to  the  magnetic  meridian,  copper 
to  the  west,  zinc  to  the  east ;  in  this  position  the  positive  face  of  the  arch  is 
to  the  north,  and  the  magnetic  lines  of  force  or  induction  which  trend  towards 
the  north  are  embraced  by  the  arch  in  the  greatest  possible  number. 

The  general  statement  of  the  phenomenon  is  :  —  A  movable  circuit  tends 
so  to  place  itself  as  to  meet  as  few  negative  or  to  have  as  many  positive 
lines  of  induction  passing  through  it  as  possible ;  a  line  passing  through  a 
circuit  being  held  positive  when,  after  passing  through,  it  emerges  from  the 
positive  face  in  a  positive  direction ;  a  line  being  held  to  be  negative  when 
its  direction  is  towards  the  positive  face.  The  position  thus  assumed  is  the 
position  of  least  potential  energy,  that  into  which  the  whole  system  tends, 
as  it  were,  to  sink.  A  circuit  in  this  position  of  least  potential  energy  embraces 
as  great  a  number  as  possible  of  positive  lines  of  induction. 

Mutual  Attraction  and  Repulsion  of  Currents.  —  Suppose  two  cur- 
rents in  the  plane  of  the  paper,  similar  in  their  directions  and  having  in  con- 
sequence their  nearer  portions  opposed  in  direction,  as  in  Fig.  243.  Let  their 
directions  be  the  same  as  those  of  the  two  currents  in  that  figure.  The  left- 
hand  current  in  that  figure  has  lines  of  induction  which  ascend  from  the 
plane  of  the  paper  and  tend  to  descend  through  the  contour  of  the  right-hand 
circuit,  meeting  its  ascending  lines.  These  descending  lines  are  therefore 
negative  to  the  right-hand  circuit,  and  that  circuit  tends  to  move  away  so  as 
to  meet  as  few  of  them  as  possible.  The  portions  of  the  currents  which  are 
nearest  one  another,  running  in  opposite  directions,  thus  seem  to  repel  one 


xvi.]  MUTUAL  ACTION  OF   CURRENTS.  703 

another.  The  area  in  which  downward  lines  meet  upward  lines  is  thus 
diminished  as  far  as  possible,  and  this  enables  us  now  to  understand  the 
propositions  illustrated  by  Figs.  229,  230. 

If  a  circuit  embracing  the  greatest  possible  number  of  positive  lines  of 
induction,  and  therefore  occupying  the  position  of  least  potential  energy,  be 
pulled  or  turned  into  any  other  position,  work  must  be  done  upon  it ;  and 
this  work  is  done  against  mutual  attractions.  This  doing  of  work  is  associ- 
ated with  diminution  of  the  number  of  positive  lines  of  induction  embraced 
by  the  movable  circuit.  As  the  circuit  moves  in  the  field,  lines  of  induction 
must  be  cut  through  by  it.  AU  cutting  through  lines  of  induction,  when 
the  number  of  lines  enclosed  by  the  circuit  is  diminished  by  the  opera- 
tion, is  effected  by  the  expenditure  of  work.  The  process  attains  its 
maximum  when  the  movable  circuit  has  been  swung  round  through  180°. 
If  it  be  still  farther  rotated,  it  comes  to  meet  fewer  negative  lines,  then 
to  enclose  an  increasing  number  of  positive  lines,  until  it  regains  its  original 
position . 

The  work  done  takes  the  form  of  the  energy  of  induced  currents,  which 
always  increase  the  resistance  to  the  actual  movement ;  if  A  and  B  repel  one 
another,  their  intensities  are  increased  when  they  are  urged  together,  dimin- 
ished when  they  are  drawn  asunder ;  if  they  attract  one  another  these  actions 
are  reversed.  This  may  be  otherwise  expressed  by  saying  that  when  a  circuit 
is  made  to  meet  a  greater  number  of  negative  lines  of  induction,  or  to  enclose 
a  smaller  number  of  positive  lines,  its  current  is  increased  in  intensity, 
or  a  new  induced  current  set  up  in  it ;  while  if  it  be  made  to  meet  a  smaller 
number  of  negative  or  to  enclose  a  greater  number  of  positive  lines,  the  inten- 
sity of  its  current  is  diminished,  or  a  new  reverse  current  is  set  up  in  it. 
The  result  is  the  same  whether  it  move  so  as  to  enclose  more  or  fewer  lines 
in  an  existing  magnetic  field,  or  whether  the  magnetic  field  itself  vary  so 
that  its  lines  either  open  out  and  become  fewer,  or  become  more  numerous 
and  approach  one  another  —  a  smaller  or  a  greater  number  of  them  conse- 
quently passing  through  the  given  circuit. 

If  a  part  of  a  circuit  of  total  resistance  r  be  movable  in  a  magnetic  field 
which  presents  b  lines  of  magnetic  induction  per  sq.  cm.,  it  will  cut  through 
all  the  magnetic  lines  in  a  certain  area  A  in  the  course  of  time  t ;  it  will 
therefore  cut  through  Ab/t  —  B/f  lines  of  induction  per  second.  The  cur- 
rent set  up  is,  firstly,  such  that  if  the  movable  part  of  the  circuit  uniformly 
diminish  the  area  of  the  circuit  as  it  moves  in  the  terrestrial  magnetic  field, 
the  current  will  run  in  the  circuit  (which  is  supposed  to  be  set  in  a  plane  at 
right  angles  to  the  magnetic  meridian)  in  a  direction  which  seems  from  the 
standpoint  of  an  observer  stationed  to  the  south  to  be  the  same  as  that  of  the 
hands  of  a  watch  :  and,  secondly,  its  intensity  is  proportional  to  B  /  rt.  In 
Electromagnetic  measure,  the  units  are  so  adjusted  that  the 
intensity  of  the  induced  current  is  equal  to  B / rt  =  Ab / rt.  If  B 
be  the  additional  number  of  lines  which  the  circuit  comes  to  enclose,  the 
intensity  of  the  induced  current  is  i  =  —  B/rt  =  —  Ab/rt.  This  relation  is 
the  same,  whatever  be  the  permeability  //,. 

When  a  block  of  copper  is  whirled  within  a  magnetic  field,  currents  are 
set  up  in  it,  which  produce  resistance  to  the  motion;  the  motion  of  the 
block  very  rapidly  ceases,  as  if  the  magnetic  field  were  highly  viscous,  and 
the  block  becomes  hot.  When  a  magnet-needle  is  suspended  immediately 
above  a  copper  plate,  any  oscillation  in  the  magnet  developes^retarding  cur- 
rents in  the  copper,  and  the  magnet  almost  immediately  comes  to  rest. 


704  ELECTRICITY    AND   MAGNETISM.  [CHAP. 

Self-induction.  —  A  current  suddenly  formed  in  a  spiral 
wire  is  retarded  by  the  mutual  action  of  the  different  turns ;  it 
does  not  flow  on,  and  its  intensity  is,  at  first,  less  than  it  would 
have  been  in  a  straight  wire  :  when  suddenly  broken  it  is  pro- 
longed and  is  as  it  were  piled  up,  so  that  the  so-called  Extra- 
Current  can  force  its  way  through  greater  resistance  than  the 
steady  current  can.  In  fact,  a  single  Daniell  cell  can  be  made  to 
electrolyse  water  by  delivering  a  part  of  the  energy  of  its  cur- 
rent, at  high  potential,  in  the  form  of  the  so-called  Extra-current. 

These  phenomena  closely  remind  us  of  the  phenomena  of 
momentum  in  a  water-pipe,  already  discussed  under  the  Hydrau- 
lic Ram ;  and  they  can  be  explained  as  phenomena  of  momen- 
tum of  the  Ether  in  the  electromagnetic  field. 

Two  wires  bearing  currents  in  opposite  directions,  and 
twisted  round  one  another,  present  no  phenomena  of  self-induc- 
tion ;  for  which  reason  the  wires  leading  to  and  from  a  galva- 
nometer should  be  twisted  together  for  some  distance  from  the 
needle. 

Coefficient  of  Mutual  Induction  of  two  Currents.  —  The  Ether  sur- 
rounding a  pair  of  current-loops  of  intensities  iy  and  in  must  possess  Energy, 
which  Clerk  Maxwell  showed  to  be  proportional  to  squares  and  products 
of  the  intensities,  and  which  may  be  written  thus  :  {1L^/2  -f  Mz/'^  +  ^L'i,/2}. 

The  second  term  vanishes  when  the  currents  are  at  an  infinite  distance 
from  one  another;  it  is  at  its  greatest  practical  when  the  two  circuits  almost 
touch  one  another,  its  greatest  theoretical  when  they  absolutely  coincide : 
at  intermediate  points  it  has  intermediate  values.  It  can  be  shown  that 
Miyi^  is,  in  any  given  position  of  A  and  B,  numerically  equal  (1)  to  the 
Mutual  Potential  Energy  of  the  two  circuits  and  (2)  to  the  Number  of 
Lines  of  Induction  which,  being  due  to  A,  pass  from  A  through  B  or, 
equally,  being  due  to  B,  pass  from  B  through  A;  and  M  is  styled  the 
Mutual  Inductance  or  the  Coefficient  of  Mutual  Induction. 
M  varies  with  the  relative  position  of  the  two  circuits. 

The  maximum  value  of  M  is  its  value  when  the  two  currents  are  made 
to  run  in  the  same  circuit ;  let  this  be  called  M0.  The  number  of  lines 
( =  MO^,)  due  to  i,  and  the  number  due  to  in  are  to  be  added  together  for 
the  conjoined  current  (it  +  i/y)  ;  for  they  all  pass  through  the  same  circuit ; 
hence  the  number  actually  threading  the  circuit  will  be  2M0i/i//.  In  this 
case  ^Ji0ilill  is  equal  to  the  part  of  the  energy  of  the  field  which  is  due  to  the 
approximation  of  the  currents  i,  and  in  from  an  infinite  distance  and  their 
coincidence ;  and  M0  is  equal,  numerically,  to  half  the  number  of  lines  of 
induction  which  pass  through  the  circuit  itself  when  it  and  itl  are  both 
unity,  that  is,  when  the  conjoined  current  has  an  intensity  =  2.  It  is  there- 
fore equal  to  the  number  of  lines  of  induction  passing  through  when  i  =  1. 

Coefficient  of  Self-induction,  or  Inductance. — We  next  see  that 
M0  =  L.  If  the  intensity  of  the  second  current  in  be  0,  the  energy  of  the 
field  is  ^Li,2  only.  A  second  current  of  the  same  intensity  in  a  circuit  of 
the  same  size,  etc.,  at  an  infinite  distance  will  have  energy  also  equal  to 


xvi.]  SELF-INDUCTION.  705 

iL*,2.  Together  the  energy  will  be  Li,2.  Now  bring  the  two  currents 
together  and  blend  them ;  the  energy  is  -^-L(2i/)2  =  2Liy2.  The  system  pos- 
sesses energy  equal  to  Li,2,  due  to  the  approximation ;  but  this  is  also  M^3 
if  both  currents  be  equal  to  it ;  whence  L  =  M0.  L  is  the  Coefficient  of 
Self-induction;  and  the  coefficient  of  self-induction  of  a  circuit  is  equal, 
numerically,  to  the  number  of  Lines  of  Magnetic  Induction  which  thread 
that  circuit  when  it  bears  a  current  whose  intensity  is  unity  in  electromag- 
netic measure.  Within  the  contour  of  a  circuit,  B  =  Li. 

In  a  solenoid  of  n  turns,  and  length  /  cm.,  the  number  of  lines  of  induc- 
tion for  unit  current  is  kirn- p,/ 1  per  sq.  cm.,  or  4irn-  A- p./ 1  across- area  A. 
If  a  second  solenoid,  of  n'  turns,  surround  the  first,  each  turn  of  it  embraces 
4r7rn-  A  •  n/l  lines  once ;  and  its  n'  turns  embrace  4?rn  •  n'-  Ap/l  lines.  Hence 
for  two  such  solenoids,  M  =  4-Tr  •  nn'-  Ap/l  ;  and  for  a  single  solenoid, 
L  =  47rn2A^/Z.  Hence  the  self-induction  of  a  coil  of  many  turns  is  very 
great, 

Extra-Currents.  —  If  the  energy  of  a  current  traversing  a  single  circuit 
be  derived  from  any  external  source,  such  as  a  battery,  which  is  independent 
of  induction,  the  energy  supplied  from  that  source  during  a  very  short  time 
8t  will  be  equal  to  ei  -  &,  where  e  is  the  E.M.D.P.  and  f  the  current-intensity. 
(All  our  measurements  in  these  paragraphs  are  supposed  to  be  made  in 
electromagnetic  measure.)  This  energy  is  divided  into  three  parts. 

(1.)  Heat  in  the  circuit.    This  is  equal  to  ri2  •  &,  where  r  is  the  resistance. 

(2.)  External  work,  mechanical,  chemical  or  other.  This  we  shall  sup- 
pose =  0. 

(3.)  Work  spent  in  imparting  energy  to  the  electromagnetic  field.  This 
is  equal  to  |L{(i  +  Si)2  —  i2},  where  Si  is  the  small  change  in  the  intensity 
produced  during  the  time  8t. 

We  thus  have  the  equation 

ei  -  &  =  ri*  •  &  +  4L  {(t  +  Si)2  -  *»} ;  (i.) 

an  equation  which  can  be  dealt  with  by  integration,  the  effect  being  that  we 
find  the  intensity,  at  any  time  t  after  the  introduction  of  a  new  E.M.D.P.  =  e 
into  the  circuit,  to  be 

it  =  e/r  -  e/r  (2-718281  -•*/!•).  (ii.) 

The  intensity  never  comes  fully  up  to  the  value  e/r;  but  it  approaches  it 
indefinitely  nearly  as  the  time  t  increases.  If,  however,  the  coefficient  L  be 
large,  as  it  is  in  a  coil  of  wire,  the  second  term  on  the  right-hand  side  is  not 
immeasurably  small,  and  it  represents  what  is  equivalent  to  a  reverse  cur- 
rent lasting  for  an  appreciable  time,  and  delaying  the  development  of  a 
current  of  full  intensity  e/r.  This  reverse  current  is  called  the  Reverse 
Extra-Current  or  the  Extra-Current  of  Closure  or  of  Making. 

When  a  circuit  is  suddenly  broken,  the  intensity  at  a  time  t  after  the 
current  has  been  stopped  is  +  (e/r}  (2-718281-*/L).  This  indicates  that 
there  is  still  an  onflow,  a"  Direct  Extra-Current  or  Extra-Current 
of  Opening  or  Breaking;  an  onflow  which  results  in  a  high  potential 
at  the  broken  extremities  of  the  wire,  and,  since  the  capacities  of  these 
extremities  are  small,  in  a  high  value  of  a  at  these  extremities. 

These  extra-currents  are  thus  associated  with  absorption  of  energy  by 
the  electromagnetic  field  while  currents,  the  energy  of  which  is  derived 
from  extraneous  sources,  are  being  produced  or  increased,  and  with  libera- 
tion of  energy  by  that  field  when  such  currents  are  brokers  or  while  they 
are  being  diminished. 

2z 


706  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

They  may  also  be  looked  at  as  phenomena  of  Induction.  When  the 
intensity  of  a  current  is  increased,  the  circuit  is  made  to  embrace  more 
lines  of  induction:  if  it  embrace  B  more  lines  of  force  in  time  &,  an 
E.M.D.P.  is  set  up,  equal  in  electromagnetic  measure,  to  —  B/&.  Where  L 
is  the  self-induction  of  a  circuit,  the  establishment  of  current  i  in  it  causes 
the  development  of  Li  lines  of  induction  embraced  by  it ;  and  this  causes  an 
E.M.D.P.  —  e,  =.  —  Li/Bt;  whence  «',,  the  mean  intensity,  =  —  Li/r-  8t ;  this 
being  the  mean  intensity  of  an  induced  current  opposed  in  its  direction  to 
the  originating  current.  When  the  main-current  ceases,  the  induced  cur- 
rent now  produced  on  the  disappearance  of  lines  of  induction  is  direct,  the 
Direct  Extra-Current. 

The  steady  intensity  i  =  e/r ;  whence  the  new  "  electromotive  force " 
e/  =  —  Le/r  •  St.  Since  8t  is  very  small  this  may  greatly  exceed  e ;  and,  for 
a  given  length  of  wire,  it  is  greatest  when  the  current  passes  through  con- 
ductors of  a  spiral  form,  in  wliich  the  value  of  L  is  great.  The  extra-cur- 
rent may  thus  be  able  to  spark  across  striking  distances  beyond  the  power 
of  the  main-current. 

The  quantity  of  electricity  in  the  extra-current  is  in  each  case  q  =  i,  •  $t 
=  eL/r2.  These  relations  are  the  same,  whatever  be  the  permeability  ju,. 

Measurement  of  Inductance  L.  —  Take  a  Wheatstone  Bridge  (Fig. 
222),  with  arms  AB,  BC,  AD,  DC,  with  a  battery  as  in  the  figure,  and  with 
a  galvanometer  between  B  and  D;  and  then  lengthen  the  wire  of  the  bat- 
tery circuit  AC,  and  of  the  galvanometer  branch  BD,  so  as  to  bring  them 
both  up  to  a  rotating  commutator  or  contrivance  which  shall,  in  repeated 
succession,  perform  the  following  cycle  of  operations,  viz. :  —  (1)  disconnect 
the  battery,  (2)  disconnect  the  galvanometer,  (3)  connect  up  the  battery, 
and  (4)  bring  in  the  galvanometer.  Then,  if  the  resistances  in  the  arms 
AB,  BC,  CD,  and  DA  be  respectively  r',  r",  r'",  and  a  resistance  r""  which 
comprises  that  of  the  loop  or  coil  which  is  to  be  tested ;  and  if  these  be  so 
adjusted  that  there  is  no  current  in  the  galvanometer;  then,  on  setting  the 
commutator  in  action,  the  balance  of  resistance  will  appear  to  be  disturbed, 
for  the  "  impedance  "  of  the  coil  (p.  722)  is  not  the  same  as  its  Resistance 
to  steady  currents ;  and  the  galvanometer-needle  will  be  deflected.  The  re- 
sistance r""  will  have  to  be  reduced  by  a  certain  amount  Br  to  restore  the 
balance ;  and  Ayrton  and  Perry  have  shown  that  if  n  be  the  number  of 
complete  cycles  per  second,  the  inductance  L  is  equal  to  8r/n.  (Ayrton 
and  Perry's  Secohm-meter.) 

Induction  Coils.  —  The  effect  of  induction  is  multiplied 
when  the  two  wires,  that  of  the  primary  and  that  of  the  secon- 
dary circuit,  though  kept  insulated  from  one  another,  are  wound 
together  round  the  same  axis.  The  secondary  current  is  then 
proportional  in  its  intensity  to  the  product  of  the  number  of 
turns  in  the  two  wires,  provided  that  the  resistances  introduced 
by  multiplying  the  coils,  or  the  differences  between  the  mutual 
distances  of  the  different  turns,  be  not  too  considerable.  In 
Induction  Coils  the  wires  of  the  two  circuits  are  wound  round 
separate  bobbins,  which  are  then  slipped  the  one  over  the  other 
to  a  greater  or  less  extent.  On  this  extent  depends  the  inten- 


xvi.]  INDUCTION-COILS.  707 

sity  of  the  secondary  current.  The  primary  current  is  made 
and  broken  with  great  frequency  by  means  of  a  Contact- 
breaker. 

This  may  be  a  mere  mechanical  contrivance,  or  it  may  be  automatic. 

In  the  latter  case  there  lies  a  bar  of  soft  iron  in  the  axis  of  the  inner, 
the  primary,  bobbin.  When  the  primary  current  passes,  this  bar  or  core 
becomes  an  electromagnet.  This  electromagnet  pulls  towards  itself  an 
armature,  a  mass  of  soft  iron,  which  is  arranged  near  one  of  its  extremi- 
ties ;  this  mass  of  soft  iron  is  an  integral  part  of  the  circuit  of  the  primary 
current,  and  by  its  movement  the  primary  current  is  suddenly  broken.  The 
electromagnet  now  loses  its  magnetic  condition;  it  ceases  to  attract  the 
armature ;  the  latter,  under  the  pressure  of  a  spring,  returns  to  its  former 
position,  and  again  completes  the  primary  circuit ;  the  electromagnet  is 
again  made,  and  the  armature  again  displaced.  The  soft  iron  armature  is 
thus  caused  to  oscillate  and  to  impart  to  the  primary  current  an  iiitermit- 
tence,  whose  frequency  depends  upon  the  intensity  of  the  current  and 
upon  the  pressure  of  the  spring. 

MAGNETIC  OR  ELECTROMAGNETIC  MEASURE. 

A  current  of  given  intensity,  I  in  electrostatic  units,  must 
be  represented  by  smaller  numbers  when  magnetic  units  are 
used ;  a  current  of  I  =  60000,000000  is  a  current  of  i  =  2 ;  for 
the  C.G.S.  magnetic  unit  of  current-intensity  or  -strength  is 
30000,000000  times  as  great  as  the  C.G.S.  electrostatic  unit. 

The  basis  of  the  Magnetic  or  Electromagnetic  system  of 
measurement  is  the  identity  of  effect,  in  air,  between  a  mag- 
netic shell  of  strength  qp  and  a  closed  current  of  the  same  con- 
tour and  of  a  particular  intensity  i  ;  the  units  of  current-intensity 
are  so  adjusted  that  i  becomes,  in  air,  numerically  equal  to  y. 

If  i  =  l,  the  current  is  equivalent  in  magnetic  effect  to  a  magnetic  shell 
whose  strength  q>  is  unity  and  whose  area  and  outline  are  the  .same  as  that 
of  the  circuit;  and  this  is  the  Magnetic  C.G.S.  Unit  of  Current-Intensity 
(Definition  i.).  [qp  =  fjd.~\ 

If  we  suppose  a  wire  bearing  a  current,  and  one  cm.  in  length,  to  be  bent 
into  a  circular  arc  whose  radius  is  one  cm.,  and  if  we  suppose  a  unit  mag- 
netic pole  to  be  placed  at  the  centre  of  the  circle  of  which  the  circular  arc 
forms  a  part ;  and  if  we  further  suppose  that  the  mechanical  force  exerted 
by  the  current  upon  the  unit  magnet-pole  is  equal  to  one  dyne  ;  —  then  the 
current  is  one  whose  intensity  is,  in  magnetic  measure,  equal  to  unity.  In 
such  a  case  i  =  1 ;  and  this  may  be  taken  as  Definition  ii.  of  the  Magnetic 
C.G.S.  Unit  of  Current-Intensity.  The  general  formula  is  F  =  tn  •  t//r2,  where 
F  is  the  force  exerted  upon  a  magnet-pole  placed  at  the  centre  of  such  an 
arc,  m  the  strength  of  that  pole,  t  the  intensity  of  the  current,  I  the  length 
and  r  the  radius  of  the  circular  arc  into  which  the  wire  is  bent.  [F  does 
not  depend  upon  /x.] 

If  the  wire  be  bent  into  a  complete  single  circular  loop  pf  radius  r,  I  — 
2irr,  and  F  =  tn  •  i  •  2ir/r. 


708  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

If  the  pole  ttt  be  in  the  axis  of  the  single  loop,  but  not  necessarily  in 
its  plane,  F  =  /-i-27r-r2/w3,  where  u  is  the  distance,  more  or  less  oblique  to 
the  axis,  between  the  pole  and  the  wire.  When  u  is  reduced  to  u  =  r ; 
that  is,  when  the  pole  comes  into  the  plane  of  the  loop,  F  =  m  •  i  -  2ir  -  r2/r3 
=  m  •  i  •  2-jr  /  r ;  which  agrees  with  the  expression  above. 

If  a  current,  =  i  in  magnetic  units,  pass  along  a  wire  across  a  uniform 
field  of  magnetic  force  or  strength  h,  in  air,  the  wire  is  acted  upon,  trans- 
versely, by  a  force  F  =  hil  dynes,  if  the  length  of  the  wire  be  I  cm. ;  and 
this  gives  us  another  definition  (Definition  iii.)  of  the  Magnetic  C.G.S. 
Unit  of  Current-Intensity.  This  also  enables  us  to  measure  the  intensity  h 
of  an  intense  magnetic  or  electromagnetic  field  ;  a  current  is  led  through  it ; 
the  wire  is  forced  in  one  or  other  direction  ;  this  force  can  be  balanced  by  a 
known  weight.  If  the  current  be  sent  through  a  column  of  mercury  in  a 
known  magnetic  field,  there  is  a  difference  between  the  manometric  pressures 
at  the  two  sides  of  the  column  (Lippmann).  [F  =  /x,-hi-/.] 

(Definition  iv.)  :  a  current  of  intensity  t,  traversing  a  straight  wire  of 
indefinite  length,  acts  upon  a  magnetic  pole  m,  at  a  distance  d  from  the 
wire,  with  a  force  F  =  nth  =  m  •  2i/d.  (Compare  p.  189,  prop.  7.)  A  unit 
current  would  therefore  act  upon  a  pole  m  with  a  force  F  =  2m/ d.  [Does  not 
depend  upon  /x,.] 

When  a  current  i  passes  into  a  long  solenoid,  of  n  turns  and  length  I 
cm.,  a  magnetic  pole  m  anywhere  within  the  solenoid  is  acted  upon  with 
a  force  F  =  h-m  =  ^ir-m-i-n/l.  Whence  a  unit  current  will  act  with  a 
force  F  =  m  •  4-rr  •  n/l,  or  (4?r  •  m)  X  the  number  of  turns  in  the  solenoid  per 
unit  of  length.  (Definition  v.)  [Does  not  depend  upon  /A.] 

The  Intensity  of  a  Current,  magnetically  measured,  and 
the  Magnetic  Strength  of  a  Shell  must  accordingly  have  the 
same  dimensions.  The  Magnetic  Strength  of  a  magnetic  shell 
is  (p.  682)  numerically  equal  to  the  product  of  the  Magnetic 
Quantity  per  unit  of  area  into  the  Thickness  of  the  shell ;  its 
dimensions  must  hence  be  those  of  Magnetic  Quantity,  divided 
by  an  Area,  and  multiplied  by  a  linear  Thickness  ;  but  the 
dimensions  of  Magnetic  Quantity  must  (since  the  imaginary 
magnetic  matter  obeys  laws  resembling  those  of  repelling  or 
attracting  electric  matter,  similarly  imaginary)  be  like  those  of 
electric  quantity,  [M*L*/T]  ;  the  dimensions  of  Magnet-Strength 
are  accordingly  [M4L*/T]  -*-  [L2]  x  [L]  =  [M*L*/T]  ;  the  Mag- 
netic Measure  of  Current-Intensity  has  the  same  dimensions, 
and  therefore  differs  from  the  electrostatic  measure,  whose 
dimensions  are  [M*L*/T2],  by  the  term  [L/T],  which  represents 
a  Velocity :  the  numerical  value  of  this  ratio  must  be  found  by 
experiment,  which  shows  it  to  be  30,000,000000  nearly,  =  V. 

Measurement  of  V.  —  The  ratio  between  the  magnetic  and  electro- 
static measures  may  be  determined  by  several  methods,  of  which  two  may 
be  taken  as  examples. 

Weber's  method.  —  Charge  a  Ley  den  jar  with  a  known  quantity  of 
electricity,  Q ;  discharge  the  jar  through  a  wire,  which  passes  round  a  gal- 


xvi.]  MEASUREMENT  OF  V.  709 

vanometer-needle.  The  quantity  of  Electricity  passing  through  the  galva- 
nometer may  be  measured  in  terms  of  the  deviation  undergone  by  the  needle 
in  consequence  of  being  thrown  by  the  instantaneous  current.  This  gives 
the  quantity,  q,  in  magnetic  measure.  These  separate  measurements  of  the 
numerical  values,  Q  and  q,  of  the  one  quantity  of  electricity  give  the  ratio 
between  the  electrostatic  and  the  magnetic  unit. 

Lord  Kelvin's  method.  —  The  two  ends  of  a  wire  of  great  resist- 
ance, R,  are  kept  at  a  constant  potential-difference,  E ;  a  constant  current 
runs  through  the  wire;  this  current  is  found  to  have  an  intensity  i  C.G.S. 
magnetic  units ;  the  difference  of  potential  is  (by  Ohm's  law)  E  =  IR  or 
e  =  ir.  E  and  e,  or  i  and  I,  bear  to  one  another  the  relation  of  1  :  V  ;  whence 
V  may  be  found  numerically. 

The  meaning  of  this  ratio  between  the  Electrostatic  and  the  Magnetic 
or  electromagnetic  units  is  frequently  found  to  be  puzzling.  Its  real  basis 
is  the  following.  Both  these  systems  of  units  are  based  on  independent 
arbitrary  conventions ;  and  neither  of  them  can  absolutely  represent  phys- 
ical truth,  though  all  calculations  will  work  out  accurately  if  we  adhere  to 
either  system.  In  the  Electrostatic  system  the  units  are  so  adjusted  that 
equal  charges  at  unit  distance  apart,  repelling  or  attracting  one  another 
through  air  with  unit  force,  are  called  "  unit  charges  :  "  but  more  generally, 
the  mutual  action  of  two  charges  depends  upon  K,  the  sp.  ind.  cap.  of  the 
medium  between  them;  F  =  QQ'/IW2;  whence  [Q]  =  [Ktal*I*t/T]  ;  and 
it  is  only  by  assuming  the  sp.  ind.  cap.  of  air  to  be  unity,  and  the  sp.  ind. 
caps,  of  other  media  to  be  Numbers  merely,  that  we  arrive  at  the  air-equa- 
tion F  =  QQ'/^2'  and  the  corresponding  Equation  of  Dimensions  [Q]  = 
[M^U/T].  Similarly,  the  Force  F  between  two  currents  i  and  i'  varies  as 
//,  •  it',  where  p  is  the  magnetic  permeability  of  the  medium  :  that  is,  [Force] 
=  [>-Q2/T2]»  and  [Q]  =  [MiLV/n*]  ;  but  by  similarly  assuming  the  mag- 
netic permeability  of  air  to  be  unity,  and  the  magn.  perms,  of  other  media 
to  be  numbers  merely,  we  arrive  at  that  air-equation  of  Dimensions, 
[Q]  =  [MW],  which  lies  at  the  basis  of  the  Magnetic  system  of  measure- 
ment. The  conventional  units  of  electric  Quantity  thus  bear  to  one  another 
the  ratio  of  [M4U/T]  to  [M*L*],  or  [L/T]  to  1;  but  this  apparent  want 
of  equality  arises  from  these  conventions  themselves.  On  a  natural  system, 
the  same  quantity  ought  to  have  the  same  Dimensions,  whether  looked  at 
from  an  electrostatic  or  a  magnetic  point  of  view ;  and  therefore,  throw- 
ing aside  these  conventions,  we  must  have  [MWKs/T]  =  [M^L*///,*],  or 
[L/T-  VK/x]  =  a  Number  merely,  and  that  number  =  1.  When  we  assume, 
in  accordance  with  our  conventions,  that  K  and  /u,  are  both  numerically  =  1, 
assumptions  which  cannot  possibly  both  be  true,  we  find  that  L/T,  the 
term  of  difference  between  the  conventional  e.-s.  and  in.  units  of  electric 
Quantity,  is  experimentally  determinable  as  numerically  equal  to  3  x  1010 ; 
and  as  from  its  Dimensions  it  is  a  Velocity,  it  is  said  to  be  a  Velocity  of 
3  x  1010  cm.  per  second.  What  the  last  equation  of  dimensions  more  truly 
shows,  however,  is  that  [K/x]  =  [T2/L2],  and  that  K/x  =  9  x  10*>.  We  do 
not  know  the  absolute  numerical  values  of  either  K  or  /w,  separately ;  neither 
do  we  know  their  Dimensions  separately.  See,  however,  p.  746. 

Dimensions  in  the  Conventional  Magnetic  Air-system. — Current 
Intensity  or  Strength,  f.  Page  670,  no.  1;  attraction  or  repulsion 
(=  mechanical  Force)  a  ii'  x  ll'/d2;  i.e.,  [ML/T2]  =  [intensity2];  .-.  [i]  = 
[MlLi/T], 

Quantity,  q,  =  Intensity  x  Time;  [?]  =  {[MW/T]  x  fT]}  =  [MIL*]. 


710  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

Potential,  or  Difference  of   Potential,  e,  =  work    done  -f-  quantity  of 
electricity     upon    which    work    is     done  ;     \_e]  =  {[ML2/T2]  -*•  [M*Li]}  = 


Electrical  Force,  the  mechanical  force  acting  on  magnetic  unit  of 
electric  quantity.  Its  dimensions  are  those  of  mechanical  force  -H-  electric 
quantity  ;  [f]  =  [F/£j  =  {[ML/T2]  -  [M*Li]}  =  [MilJ/T2].  Otherwise, 
this  is  Potential-Slope,  [WLt/T*]  •*•  [L]  =  [MlLl/T^. 

Resistance  =  difference  of  potential  -f-  current-intensity  ;  {[M^Li/T2] 
-i-  [MAL5/T]}  =  [L/T],  a  Velocity  ;  *  whence  the  resistance  of  an  Ohm  wire 
(109  magnetic  C.G.S.  units  of  resistance)  is  said  to  be  1C9  cm.  per  sec.  ;  and 
so  on  in  proportion. 

Capacity  is  quantity  of  electricity  stored  up  per  unit  potential-differ- 
ence produced  by  it  ;  its  dimensions  are  {[M*L4]  -5-  [M^Li/T2]}  =  [T2/L]. 

Conductivity:  the  intensity  of  current  passing  across  unit  area 
under  the  action  of  unit  electrical  force.  Its  dimensions  are  those  of  cur- 
rent-intensity -T-  (electrical  force  x  area),  viz.,  {[M*L*/T]  +  [(MiLi/T2  x 


Resistivity,  the  reciprocal  of  the  Conductivity  ;   [L2/T]. 

Coefficients  of  Self-Induction  and  Mutual  Induction  of 
Currents:  Ratios  between  E.M.D.P.  produced  and  the  rate  of  change  of 
current-intensity  producing  it  :  the  dimensions  of  the  former  are  [MsU/T2]  ; 
those  of  the  latter  are  [Intensity  -=-  Time]  =  [MiJ>/T]  -i-  [T]  =  [M*U/T2]  ; 
the  ratio  therefore  has  the  dimensions  [M*lJ/T2]  -*-  [MiJJ/T2]  =  [L]. 

Magnetic  Dimensions  in  any  medium.  —  Current-intensity,  [M*Li/ 
T/A*];  Electric  Quantity,  [M^Li//*4]  5  Electric  Potential,  j>*M4U/T];  Elec- 
tric Force,  [/^MsLs/T2]  ;  Resistance,  |>L/T]  ;  Conductance,  [T/L/x]  ; 
Resistivity,  [>L2/T];  Conductivity,  [T/L2/jJ]  ;  Capacity,  [T2/L/x]  ;  Coeffi- 
cients of  Induction,  [/xL]. 

From  these  dimensions  we  find  that  that  which  is  measured 
electrostatically  as  a  current  of  intensity  I  e.-s.  units  is  magneti- 
cally a  current  of  intensity  i  =  (I/V)  magnetic  units.  Similarly, 
by  comparison  with  the  electrostatic  measures,  we  find  that 
electrostatic  quantity,  Q  e.-s.  units,  is  numerically  expressible  as 
2=(Q/V)  magnetic  units;  potential-difference,  E  in  electrostatic 
measure,  as  e  =  (EV)  in  magnetic  ;  resistance,  R  electrostatic 
units,  as  r  =  (RV2)  magnetic  units;  capacity,  C,  as  (C/V2) 

*Resistance  a  Velocit  y.—  In  the  Tangent-Galvanometer,  p.  712,  and  by  defini- 
tion (ii.)  ,  p.  707,  h  =  fy  tan  e  =  il/rad.2  ;  .-.  i  =  fj  .  tan  0  •  rad.^/l.  By  definition,  p.  703, 
i  =  A  •  lo/rt,  where  r  is  the  resistance  ;  let  (Ab/i)  lines  of  induction  per  second  be 
cut  by  a  vertical  slider,  connecting  two  parallel  horizontal  rails  which  lie  East  and 
West,  one  vertically  above  the  other,  at  a  mutual  distance  of  d  cm.,  and  which,  with 
the  aid  of  the  slider,  form  part  of  a  circuit;  then,  in  order  to  cut  this  exact  number 
of  lines  per  second,  the  slider  must  travel  with  a  particular  mean  velocity  v.  The 
horizontal  component  of  intensity  of  the  terrestrial  magnetic  field  is  jjj  ;  the  number 
of  lines  of  induction  cut  per  second  is  thus  Ab/£  =  fy  •  v  •  d',  whence  i,  the  mean 
intensity  of  the  current  induced  in  the  circuit,  =  JjueZ/r.  Then  {j  •  v  •  d/r  =  i  = 
fy  -tan0  -rad.z/l;  whence  r  =  v-d'l/(rad.2t&n0).  Now  impose  two  conditions; 
first,  the  wire  coiled  in  the  galvanometer  is  to  be  of  length  I  =  rad.2/d;  and  second, 
the  velocity  is  to  be  such  as  to  produce  a  deflection  0  =  45°.  Then  r  =  v  ;  the  Resist- 
ance is,  in  this  case,  numerically  equal  to  the  Velocity  of  the  slider  ;  and  it  is  always 
some  merely  numerical  multiple  of  the  slider-velocity. 


xvi.]  MAGNETIC    MEASURE. 

magnetic  units  ;  conductivity  and  resistivity,  equal  to  D  and  R 
electrostatic  units,  as  respectively  equal  to  (DV2)  and  to  (R/V2) 
magnetic  units. 

Practical  Units.  —  Some  of  the  units  of  the  C.G.S.  Mag- 
netic System  are  inconveniently  large  or  small.  It  is  therefore 
the  practice  not  to  use  the  C.G.S.  magnetic  units  of  electrical 
quantity,  intensity,  resistance,  etc.,  but  to  build  up  a  magnetic 
system  based  on  new  units  of  length,  Z,  and  of  mass,  m.  These 
are  respectively  1000,000000  cm.  (the  earth's  quadrant)  and  the 
100,000,000000th  part  of  a  gramme.  The  unit  of  current- 
intensity  is  then  [w¥/T]  =  [(M/ 100,000,000000)*  .  (L  x 
1000,000000) */TJ  =  JQ  [M4L*/T].  The  new  unit  of  intensity, 
the  Ampere,  is  thus  equal  to  -fa  C.G.S.  Magnetic  Unit;  and 
the  new  unit  of  quantity,  the  Coulomb,  is  similarly  equal  to 
-^Q  C.G.S.  Magnetic  Unit.  In  the  same  way  we  find  the  unit 
of  resistance,  the  Ohm,  =  109  C.G.S.  Magnetic  Units.  The 
Megohm  =  1  million  Ohms ;  the  Microhm  =  one-millionth  Ohm. 
The  unit  of  difference  of  potential,,  the  Volt,  =  108  C.G.S. 
Magnetic  Units ;  the  Megavolt  =  1  million  Volts  ;  the  Micro- 
volt =  one-millionth  Volt.  The  unit  of  capacity  [T2//]  = 
[T2/ 1000,000000 L]  =  {[T2/L]  •*- 1000,000000|  =  10~9  C.G.S. 
Magnetic  Unit  =  1  Farad.  The  Farad  =  10~9  x  one  C.G.S. 
Magnetic  Unit  of  Capacity ;  but  the  latter  unit  is  equal  to  the 
electrostatic  unit  x  V2,  or  to  9  x  1020  Electrostatic  Units;  the 
Farad  is  therefore  equal  to  10~9  x  (9  x  1020)  =  (9  x  1011)  Elec- 
trostatic Units  of  Capacity.  The  electrostatic  capacity  of  a 
sphere  is  equal  to  its  radius  ;  a  Farad  is  therefore  the  electro- 
static capacity  of  a  sphere  of  (9  x  1011)  cm.  radius;  and  for 
convenience  the  standard  in  use  is  the  Microfarad,  the  millionth 
of  a  Farad.  The  coefficient  of  self-induction,  the  Henry  or 
Secohm  or  Quadrant,  is  [1000,OOOOOOL]=  109  Magnetic  C.G.S. 
units. 

In  this  system  the  quantity  i  •  n,  which  so  often  occurs  in  our  equations, 
is  known  as  the  Ampere-turns. 

The  heat  developed  in  a  wire,  per  second,  by  a  steady 
current  of  A  Amperes,  under  a  potential-difference  of  V  Volts, 
is  (V  x  108)  x  (A  x  10-1)  ergs  =  107-  VA  ergs  =  0-24  FJ.  ca. 

In  electric  lighting  a  certain  unit  is  commonly  made  use  of  as  a  con- 
ventional basis  for  estimating  the  sum  due  by  the  consumer.  This  unit 
represents  1000  A  rape  re- Volt-Hours,  and  is  equivalent  to  the  Energy  con- 
veyed by  a  current  of  one  Ampere  intensity,  passing  down  a  fall  of  poten- 
tial of  one  Volt,  and  sustained  for  1000  hours.  This  amount  of  Energy  = 


712  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

1  Ampere- Volt  or  Watt  x  3,600,000  sec.,  and  is  therefore  equal  to  {10,000000 
Ergs  per  sec.  x  3,600000  sec.}  =  36,000000,000000  Ergs  or  2,654,340  foot- 
pounds, or  about  865,000  ca,  an  amount  of  heat  which  would  convert 
2-95  Ibs.  of  ice-cold  water  into  steam  at  100°  C. ;  and  the  commercial  unit 
of  current  is  a  current  of  any  intensity  continued  until  this  quantity  of 
energy  has  been  transmitted  through  the  consumer's  apparatus. 

The  accompanying  table,  pp.  714,  715,  gives  a  conspectus 
of  the  relations  between  the  quantities  dealt  with  in  this  chap- 
ter, and  of  the  numerical  data  which  are  requisite  in  order  to 
transform  a  quantity,  numerically  stated  in  terms  of  one  of 
the  three  systems  of  conventional  units  described,  into  the  same 
quantity  numerically  stated  in  terms  of  either  of  the  other  two 
systems. 

Magnetic  Measurement  of  Current-Intensity.  —  The  mag- 
netic units  of  measurement  have  all  been  derived  in  theory  from 
the  magnetic  measurement  of  intensity  of  a  steady  current: 
the  magnetic  measurement  of  a  steady  current  is  therefore  a 
fundamental  measurement.  It  is  effected  by  the  use  of  galva- 
nometers and  electrodynamometers. 

A  magnet  surrounded  by  a  coil  of  wire  will,  when  a  current  is  passed 
through  the  wire,  tend  to  place  itself  at  right  angles  to  the  plane  of  the 
current,  that  is,  to  place  its  axis  along  the  Lines  of  Induction.  If  the  coil 
be  placed  vertically  in  the  plane  of  the  magnetic  meridian,  and  if  the  needle 
be  suspended  horizontally  at  its  centre,  so  that  it  can  swing  round  a  vertical 
axis  of  rotation,  then,  on  the  supposition  that  both  poles  of  the  magnet  are 
at  the  centre  of  the  coil  —  an  ideal  approximated  to  when  the  coil  has  an 
extremely  large  diameter,  or  when  the  needle  is  extremely  short  —  the 
deflection  of  the  needle  from  the  magnetic  meridian  is  such  that  its  tangent 
is  proportional  to  the  current  passing.  Such  an  arrangement  is  called  a 
Tangent-Galvanometer. 

In  a  tangent-galvanometer  in  which  the  coil  consists  of  only  one  turn, 
of  radius  r,  the  force  acting  upon  a  pole  tn  very  near  the  centre  is  F  =  hm 
=  mil/r2  =  mi  -  2irr/r2  =  mi  -  27r/r,  when  i  is  measured  in  C.G.S.  Magnetic 
Units,  and  where  h  is  the  force  acting  on  a  unit-pole.  The  mechanical 
force  exerted  by  the  horizontal  component  of  the  earth's  magnetism  is  fj 
on  a  unit-pole,  fym  on  a  pole  m.  The  deflection  of  the  needle  is  0.  The  mag- 
netic couple  is  ij  •  ml-  sin  0  if  the  moment  of  the  magnet  be  /•  m.  From  the 
"  Equilibrium  of  Couples,"  page  160,  prop.  2,  we  learn  that  F  :  fntt  : :  tan  0  :  1. 
Therefore  i  •  Zir/r  =  fj  tan  0,  or  i  =  fj  tan  0  -  r/2-Tr.  fj  can  be  found  as  on  page 
681,  or  turned  up  in  observational  tables  of  local  magnetic  intensities ;  6  can 
be  observed ;  r  can  be  measured ;  whence  { can  be  found  numerically  in  mag- 
netic C.G.S.  measure  :  and  it  will  be  observed  that  it  is  independent  of  vari- 
ations in  the  strength  m  of  either  pole  of  the  magnet.  It  is  also  the  same 
whatever  the  value  of  /x. 

If  the  coil  consist  of  n  turns,  whose  mean  radius  is  r',  the  force  h  acting 
on  unit-pole  (the  '  intensity  of  the  electromagnetic  field ')  at  the  centre  is 
i  -  n  -  27r/r/.  If  it  be  a  coil  of  rectangular  section  with  inner  and  outer  radii 
rt  and  rlfl  and  of  length  I  cm.,  with  n  turns  in  it  on  the  whole,  the  intensity 
of  field  is 


xvi.]  GALVANOMETERS.  713 


The  Tangent-Galvanometer  is  most  sensitive  when  0  is  45°. 

If  the  coil  be  placed  parallel  to  the  deflected  needle,  the  sine  of  the 
deflection  becomes  proportional  to  the  current-intensity  :  we  then  have  the 
Sine  Galvanometer,  in  which  a  long  needle  is  used.  (Compare  Figs.  83  a 
and  83  &.) 

In  Galvanometers,  in  which  a  passing  current  produces 
an  electromagnetic  field  in  which  a  magnetic  needle  is  deflected, 
the  amount  of  this  deflection  indicates  the  strength  of  the  cur- 
rent. It  is  well  in  all  cases  to  produce  as  uniform  a  field  of 
force  as  possible.  This  is  effected  by  arranging  a  number  of 
coils  so  as  to  surround  the  field,  not  wholly  but  in  outline.  In 
von  Helmholtz's  Galvanometer,  for  example,  there  are  two  parallel 
coils,  between  which  the  needle  is  placed,  at  a  mean  distance 
from  each  equal  to  half  the  mean  radius  of  either.  For  sensi- 
tiveness, each  winding  of  the  wire  is  made  to  come  as  near  the 
magnet  as  is  practicable. 

Galvanometer-Constant.  —  When  a  current  passes  through  the  wire 
of  a  galvanometer,  the  needle  is  in  a  magnetic  field  of  a  certain  intensity  or 
strength,  measured,  as  usual,  by  h,  the  force  locally  acting  upon  a  unit  mag- 
netic-pole. If  i  =  1,  h  has  a  certain  numerical  value  which  involves  only 
measurements  derived  from  the  construction  of  the  galvanometer  itself  :  this  is 
known  as  the  Galvanometer  -Constant,  and  gives  the  numerical  value 
of  the  strength  of  the  field  when  the  current  traversing  the  instrument  is  of 
unit  intensity.  It  is  distinctively  represented  by  the  symbol  F,  and  the 
force  h  acting  on  a  unit-pole,  when  the  intensity  of  the  current  is  i,  is  equal 
to  Ti  ;  acting  on  a  pole  tn,  the  mechanical  force  is  F  =  htn  =  Fmt-  For  exam- 
ple, in  a  tangent-galvanometer  of  one  turn  the  force  acting  is,  as  above, 
F  =  mi'2ir/r;  whence  F  =  2?r/r.  The  dimensions  of  F  are  [1/L]. 

Ballistic  Galvanometer.  —  If  a  tangent-galvanometer  be  constructed 
with  a  short  heavy  needle  of  length  I,  and  if  a  very  brief  current,  enduring- 
only  for  the  exceedingly  small  interval  8t,  be  passed  through  it,  the  needle 
will  receive  a  twitch  and,  after  the  current  has  passed,  will  swing  through 
an  angle  0.  The  last  equations  under  Ballistic  Pendulum  (p.  215)  were 
I  N<o2  =  (M  +  m)gh  •  2  sin2  0/2  and  <o  =  2  V(M  +  m)gh/N  •  sin  0/2.  The  prob- 
lem here  corresponds  closely,  but  instead  of  (M  +  m)h  we  have  the  magnetic 
moment  of  the  needle  (  =  /-m),  which  we  write  as  |H;  instead  of  g  the  local 
intensity  of  gravity  {i.e.  the  force  acting  upon  a  unit-mass),  we  have  ij  the 
effective  component  of  the  local  intensity  of  the  field  within  which  the  needle 
moves  after  the  current  has  passed,  —  that  is,  of  the  terrestrial  mag- 
netic field.  The  equations  of  that  paragraph  therefore  become  for  our  pres- 
ent purposes 


which  represents  the  energy  imparted  to  the  needle,  and 

o>  =  2V£Hij/X.sin072 
which  represents  the  angular  velocity  imparted  to  it. 


714 


TABLE   OF   ELECTRICAL 


DIMENSIONS   OF   UNITS. 

ELECTROSTATIC. 

MAGNETIC. 

Derivation. 

Dimensions. 

Derivation. 

Dimensions. 

1  Electric  Quantity,  Q,  q     ..... 

2  Electric  Surface-Density,  a  .     .     . 
3  Electric  Force  on  Unit  Quantity; 
Electromotive  Intensity  ;  Inten- 
sity of  Electrostatic  Field  ;  Im- 
pressed   Electromotive    Force  • 
Potential-Slope,  <}>,£.... 
4  Total  Electrostatic  Induction,  I 
5  Induction  per  sq.  cm.,  i    .     .     .     . 
6  Sp.  Ind  .  Capacity,  or  Permittivity,  K 
7  Electrostatic  Potential  V  ;   Differ- 
ence of  Potential,  E,  e      .     .     . 
8  Electrostatic   Capacity  or  Permit- 
tance, C     

VForce-cPK 
Q/Area 

Force/  Q 

47rQ 
47T(r 
47r<r/4> 

Work/Q 

Q/V 

Q/Time 
I/Area 
I/E 

V4» 

1/D 

1/D 

F=mil/d2 

K*M*L*/T 
K*M*/L*T 

M*/L*TK* 

K*M*L*/T 

K*MS/L*T 

K 
M*L*/TK* 

KL 
K*M*L*/T2 
K5M*/L*T2 
KL/T 
K/T 
T/LK 
T/K 

M*L4/K* 

RiMiLi/T2 
M*L*/K* 
M*/L*R* 
K*M*L*/T2 
M^/LfK^ 
M*/L*K* 

M5/L*K* 

M*L*/K* 
T2/L2R 
T2/L2K 

T2/LK 

TV^K 

K^M^LVT2 
KL/T2 

KLVT2 

i  x  Time 
*=9VV 

M'L^/Ai* 
M»/LM 

^M^L^/T2 
M^L*//x* 

M*/LV 
TVL2^ 

^MW/T2 

TYI^ 

M*L*/Tf** 

MVL^T/i* 

T/L/, 

T/L2^ 
ML/T 
A*L2/T 
M*M*L*/T 

M*/L*TAI* 

^M^/T 

/JMS/I^T 

M*L4/T/t* 
ffotf/IJT 

At*MiL*/T 

^M^/L^T 

^MJLf/T 
/x 
pi 

ML 

ML 
M*L*/T/MJ 

I/ML 

VA; 

9  Current-Intensity  or  Strength,  I,  i 
10  Current-Density   A                     . 

11  Conductance,  D  
12  Conductivity,  D  .     
13  Resistance,  K,  r      
14  Resistivity,  B      

15  Magnetic  Quantity,  m      •     •     •     . 
16  Magnetic  Force  ;  Mechanical  Force 
on  Unit  Quantity  ;  Intensity  of 
Magnetic  Field;  Potential-Slope; 
Number  of  Lines  of  Force  per 
SQ    cm  '  h 

VForce  •  ^  •  /* 

Force  /m 
m  x  Length 
/vol. 
Work/m 
tn/Area 
s  •  thickness 

47rm/Area 

47rtn 
i/fe 

e+M/dt 

e+di/dt 
4:Tri  •  n 
I/Aft 

(I/A^)>(I/A) 

17  Magnetic  Moment,  fSi      .... 
18  Intensity  of  Magnetisation,  K    .     . 
19  Magnetic  [scalar]  Potential,   ft      . 
20  Magnetic  Surface-Density,  s     .     . 
21  Strength  of  Magnetic  Shell,  <jp  .     . 
22  Magnetic   Induction  or  Flux  per 
so    cm    b 

23  Magnetic  Induction  or  Flux  within 
a  Contour,  B  . 

24  Magnetic  Susceptibility,  K    .     .     . 
25  Permeability  or  Inductivity,  /x,  . 
26  Coefficient    of    Self-induction,    or 
Inductance,  L 

27  Mutual  Induction,  Mutual  Induct- 
ance, M      

28  Magneto-  motive  Force  across  area 
29  Reluctance  of  Magnetic  Circuit     . 
30  Reluctivity   (Reluctance  per  unit 
volume)          .              . 

Example.  —  The  electrostatic  capacity  of  a  certain  small  Leyden  jar  is  found,  by  the  formula 
is  {(300/71-)-=- (9  xlO11)}  practical  magnetic  units  of  electrostatic  capacity,  or  l/3000,0000007r 


UNITS   AND   MEASURES. 


715 


REDUCTION-FACTORS,  to  transform 

1 
2 

CONVENTIONAL. 

E.8.C.G.S.  to 

M.C.G.S.  to 

Practical  Magnetic  to 

E.S. 

Magn. 

M.C.G.S. 

P.M. 

E.8.C.G.8. 

P.M. 

E.8.C.G.8. 

M.C.G.S 

Ji*L*/t 

M*/L» 

•*•  (3  x  101°) 

+-  (3  x  109) 
-H  (3  x  10~9) 

x  (3  x  101°) 
X  (3x101°) 

xlO 
xlO19 

Coulombs,  x  (3  x  109) 
x(3xlO~9) 

-10 

M*L*/T 

M*/L*T 
(No.) 

T2/L2 

x  (3x101°) 
-  (3  x  101°) 
-  (3  x  101°) 
-  (9  x  1020) 

-(3  xlO9) 
-(3  xlO-9) 
-K9  x  10*) 

-  (3  x  101°) 
x  (3  x  101°) 
x  (3  x  101°) 
x(9x!020) 

xlO 
xlO 
xlOi9 

x  (3  x  109) 
x  (3  x  10-9) 
x(9x!02) 

-10 
-10 

-1019 
-lOis 

:) 

4 

5 
6 

M»L*/T 

MM/T2 

x  (3  x  101°) 

x  (3  x  102) 

-  (3  x  101°) 

-108 

Volts,  -(3xl02) 

xlO8 

7 

L 
M*L*/T* 

L/T 
1/T 
T/L 
T 

M*L^ 

T2/L 

T/L 
T/L2 
L/T 
L2/T 
M*L*/T 

-  (9  x  1020) 
-  (3  x  101°) 
-  (3  x  IQio) 
-  (9  x  1020) 
-  (9  x  10*>) 
x  (9  x  1020) 
x  (9  x  1020) 
x  (3  x  101°) 

-4-  (3  x  109) 
-  (3  x  10-9) 
-  (9  x  ion) 
-  (9  x  102) 
x  (9xlOU) 
x  (9  x  102) 
x  (3  x  102) 

x  (9  x  1020) 
x  (3  x  101°) 
x  (3  x  101°) 
x  (9  x  10») 

x  (9  x  1020) 
-(9x1020) 

-  (3  x  101°) 

xlO9 
xlO 
xlOi9 
xlO9 
xlOi8 
-109 
-lOi8 
-108 

Farads,  x(9xlOH) 
Amperes,  x  (3  x  109) 
x  (3  x  10~9) 
Mhos,  x(9xlOU) 
x(9x!02) 
Ohms,  -(9xlOU) 
-(9  xlO2) 
H-(3xl02) 

-id9 

-10 
-10i9 
-109 

xlO9 
xlO8 

8 
9 
10 
11 

18 

14 

15 

M*L*/T2 

M*/Lf 
lf*L*/T» 

pt/L* 

M*/L* 

M*/L*T 
M*/L*T 
M*/L*T 

-  (3  x  101°) 
x  (3  x  101°) 
x  (3x101°) 
-(3x101°) 
x  (3  x  101°) 
x  (3  x  101°) 

-3 

x(3xlO-7) 
x(3x!023) 
-  (3  x  109) 
x  (3  x  102)) 
x  (3  x  ion) 

x  (3  x  101°) 
-(3xlOio) 
-  (3  x  IQio) 
x  (3  x  10i°) 
-  (3  x  101°) 
-(3  xlO10) 

xlOio 

xlOio 
xlO 
xlOio 
xlO 

Gausses,  x3 
-  (3  x  10~7) 
^  (3  x  1020) 
x  (3  x  109) 
-(SxlO20) 
^-(3xlOU) 

xlO" 
-10 
-10 

in 
17 
18 
19 
20 
21 

MVL* 

M*/L*T 

x  (3xlOio) 

x(SxlO-) 

-  (3  x  101°) 

xlOio 

-  (3  x  1020) 

-101° 

22 

Hit* 

T2/L2 
T2/L2 

M*L?/T 

(No.) 
(No.) 

x  (3  x  101°) 
x  (9  x  1020) 
x  (9  x  1020) 

x  (3  x  102) 
x  (9  x  1020) 

-  (3  x  lOio) 
-  (9  x  1020) 
-(9x1020) 

-108 
Same 
Same 

Webers,  -(3xl02) 
-(9x1020) 
H-(9xl020) 

xlO8 
Same 
Same 

L>3 

24 
26 

T2/L 

L 

x  (9  x  1020) 

x  (9xlOH) 

-(OxIO20) 

-109 

Henries,  -(9x10") 

xlO9 

26 

T2/L 
L/T2 

L 
M*L*/T 
1/L 

x  (9  x  1020) 
-  (3  x  lOio) 
-  (9  X  1020) 

x  (9  x  1011) 
-  (3  x  10°) 
-  (9  x  10") 

-  (9  x  1020) 
x  (3  x  101°) 
x  (9  x  1020) 

-109 
xlO 
XlO9 

+  (9  x  ion) 
Gilberts,  x(3x!09) 
Oersteds,  x  (9  x  1011) 

xlO9 
-10 
-f-109 

27 
28 

L2/T2 

(No.) 

-  (9  x  1020) 

-  (9  x  1020) 

X  (9  X  1020) 

Same 

x  (9-x  1020) 

Same 

80 

C  =(K/<Z)  •  (surface/47r),  to  be  300/Tr  C.G.S. 
Farad,  or  1/9425  microfarad.     See  note,  p.  635. 


electrostatic  units;  this,  by  the  above  table, 


716  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

But  again,  we  can  in  other  terms  express  the  work  done  on  the 
needle  during  the  brief  period  of  the  current ;  as  the  product,  namely,  of 
half  the  twisting  moment  into  the  angle  of  twist  imparted  during  that 
period  (p.  263).  The  twisting  moment  is  Tim  x  I;  the  angle  of  twist  is  an 
exceedingly  small  angle  £  =  Qw  x  &)>  where  w  is  the  angular  velocity  imparted 
to  the  needle,  and  Bt  the  time  during  which  the  current  lasts.  The  work 
done  is  therefore  %Tim  - 1  -  $ti>8t  =  ^T-ml-iBt-^  =  ?r|BQ  •  i  <o,  where  Q  is  the 
whole  quantity  of  electricity  passing  in  time  Bt.  This  is  equal  to  AN"u>2. 

Now  equating  these  two  values  of  |Nco2  the  energy,  we  have  ^FfHQ  •  to 
=  ffilfr.2siii2(0/2);  or  Q  =  4fr/r.2  sin2  ((9/2).  /o>  =  4fr/r«2  sin2  (0/2)  -s- 
[2>/iflfHj/N".siii  (0/2)]  =  4ij/r-  VN/jmij.sin0/2.  But  T,  the  time  of  a  com- 
plete to-and-fro  oscillation  of  a  needle  swinging  freely  in  the  terrestrial  field,  is 
T  =  2uVN/jafj;  whence  Q  =  2Jj/F  •  T/TT-  sin  (0/2),  in  which  aU  the  terms 
are  measurable. 

When  the  current  ceases,  the  magnet  tends  to  oscillate  for  some  time, 
like  a  pendulum ;  but  if  it  oscillate  in  a  strong  magnetic  field  of  force  —  as, 
for  example,  in  the  neighbourhood  of  a  strong  magnet  —  its  oscillations  will 
be  very  rapid  and  of  small  amplitude.  If  masses  of  metal  be  so  arranged 
that  any  oscillations  of  the  magnet  tend  to  produce  retarding  induced-cur- 
rents in  these  masses,  then,  especially  if  the  needle  be  light,  the  oscillations 
of  the  magnet  rapidly  cease,  as  if  it  were  immersed  in  a  viscous  medium, 
and  the  magnet  is,  without  further  oscillation,  restored  to  its  position  of 
repose.  A  galvanometer  arranged  on  this  principle  is  a  Dead-Beat  gal- 
vanometer. The  same  dead-beat  effect  is  mechanically  produced  by  making 
the  magnet  move  in  a  small  closed  chamber  of  air  which  it  nearly  fills ;  it 
thus  moves  against  air-resistance. 

In  Differential  Galvanometers,  two  equal  and  separate  wires  are 
similarly  coiled  round  the  same  needle ;  through  these  wires  currents  may 
be  sent  in  opposite  directions ;  if  the  two  currents  be  equal,  the  needle 
remains  at  rest ;  if  either  predominate,  the  needle  moves. 

In  Electrodynamometers  the  current  is  passed  through 
a  coil  which  is  suspended  within  a  strong  and  uniform  magnetic 
field,  such  as  that  produced  by  powerful  electromagnets  actuated 
by  a  second  current,  or  again  by  fixed  coils  through  which  a 
second  current  is  passing.  The  deflection  of  the  suspended  coil 
depends  upon  the  strength  of  the  current  passing  through  it, 
and  also  upon  the  strength  of  the  magnetic  field  surrounding  it. 
If  the  same  current  traverse  both  the  fixed  and  the  suspended 
coils,  the  rotating  couple  is  proportional  to  z2,  and  therefore  to 
the  energy  of  the  current ;  and  it  is  independent  of  its  direction. 

The  two  coils  have  the  respective  mean  radii  r  and  r,,  r  the  greater, 
r,  the  less ;  the  respective  numbers  of  turns  are  n  and  nt ;  when  the  plane 
of  the  suspended  coil  makes  with  the  plane  of  the  larger  coil  an  angle  0, 
the  couple,  tending  to  bring  the  two  coils  into  the  same  plane  with  their 
currents  opposed,  is  />u2  •  2?r2  •  r,2  •  rm^sin  0.  /r.  If  I  and  I,  be  the  lengths 
of  wire  in  the  two  coils  respectively,  this  expression  may  be  written  as 
^u'2. //,./- sin  0./r2. 

If  the  current  to  be  measured  be  a  rapidly  alternating  one,  the  result  is 
the  same  as  if  it  were  constant ;  it  is  reversed  in  both  coils  at  the  same  time, 
and  the  algebraic  sign  of  i2  is  always  positive. 


xvi.]  ELECTRODYNAMOMETERS.  717 

In  the  Siemens  electrodynamometer,  used  for  measuring  electric- 
lighting  currents,  the  same  current  is  made  to  pass  in  succession  through 
two  thick-wire  loops.  These  tend  to  place  themselves  in  the  same  plane ; 
but  by  the  torsion  of  a  spring,  they  are  forced  into  a  standard  position  at 
right  angles  to  one  another.  This  torsion  is  measured  by  the  angle  of 
rotation  of  a  pointer  connected  with  the  spring;  and  it  is  proportional 
to  the  square  of  the  intensity  of  the  current. 

If  a  movable  coil  be  free  to  slip  up  and  down  the  axis  of  a 
fixed  coil  in  which  a  current  is  passing,  the  inner  coil  may  be 
sucked  in  or  repelled  with  a  force  which  may  be  balanced  and 
measured  by  known  weights  or  elastic  tensions  or  torsions ;  or 
if  the  current  in  the  suspended  coil  be  variable,  the  tension 
tending  to  draw  it  in  to  the  fixed  coil  may  be  made  to  act 
against  a  spring,  and  graphically  to  record  its  own  variations 
upon  a  uniformly-moving  piece  of  paper. 

If,  instead  of  a  movable  coil  free  to  slip  up  and  down  the  axis  of  a 
fixed  coil,  we  have  a  bar  of  soft  iron,  it  will  also  be  sucked  in  or  repelled ; 
but  in  this  case  the  magnetisation  of  the  bar  is  not  strictly  proportional  to 
the  intensity  of  the  current :  whence  the  law,  that  the  force  of  suction  or 
of  repulsion  is  proportional  to  the  square  of  the  current-intensity,  fails  us. 
If,  however,  the  bar  be  reduced  to  a  very  thin  soft-iron  tube,  it  rapidly 
becomes  saturated  and  soon  becomes  practically  constant  in  strength.  When 
this  limit  has  been  reached,  the  force  is  directly  proportional  to  the  intensity 
of  the  current.  This  is  the  principle  of  Ayrton  and  Perry's  Ammeter 
(  =  "  Ampere-meter  ").  A  slender  piece  of  soft  iron  tends  to  move  towards 
the  centre  of  a  current-bearing  coil;  it  may  thus  be  made  to  rotate  against 
gravity  round  a  fixed  pivot:  a  pointer  attached  to  it  will  indicate  the 
amount  of  rotation:  and  this  is  the  principle  ofSchuckert's  Ammeter. 
A  short  thin  bar  of  soft  iron  tends  to  be  pulled  so  as  to  lie  in  the  strongest 
part  of  the  field  between  two  electromagnet-poles  (Ever shed's  Amme- 
ter). The  tendency  towards  suction  of  a  suitable  electromagnet  into  a  coil 
can  also  be  measured  by  balancing  it  against  weights,  as  in  the  Electric 
Power  Storage  Co.'s  Steelyard  Ammeter.  Ampere-meters,  as  their  name 
indicates,  are  graduated  in  Amperes,  not  in  magnetic  C.G.S.  units. 

The  principle  of  the  differential  galvanometer  may  be  here  applied,  as 
in  Prof .  Langley's  Thermic  Balance  or  Bolometer.  The  suspended  coil 
is  composed  of  two  separate  wires  wound  together,  but  insulated  from  one 
another :  a  single  current  is  divided  into  two  equal  moieties  which  run  in 
opposite  directions  through  the  two  wires  of  the  coil;  there  is  no  effect. 
The  least  variation  in  one  of  these  moieties,  as  when  the  conductivity  of  its 
path  is  affected  by  the  local  application  of  heat,  causes  imperfect  compensa- 
tion, and  practically  a  small  uncompensated  current  passes :  however  feeble 
this  may  be,  it  can  be  rendered  manifest  and  measurable  by  increasing  the 
strength  of  the  magnetic  field  within  which  the  double  coil  is  suspended. 

The  part  of  the  divided  circuit  to  which  heat  may  be  locally  applied 
may  be  an  exceedingly  thin  strip  of  platinum.  This  may  be  moved  up  and 
down,  say,  in  the  dark  region  of  the  spectrum.  In  some  places  it  is  heated, 
in  others  —  dark  lines  —  it  is  not.  By  thus  groping  in  the  dark  it  discovers 


718  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

the  dark  lines  and  the  specially  "  bright "  lines  of  the  heat-spectrum.  An 
instrument  of  this  kind  is  sensitive  to  differences  of  temperature  of  TT^TFTF  F°- 

Magnetic  Measurement  of  Resistance.  —  If  a  circle  of  wire,  radius 
r,  stand  at  right  angles  to  the  magnetic  meridian  it  will  embrace  /xfy  lines 
of  terrestrial  magnetic  induction  (yx  =  1)  per  sq.  crn.  or  yuij  •  rrr2  lines  over  its 
whole  area;  if  it  be  turned  round  a  vertical  axis  through  180°,  it  will  come 
to  embrace  /xij  •  -jrr2  lines  oppositely  directed  with  reference  to  it :  the  number 
of  lines  of  induction  passing  through  it  has  therefore  been  increased  or 
decreased  by  2/xfj  •  Trr2.  The  circle  of  wire  thus  rotated  (E  a  r  t  h  -  i  n  d  u  c  t  o  r) 
becomes  the  seat  of  a  current  whose  mean  E.M.D.P.  is  numerically  equal 
to  2/xfy  •  7rr'2/t  in  magnetic  units,  and  whose  mean  intensity  is  i  =  2/xfj  •  7rr2/R£, 
where  R  is  the  resistance  (also  measured  in  magnetic  units)  and  t  the  time 
occupied  in  the  rotation  through  180°.  If  a  small  needle  be  suspended  at 
the  centre  of  this  rotating  circle,  that  needle  will  be  deflected ;  the  rotating 
circle  acts  somewhat  as  if  it  were  its  own  Tangent-galvanometer;  but  instead 
of  a  deflection  0  such  that  tan  0  =  i-27rn/rfy,  where  n  is  the  number  of  coils 
and  r  their  mean  radius,  we  have  (approximately)  tan  20  =  i-ir2-  n2/rty. 
But  i  =  2/4 •  7rr*/Rt ;  whence  tan  20  =  2/x  •  ^n^r/Rt.  If  t  be  the  2Nth  part 
of  a  second,  the  coil  will  make  N  complete  turns  per  second,  and  tan  20 
=  fyNTr^V/R.  Therefore  R,  =  4/xN7r8n2>-/tan  20.  Of  these  quantities,  n 
the  number  of  coils  and  r  their  mean  radius  are  obtained  by  measurement ; 
tan  20  is  the  ratio  between '  the  scale-reading  (straight  scale)  and  the  dis- 
tance of  the  scale  from  the  mirror  fixed  to  the  centre  of  the  deflecting 
needle ;  N  can  be  read  off  on  a  speed-indicator ;  and  /z  =  1  in  air.  When 
the  resistance  is  so  adjusted  that  to  a  speed  N  there  corresponds  a  deflection 
0  such  that  the  product  above  (with  due  corrections)  is  numerically  equal 
to  1000,000000,  the  resistance  employed  is  equal  to  one  Ohm.  This  is  the 
principle  of  the  method  by  which  the  British  Association  Committee  on 
Electrical  Standards  constructed  the  original  standard  Ohm. 

Measurement  of  the  Capacity  of  a  Conductor  or  Condenser. — 
The  capacity  is  C  =  Q/V,  and  therefore  we  can  find  the  value  of  C  if  we 
find,  in  terms  of  units  of  the  same  system,  the  quantity  Q  with  which  a 
body  is  charged,  and  the  potential  V  to  which  this  charge  raises  it.  This 
potential  V  may  be  the  difference  between  the  potential  of  the  body  charged 
and  that  of  the  earth,  or  it  may  be  the  difference  between  the  potentials  of 
the  opposed  plates  of  a  condenser. 

This  is,  however,  not  a  convenient  method ;  and  the  practical  method 
is  first  to  construct  standard  condensers  of  known  capacity,  and  then  to 
compare  the  capacity  of  the  body  examined  with  that  of  these  standards. 

Standard  Condensers.  —  Suppose  a  conductor  of  capacity  C  and 
bearing  a  charge  Q  to  be  discharged  through  a  known  resistance  which 
includes  a  galvanometer ;  the  resistance  being  so  considerable  that  the  dis- 
charge is  far  from  instantaneous.  The  initial  potential  of  the  conductor  is 
V  =  Q/C.  A  current  will  pass  through  the  galvanometer,  but  will  con- 
tinuously diminish  in  intensity.  At  the  end  of  time  t  let  the  potential  have 
sunk  to  Vy  which  is  the  nth  part  of  V,  and  the  charge  to  Q, ;  and  at  the 
end  of  a  very  small  further  interval  &,  let  these  have  farther  sunk  by  the 
amounts  Vy  and  Q,  respectively.  Then  the  quantity  which  has  escaped  in 
time  &  is  Q,,  which  is  necessarily  equal  to  C  V, ;  it  is  also  equal  to  the  instan- 
taneous intensity  I  multiplied  by  the  time  §/,  and  this  product  is  equal,  by 
Ohm's  law,  to  (Vy/R)  x  &.  Hence  (V,/R)  •  &  =  CV,,  or  &  =  CR-  V,/Vy. 
From  this  we  find,  by  means  of  the  Integral  Calculus,  that  the  time  which 


XVI.] 


MEASUREMENT   OF   CAPACITY. 


719 


must  have  elapsed  between  the  initial  instant  at  which  the  potential  had 
been  V  and  that  instant  at  which  the  potential  had  sunk  to  V,  is  equal  to 
CR  log  (V/V,).  But  by  our  supposition  this  time  is  t,  and  the  ratio  V/Vy 
is  equal  to  n.  Whence  t  =  CR  log  n,  or  C  =  t/R  •  log  n.  If  we  observe  the 
successive  values  of  the  current-intensity  at  equal  intervals  of  time  we  can 
find  the  value  of  n,  and  then,  knowing  the  value  of  R  in  electrostatic,  in 
magnetic,  or  in  practical  units,  we  can  find  the  corresponding  value  of  C, 
the  capacity,  measured  in  units  of  the  corresponding  system. 

Comparison  of  Capacities.  —  1.  De  Sauty's  Bridge-method,  appli- 
cable to  small  capacities.  Two  condensers  of  capacities  C,  and  C7/,  if 
charged  to  the  same  potential  V,  must  be  charged  with  the  respective 
quantities  C/V  and  Cy/V.  If  one  of  these  be  discharged  through  a  resist- 
ance R;  for  time  &,  the  current  which  is  set  up  is  of  mean  intensity  I,,  and 
the  quantity  passing  in  time  &  is  I,  •  &.  But  this  is  equal  to  the  fall  in  the 
value  of  Q  during  the  time  &;  that  is,  to  Q.  But  Q  =  C/V",,  where  V, 
is  the  fall  of  potential  in  the  condenser  Cy.  Whence  I,  =  C/^/&;  and  R,, 


which  varies  inversely  as  I/5  is  proportional  to 


Similarly,  if  C/y  be 


discharged  by  a  current  of  intensity  Iy/  through  a  resistance  R/y,  that  resist- 
ance must  bear  the  same  proportion  to  &/€,/$"„  ;  and  if  Vy/,  the  fall  of 
potential  in  C/y,  be  the  same  as  in  condenser  C,,  that  is,  if  V/y  =  V,  ;  then  the 
equation 


shows  that  the  Resistances  through  which  the  two  condensers  charged  to 
equal  potentials  must  be  discharged,  in  order  that  the  potentials  of  the  two 
condensers  may  fall  concurrently  and  remain  persistently  equal  to  one 
another,  must  be  inversely  proportional  to  the  respective  Capacities  of  the 
bodies  discharged  through  them. 

This  being  postulated,  the  arrangement  of  the  apparatus  is  indicated  by 
Fig.  254.  A,  a  battery;  K,  a  key  with  which  the  wire  B  may  be  at  will  con- 
nected either  with  the  battery  A  or  directly  with  the  earth,  or  else,  as  in  the 
figure,  isolated  from  these.  When 
the  battery  is  connected  with  B,  the 
condensers  Cy  and  C7/  are  charged 
through  the  resistances  R,  and  R/y. 
Connect,  then,  A  with  B  for  a  cer- 
tain time;  disconnect.  The  two 
condensers,  if  not  already  at  equal  K" 
potentials,  soon  become  so,  for  an 
equalising  current  traverses  EGF; 
when  equalisation  is  complete,  the 
needle  of  the  galvanometer  G  re- 
turns to  rest.  Now  put  B  to  earth. 
The  charges  of  C,  and  C7/  escape 
through  R,  and  R/y  respectively.  If 
the  resistance  R/  be  disproportion- 
ately great,  the  outflow  through  it 
is  disproportionately  small,  and  the  potential  at  F  sinks  faster  than  that  at 
E  ;  a  current  therefore  passes  from  E  to  F,  and  the  galvanometer-needle  in 
G  is  deflected.  If  R,  :  R,,  :  :  C/y  :  Cy,  the  galvanometer-needle  regains  at  rest, 
for  the  potentials  at  E  and  F  as  they  sink,  sink  together  and  are  concur- 


EARTH 


EARTH 


720 


ELECTRICITY  AND   MAGNETISM. 


[CHAP. 


rently  equal  to  one  another.  If  therefore  we  adjust  the  resistances  Ry  and  Ryy 
until  we  find  that  on  effecting  the  three  operations  —  (1)  connecting  B  with 
the  battery  A ;  (2)  isolating  B  from  A  until  the  galvanometer-needle  comes 
to  rest ;  (3)  putting  B  to  earth  —  the  last  of  these  is  followed  by  no  deflec- 
tion of  the  galvanometer-needle,  then,  since  we  know  the  relative  values 
of  the  resistances  Ry  and  R//?  we  know,  inversely,  the  relative  values  of 
the  capacities  C/7  and  C,;  and  as  one  of  these  capacities  is  a  standard,  we 
are  thus  enabled  to  state  absolutely  the  actual  value  of  the  capacity  to  be 
measured. 

2.   Compensation-method.     If  a  wire  Cu  Zn  (Fig.  255),  connecting  two 
poles  of  a  battery,  be  connected  at  any  one  point  with  the  earth,  the  potential 


Fig. 255. 
O  WIRE 


Zn, 


Cu. 


EARTH  t» 

of  that  point  must  become  equal  to  zero ;  but  the  difference  of  potential  be- 
tween the  extremities  of  the  wire  remains  unaffected.  The  positive  potential 
at  Cu  (Fig.  255)  bears  to  the  negative  potential  at  Zn  a  numerical  ratio,  the 
same  as  that  between  CuO  and  OZn ;  for  obviously  CuA  :  ZnB  : :  CuO  :  OZn. 
Let  now  between  the  points  Cu  and  O  a  resistance  of  reduced  length  R; 
be  placed,  and  between  the  points  O  and  Zn  a  resistance  of  reduced  length 
Ry/.  The  potential  at  a  point  just  between  Ry  and  Cu,  and  the  potential  at 
a  point  just  between  R/y  and  Zn,  bear  to  one  another  the  ratio  of  Ry :  R/7.  If 
these  potentials  be  +  Vy  and  —  V,,  respectively,  we  have  Vy :  Vy/ : :  Ry :  R/y. 
Let  these  two  points,  at  potentials  Vy  and  —  V/y  respectively,  be  connected 
with  two  condensers  of  which  the  one  has  standard  capacity  Cy,  the  other 
the  capacity  C/y  to  be  determined.  The  two  condensers  will  become  charged 
to  the  respective  potentials  V,  and  Vy/;  but  the  aim  is  so  to  adjust  the  poten- 


Fig.  266. 


+X  Viff  w&m  tials  that  these  condensers 

shall  become  charged  with 
equal  but  opposite  quan- 
tities of  electricity.  Sup- 
pose this  adjustment  to 
have  been  effected.  Then 
Fig.  256  illustrates  the 
successive  operations. 

(i.)  Connect  at  D 
and  E.  The  condensers 
become  charged  to  po- 
tentials +  V,  and  -  V/y 
respectively.  They  are 
therefore  charged  with 
quantities  +  CyVy  and 
—  C//V//.  Disconnect  at 
D  and  E. 

(ii.)  Connect  at  F. 
The  two  charges  +  C,V, 
and  —  Cy/Vy/  blend,  and  there  remains  in  the  conjoined  condensers  a  resid- 
ual charge  of  (CyVy  -  Cy/V/y),  which,  if  CyVy  =  Cy/Vy/,  is  equal  to  zero. 


EARTH 


EARTH 


xvi.]  MEASUREMENT   OF   CAPACITY.  721 

(iii.)  Connect  at  H.  The  residual  charge,  if  any,  runs  to  earth  and 
deflects  the  needle  of  the  galvanometer;  if  none,  there  is  no  deflection. 
There  is  no  deflection  when  C,V,  =  C,,V/y.  But  V, :  V;/ : :  R, :  R/y.  There- 
fore, when  there  is  no  deflection  on  making  contact  at  H,  C/R/  =  C^R,,;  and 
the  capacities  C,  and  Cy/  are  inversely  as  the  corresponding  resistances. 

Adjust  therefore  the  resistances  Ry  and  Ry/  until  there  comes  to  be  no  de- 
flection of  the  galvanometer-needle  after  operation  (iii.),  and  from  the  known 
ratio  of  the  resistances  we  find  that  of  the  capacities,  for  R//R//  =  Cy//Cy. 


OSCILLATING  OR  ALTERNATING  CURRENTS. 

If  the  potential,  which  tends  to  give  rise  to  a  current,  itself 
fluctuate  between  positive  and  negative  values,  its  variation 
being  simple-harmonic,  we  have  an  oscillating  or  alternating 
current  produced  in  the  circuit.  Oscillating  currents  differ 
in  some  important  respects  from  the  steady  currents  hitherto 
discussed.  If  their  frequency  be  small,  they  approximate  to 
steady  currents  in  their  character,  and  merely  fluctuate  in  their 
intensity  and  direction :  if  it  be  great,  they  present,  as  it  were, 
nothing  but  the  Variable  Period,  and  never  arrive  at  the  Steady 
State.  We  shall  mention  the  main  characteristics  of  those  cur- 
rents in  which  the  frequency  is  great,  say  a  million  oscillations 
per  second.  How  such  currents  are  produced  we  shall  learn 
later. 

The  phenomena  of  the  current  tend,  as  the  frequency 
increases,  to  be  entirely  confined  to  the  dielectric,  so 
that  only  a  thin  skin  of  the  wire  is  concerned  in  the  cur- 
rent :  but  the  less  the  magnetic  permeability  of  the  conducting 
wire,  the  thicker  is  that  conducting  skin ;  and  the  smaller  the 
frequency  of  alternation,  again  the  thicker  is  that  skin.  Even 
when  the  frequency  is  as  low  as  100  per  second,  the  skin,  in  the 
case  of  iron,  is  practically  not  more  than  0*3  cm.  thick,  while 
with  a  Leyden-jar  discharge  it  is  less  than  0-001  cm.  in  thickness. 
In  the  dielectric  there  are  Waves  of  Propagation  of  Lines  of 
Force ;  these  lines  travel  back  and  fore  with  the  velocity  of  Light, 
with  their  ends  on  the  conducting  wires,  and  approximately  at 
right  angles  to  these.'  In  the  dielectric  or  the  electromagnetic 
field  surrounding  the  wire,  the  lines  of  magnetic  induction  have 
the  same  direction  as  in  the  case  of  a  steady  current,  but  the 
field-intensity  fades  away  very  rapidly  as  the  wire  is  receded 
from.  The  Electrostatic  Attraction  between  the  two  sides  of  the 
circuit  tends,  as  the  frequency  increases,  towards  equality  with 
the  Electromagnetic  Repulsion  between  them.  The  Transmis- 
sion of  Energy  through  the  dielectric  is  approximately  parallel 

3  A 


722  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

to  the  wire.  The  Apparent  Resistance,  or  Impedance,  r,  of  the 
wire  tends  towards  a  value  inversely  proportional  to  the  cir- 
cumference, instead  of  to  the  cross-section  of  the  wire ;  and  it 
is  greater  the  greater  the  magnetic  permeability  of  that  wire, 
for  the  conducting  skin  is  in  that  case  all  the  thinner.  As  a 
wave  of  potential  passes  along  the  wire,  its  amplitude  wanes ; 
and  the  effect  of  this  is  more  marked  the  greater  the  frequency ; 
so  that  after  passing  over  a  long  wire  a  complex  harmonic  dis- 
turbance may  have  its  higher  harmonics  considerably  more  atten- 
uated than  the  lower  components.  Besides  this,  disturbances  of 
different  wave-lengths  produce  their  maximum  effect,  at  a  given 
distant  point,  at  different  times. 

The  conducting  skin  screens  the  interior  of  the  wire  from  the  inductions 
which  would  otherwise  be  set  up  in  it  by  the  alternating  field ;  and  generally, 
a  film  of  metal  acts  as  a  screen,  or  is  opaque,  to  the  alternating  condition  of 
the  field,  while  an  insulator  is  not.  An  electrolyte  is  not  quite  opaque 
in  this  sense,  but  has  a  high  absorptive  coefficient,  and  a  thickness  of  some 
millimetres  acts  like  a  film  of  metal. 

Suppose  that  the  E.M.D.P.  is  itself  subject  to  variation  which  follows 
the  S.H.M.  law;  that  is,  with  a  frequency  n  times  per  second,  it  oscillates 
between  the  extremes  +  e  and  —  e,  and  is  at  any  given  time  t  equal  (assum- 
ing it  to  be  zero  at  the  initial  instant  when  t  =  0)  to  e-sin  ^imt.  Then, 
instead  of  e  in  equation  i.,  p.  705,  we  have  (e  -  sin  27rnt)  ;  and  that  equation, 
when  so  modified,  upon  being  integrated,  gives  it  =  (e/  Vr'2  +  4LV27r2) ; 
sin  (lirnt  —  tan"1  27rnL/r),  a  S.H.  function  :  that  is,  the  intensity  of  the  cur- 
rent varies  as  time  goes  on,  according  to  the  simple  harmonic  law,  and  its 
maximum  alternating  value  is  ±e/^/r2  +  4L2nV2.  Observe  from  this  expres- 
sion that  if  the  coefficient  L  be  comparatively  great,  as  in  a  coil  of  wire, 
the  maximum  values  of  the  current  may  fall  considerably  short  of  the  value 
e/r  ultimately  attained  by  a  steady  current,  but  will  tend  to  be  little  affected 
by  changes  in  the  resistance  of  the  circuit ;  and  that  as  the  frequency  n  of 
oscillation  increases,  the  maximum  intensities  fall  off.  The  apparent  resist- 
ance, or  Impedance,  is  thus  vV2  -f-  4LW  =  r.  Hence,  when  an  alter- 
nating current  of  high  frequency  is  sent  through  a  coil  of  many  turns  round 
a  soft-iron  core,  the  impedance  may  have  a  very  high  value,  and  the  current 
round  the  coil  may  be  reduced  to  a  minimum.  The  current  may  thus  be 
"throttled,"  or  choked  down.  Observe  also  that  in  the  equation  arrived 
at  above,  the  S.H.  variation  of  the  current-intensity  does  not  keep  step  with 
the  variations  of  the  E.M.D.P.,  as  it  would  do  if  the  last  term  were  simply 
sin27rw£;  there  is  a  lag;  the  variations  of  current  fall  behind  those  of  E.M.D.P. 
by  an  amount  represented  by  an  angle  whose  tangent  is  ZirnL/r.  This 
Lag  is  greater  the  greater  the  frequency  n  of  the  variations  of  the  applied 
E.M.D.P.,  or  the  greater  the  self-induction  L  of  the  circuit  or  coil,  or  the  less 
its  resistance  r;  but  the  angle  cannot  exceed  90° ;  hence  the  zero  current  is 
delayed  after  the  zero  E.M.D.P.,  by  an  interval  of  time  corresponding  to 
not  more  than  a  quarter  of  a  revolution  in  the  circle  of  reference  (see 
S.H.M.,  Fig.  29) ;  that  is,  it  is  in  arrear  by  an  interval  of  time  not  greater 
than  a  quarter  period,  T/4,  or  l/4n. 


xvi.]  ALTERNATING   CURRENTS.  723 

Since,  during  any  S.H.  variation  of  potential  or  of  current-strength,  etc., 
the  average  value,  taken  throughout  the  whole  of  a  positive  or  a  negative 
phase,  is  2/?r  x  the  maximum  value  (see  p.  85),  the  actual  mean  intensity  of 
an  alternating  current  is  2/7r  or  0-6368  times  the  maximum  intensity. 

Alternating  currents  produce  Heat,  and  therefore  also  Light, 
as  in  incandescent  and  arc  lamps;  and  they  can  light  up  Geissler- 
tubes. 

A  Cardew's  Ammeter,  or  an  Electrodynamometer,  produces  Heat,  or  a 
Torque,  proportional  to  the  square  of  the  current  at  any  instant;  such 
instruments  therefore  indicate  the  mean  value  of  i2,  not  of  i;  and  in  S.H. 
variations  of  i,  the  mean  value  of  i2  is  half  the  square  of  the  maximum  inten- 
sity ;  whence  the  mean  jalue  of  «,  as  given  by  such  instruments,  is  the 
maximum  intensity  x  V^.  This  differs  from  the  true  arithmetical  mean  in 
the  ratio  V£  :  2/Tr,  or  -707/-637  ;  and  it  is  called  the  effective  or  virtual 
mean  intensity.  Similarly  for  the  voltages.  Hence  the  Heat  produced 
by  an  alternating  current  whose  maximum  intensity  is  i/  is  \i?r. 

If  an  object  of  some  capacity  —  a  small  porcelain  ball  in  a 
vacuum  chamber  —  be  connected  with  one  terminal  only  of  the 
secondary  coil  of  an  induction  coil  subjected  to  alternations  of 
extreme  frequency,  it  may  itself  become  very  hot ;  and  if  it  be 
itself  surrounded  by  air,  it  may  subject  the  molecules  of  that 
air  to  collisions  and  shock,  so  that  the  air  glows  with  a  phospho- 
rescent light.  A  Geissler-tube  connected  in  the  same  way  will 
light  up.  The  human  body  may  be  charged  in  this  extremely- 
rapidly-alternating  manner  by  being  connected  in  the  same  way ; 
when  this  is  done,  a  Geissler-tube  held  in  the  hand  will  light 
up ;  while  the  current  then  passing  in  the  body,  though  of 
enormous  voltage,  appears  to  do  no  more  harm  than  the  impact 
of  light-waves  does.  Mr.  Nicola  Tesla  has  recently  devised  a 
series  of  extraordinary  experiments  on  these  lines.  He  has  con- 
structed lamps  connected  only  by  one  wire  with  the  terminals 
of  secondary  coils;  and  he  has  obtained,  at  these  terminals, 
brushes  in  the  form  of  veritable  flames,  consisting  merely  of 
air-molecules  subjected  to  collision  and  shock. 

Since  the  current  is  alternating,  there  can  be  no  electrolytic 
effect,  except,  apparently,  a  small  residual  decomposition  in  some 
cases,  which  is  possibly  due  to  greater  ease  of  charging  particular 
elements  with  one  kind  of  electricity  than  with  another. 

When  an  alternating  current  is  sent  through  a  loop,  as  in 
Fig.  221,  the  derived  currents  are  not  so  distributed  as  to  produce 
the  minimum  amount  of  Heat,  as  is  the  case  in  steady  currents ; 
but  they  are  so  distributed  as  to  keep  the  kinetic  energy  of  the 
field  down  to  a  minimum,  and  to  neutralise  each'other's  pro- 


724  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

duction  of  an  electromagnetic  field  as  far  as  possible.  The 
currents  in  the  two  branches  of  a  loop  may  thus  be  opposed, 
and  even  be  each  much  greater  than  the  leading  current ;  but 
their  mean  difference  will  be  equal  to  the  mean  intensity  of  that 
current. 

The  preponderating  branch-current  goes  in  the  direction  of  the  leading 
current  along  that  branch  which  has  the  less  self-induction ;  that  is,  along 
that  branch  the  passage  of  a  current  along  which  would  give  rise  to  the 
smaller  amount  of  total  magnetic  induction  in  the  part  of  the  field  affected 
by  that  branch;  but  if  the  electrostatic  capacity  of  the  other  branch  be 
increased,  this  may  neutralise  the  effect  of  its  superior  self-induction. 

When  an  alternating  current  is  used  to  excite  an  electro- 
magnet, there  is  a  strong  tendency  to  the  formation  of  Eddy- 
Currents,  parallel  to  the  wire  of  the  inducing  coils.  This  is 
combated  by  building  up  the  core,  of  laminae  or  of  thin  wires. 

If  an  alternating  current  be  used  to  excite  a  long  electromagnet,  the 
alternating  magnetising  effect  falls  off  very  rapidly  at  a  distance  from  the 
exciting  coil,  and  does  so  the  more  rapidly  the  greater  the  frequency; 
the  lines  of  magnetic  induction  leak  out  laterally  from  the  iron,  and  find 
closed  return-paths  through  the  air. 

In  a  non-uniform  magnetic  or  electromagnetic  field,  in  a 
case  where  a  conductor,  say  a  coil,  bearing  a  steady  current, 
would  act  like  a  magnet,  one  bearing  an  alternating  current  acts 
like  a  diamagnetic  body ;  it  is  repelled  into  the  weakest  part  of 
the  field. 

Two  alternating  currents  in  the  same  direction  and  in  the 
same  phase  attract  one  another ;  if  in  opposite  phases,  they  repel 
one  another ;  if  their  phases  differ  by  ?r/2,  they  have  no  effect 
upon  one  another.  Hence  the  mutual  attraction  or  repulsion 
of  two  alternating  currents  will  depend  upon  their  relative 
amounts  of  Lag. 

When  an  alternating  current  acts  by  induction  on  a  coil  laid  parallel  to 
the  inducing  coil,  in  the  absence  of  self-induction  the  induced  current  would 
be  strongest  when  the  inducing  current  passed  through  its  zero  value,  for 
the  strength  of  that  current  would  then  vary  most  rapidly.  But,  when  the 
secondary  coil  is  excited  by  an  electromagnet,  itself  excited  by  an  alter- 
nating current,  there  is  Lag  in  the  secondary  coil ;  there  is  now  repulsion 
between  the  two  coils ;  and  the  secondary  coil  tends  to  fly  off  the  electro- 
magnet. Similarly,  an  alternate-current  electromagnet  may  repel  copper  by 
its  action  on  the  currents  induced  in  that  copper. 

Transformers.  —  An  alternating  current  of  high  voltage 
and  few  Amperes  can  be  sent  to  a  great  distance,  for  there  is 
comparatively  little  loss  by  transformation  into  heat.  But  for 
use  in  houses,  etc.,  its  voltage  must  be  reduced.  This  is  effected 


xvi.]  TRANSFORMERS.  725 

by  Transformers,  which  are,  in  effect,  Induction  Coils  reversed 
in  their  action.  The  current  of  high  voltage  is  sent  through  the 
coil  of  many  turns  (inside  the  other  coil),  and  an  induced 
alternating  current  of  lower  voltage  and  correspondingly  greater 
quantity  is  induced  in  the  coil  of  fewer  turns.  The  soft-iron 
wire  or  laminated  core  may  form  either  a  closed  or  an  open 
"  magnetic  circuit."  An  interrupter  or  contact-breaker  is  not 
necessary.  When  the  core  is  large  and  the  alternations  rapid, 
the  effective  current-intensities  are  inversely  proportional  to  the 
number  of  turns  in  the  respective  coils.  The  induced  currents 
are  opposite  in  phase  to  the  inducing  currents ;  they  thus  tend 
to  demagnetise  the  soft-iron  core  when  the  house-circuit  is  closed. 
They  thus  tend  to  diminish  the  Impedance  of  the  main 
circuit ;  and  they  do  this  the  more  completely,  the  less  the  resist- 
ance in  the  house-circuit.  When  the  house-circuit  is  broken, 
little  or  no  current  passes  through  the  main  coil,  on  account  of 
the  impedance  of  that  coil,  with  its  core ;  but  as  the  resistance 
in  the  house-circuit  is  reduced  by  increasing  the  number  of 
paths  along  which  the  house-current  may  pass,  that  house-cur- 
rent is  allowed  to  gain  in  strength  in  proportion  to  the  reduc- 
tion, that  is,  in  proportion  to  the  work  to  be  done.  From  3  to 
6  per  cent  of  the  energy  supplied  is  usually  lost  in  eddy-currents 
and  hysteresis. 

PRODUCTION  OF  ALTERNATING  CURRENTS. 

There  are  two  main  methods  of  producing  these ;  (1)  by 
the  action  of  dynamo-electric  machinery,  the  frequencies  pro- 
duced by  which  range,  say,  from  40  to  150  per  second ;  and  (2) 
by  means  of  the  discharge  of  a  Leyden-jar  or  other  electrostatic 
condenser,  the  frequencies  produced  by  which  may  amount  to, 
say,  10,000000  per  second.  We  shall  deal  with  the  latter  first. 

Oscillation  in  Leyden-jar  Discharge.  —  Assume  the  source  of  elec- 
tricity in  a  circuit,  —  which  circuit  may  have  an  air-gap  in  it  equivalent  to 
an  interposed  resistance,  —  to  be  a  Ley  den  jar  charged  with  a  quantity  Q; 
the  E.M.D.P.  is  E  =  Q/C,  where  C  is  the  Capacity  of  the  jar.  The  equa- 
tion (i.),  p.  705,  is  easily  reduced  —  neglecting  squares  of  SI,  which  is  a 
proper  omission  in  a  calculation  involving  subsequent  integration  —  to  the 
form  E  =  RI  +'  L  •  81 ;  or,  since  I  is  itself  equal  to  -  Q,  we  have  Q/C  = 
_  (RQ  +  LQ) ;  R  being  the  total  Resistance  of  the  circuit.  This  equation 
is  a  Differential  Equation ;  and,  starting  from  a  charge  Q  and  no  current  at 
the  initial  instant,  this  equation  is  reduced,  by  appropriate  mathematical 
treatment,  to  the  statement  that  at  the  end  of  any  given  time-interval  f, 
the  charge  Q,  left  in  the  Leyden  jar  is  Q,  =  Q  •  e-^/2L  •  {((2La  '+  R)/4La)  •  «<* 


726  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

-f  ((2La  —  R)/4La)  •  e~a*},  in  which  expression  the  letter  a  is  made  to  do 
duty  for  V(R/2L)2  -  1/CL,  and  c  is  equal  to  2-718281.  On  interpreting 
this  statement  we  find  that  there  are  two  cases,  in  which  the  consequences 
are  different;  (1)  when  L  is  less  than  R2C/4,  a  is  a  possible  positive  quan- 
tity, and  the  charge  gradually  diminishes  as  t  increases,  that  is,  as  time  goes 
on,  so  that  the  Leyden  jar  steadily  discharges  itself;  but  (2)  when  L  is 
greater  than  R2C/4,  a  becomes  the  square  root  of  a  minus  quantity,  and 
being  itself  therefore  impossible,  renders  the  expression  an  unintelligible 
one  as  it  stands.  But  if  we  transform  it  by  making  aV—  1  =  a',  we  find 
the  result  to  be  that  the  expression  takes  the  form  Q,  =Q  •  €-R«/2L.  (Cos  a't 
—  (R/2Lo/-sina/£)).  That  is  to  say,  the  charge  in  the  jar  oscillates  in 
amount,  being  nothing  when  (sin  a't  /cos  a't)  =  2La'/R>  and  attaining  a 
succession  of  rapidly-decreasing  maxima  (alternately  positive  and  negative), 
which  occur  whenever  a't  ==  mr,  where  n  is  any  whole  number.  The  current- 
intensity  is  a  maximum,  positive  or  negative,  when  the  charge  in  the 
jar  is  passing  through  a  zero  value ;  when  this  is  the  case,  the  intensity 
is  equal  to  —  Qt  =  Q/CLo/-  e-R*/2L.  sin  a't;  and  it  thus  oscillates  in  value, 
rapidly  diminishing,  with  a  period  T  of  complete  oscillation  =  2-jr/a!  = 
27T/  Vl/CL  —  (R/2L)2.  This  period  T,  when  R  is  very  small  in  compari- 
son with  L,  is  approximately  T  =  27rVCL.  Whether,  therefore,  a  steady  or 
an  oscillating  discharge  will  be  obtained  depends  on  the  relations  between 
the  capacity  C  of  the  jar,  the  self-induction  L  of  the  circuit,  and  its  resist- 
ance R :  diminish  C  or  R  or  increase  L  sufficiently,  and  an  oscillating  dis- 
charge may  be  obtained ;  while  by  increasing  L  very  much,  as  by  interposing 
very  large  coils  in  the  circuit,  the  rate  of  oscillation  may  be  greatly  reduced. 

Since  a  maximal  positive  value  of  the  current-intensity  occurs  once  in 
each  period,  the  time-interval  between  any  two  such  positive  maxima  is  T  = 
2-7T  VCL.  The  successive  maximal  positive  values  of  the  current-intensity  are 
thus  Q/CLa'-e-^/2L  =  Q/CLa'  •  c~ BrVc/L f or  the  first;  Q/CLa'  •  €-R2,i>/c7L 
for  the  second,  and  so  on ;  the  value  being  Q/CLa'  •  e-R'*7rV'c/L  for  the  nth 
positive  maximum. 

Numerical  Example.  —  In  a  solenoid  of  n  turns  and  of  length  I  cm., 
L  =  47m2A//,/£.  Let  n  =  100,  and  I  =  20 ;  and  let  r  =  1  cm.,  and,  as  the 
medium  is  air,  /*  =  1.  Then  L  =  19,739,  the  number  of  lines  of  induction, 
in  magnetic  measure,  which  thread  the  solenoid  when  unit-current  passes. 
The  Resistance  of  this  wire  may  be  taken  as  0-0045  Ohm  per  metre,  or  45000 
magnetic  units  per  cm. ;  R  =  45000  '2-nr  •  n  =  (nearly)  28,275,000  magnetic 
units.  Next,  suppose  a  Leyden  jar  to  have  opposed  surfaces  of  250  sq.  cm. 
each,  at  a  mutual  distance  0-3  cm.  across  glass  whose  sp.  ind.  cap.  is,  say, 
K  =  2-5 :  then  its  Capacity  C  will  be  K/47T  •  surface/ d  =  165-78  C.G.S.  electro- 
static units,  or  (165-78  -r-  9.1020)  magnetic  units.  Now  discharge  the  Leyden 
jar  through  this  solenoid,  it  being  assumed  that  there  is  no  other  part  of  the 
circuit  to  be  taken  into  consideration.  Then  a'  —  Vl/CL  —  (R/2L)2  = 
16,583500,  in  magnetic  units ;  and  the  period  of  complete  oscillation 
T  =  27r/a'=-27p^  second.  _ 

To  ascertain  the  progressive  decrease  of  the  successive  maxima  of  cur- 
rent-intensity, put  these  numerical  values  of  C,  R,  and  L  in  the  expressions 
given  above  for  that  intensity :  then  we  find  that  the  current-intensity  is,  at 
the  first,  the  10th,  the  100th,  the  1000th,  and  the  10000th  maxima  respec- 
tively, in  the  ratios  of  0-99973  :  0-99742  :  0-97323  :  0-76237  :  0-06632  ;  and  the 
maxima  of  current-intensity  are  reduced  to  a  millionth  in  0-02428  second. 


xvi.]  LEYDEN-JAR   DISCHARGE.  727 

The  discharge  of  a  Leyden  jar  is  thus  practically  instantaneous ;  and  in 
order  that  it  may  keep  up  a  continued  discharge,  the  jar  must  be  fed  from  a 
machine  or  an  induction-coil. 

If  the  apparatus  of  the  above  example  be  reduced  to  half  the  size,  line- 
arly, the  time  of  oscillation  will  be  reduced  to  one-half;  and  so  on  in  pro- 
portion. A  Leyden  jar  of  molecular  size  would  give  an  oscillating  discharge 
whose  frequency  is  of  the  same  order  as  that  of  light-waves  ;  but  if  the  mole- 
cule be  simple  in  its  structure,  the  frequency,  thus  calculated,  will  lie  beyond 
the  violet. 

It  has,  however,  been  assumed  in  the  above  that  the  current  is  uniform 
all  along  the  wire ;  that  is,  that  the  wave-length  is  great  in  comparison  with 
the  length  of  the  circuit ;  and  that  the  current-density  is  uniform  across  the 
cross-section  of  the  wire.  If  we  take  into  account  that  these  things  are  not 
true  at  high  frequencies,  we  find  that  we  have  to  replace  the  Resistance  by 
the  Impedance,  and  that  the  value  of  L  is  affected  by  the  frequency ;  which 
modifies  the  numerical  results. 

If  the  condenser  be  reduced  to  the  mere  tips  of  the  wire  of  a  loop  or  coil, 
the  to-and-fro  reflexion  of  the  disturbances  in  the  wire  will  occur  at  a  definite 
frequency,  which  depends  on  the  size  of  that  loop  or  coil.  Similar  phe- 
nomena of  to-and-fro  reflexion  occur  whenever  abrupt  signals  are  sent  over 
a  short  circuit,  but  on  a  long  one  they  die  out ;  they  are  due  to  the  electro- 
static charge  on  the  wire. 

The  second  -method  involves  the  use  of  Magneto-Electric 
and  Dynamo-Electric  Machines.  The  former  are  now  seldom 
seen,  except  in  small  apparatus  such  as  that  used  for  medical 
purposes :  the  latter  have  assumed  great  importance  in  Electric 
Lighting,  Electrolysis,  Transmission  of  Power  and,  to  a  smaller 
extent,  in  Heating. 

Given  an  existing  magnetic  field :  then,  if  a  loop  of  wire  be 
moved  in  this  field,  so  as  to  embrace  more  or  fewer  lines  of  mag- 
netic induction,  a  current  will  be  set  up  in  that  wire.  In  the 
former  case,  Work  has  to  be  done  upon  the  loop ;  in  the  latter, 
the  field  does  work  upon  the  loop ;  and  both  these  amounts  of 
Work  appear  as  the  Energy  of  Electric  Current  in  the  loop. 
The  direction  of  that  current  depends  upon  whether  work  is 
being  done  by  or  against  the  field  at  the  moment. 

We  have  seen  that  if  the  lines  of  magnetic  induction  point  eastwards, 
and  the  direction  of  motion  of  the  lines  of  force  in  the  field  be  northward, 
the  corresponding  lines  of  Electric  Force  will  themselves  point  upwards. 
Here  we  have  the  converse  case :  if  any  small  part  of  the  wire  stand  vertical 
and  move  broadside-on,  towards  the  south,  across  lines  of  magnetic  induction 
whose  trend  is  towards  the  east,  there  will  be  set  up  in  that  part  of  the  wire 
a  current  whose  direction  is  upward.  In  the  former  case,  the  lines  moved 
up  to  the  wire ;  in  the  latter  the  wire  moves  in  the  opposite  direction  towards 
them. 

If  a  loop,  or  a  coil,  be  flashed  past  the  two  poles  of  a  per- 
manent magnet,  so  as  alternately  to  embrace  the  magnetic  lines 


728  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

radiating  from  these  two  poles,  it  will,  as  it  approaches  the  one 
pole,  travel  into  a  field  whose  strength  it  finds  to  increase  to  a 
maximum,  and  then  to  fall  away  as  it  recedes  from  that  pole.  As 
the  coil  goes  farther,  it  goes  through  a  region  of  zero  and  then 
into  one  of  opposite  potential,  which  in  its  turn  again  reaches  a 
maximum  and  then  falls  away.  The  currents  produced  in  any 
given  part  of  the  wire  are  thus  alternately  in  opposite  directions. 

In  Pixii's  machine,  permanent  magnets  were  themselves  flashed  past 
the  coils,  of  which  there  were  two,  parallel  and  wound  on  bobbins,  with  a 
soft-iron  core  in  each.  These  cores  assumed,  in  rapid  alternation,  opposite 
magnetic  characters,  and  exposed  the  wire  to  a  more  intense  alternating 
magnetic  field  than  they  would  have  been  exposed  to  had  there  been  no 
cores.  In  other  cases  —  Clarke's,  etc. — the  magnets  were  fixed,  and  the 
bobbins  were  made  to  pass  their  poles.  This  kind  of  machine,  by  multipli- 
cation of  the  magnets  and  of  the  rotating  bobbins,  has  been  made  in  very 
large  sizes ;  and  with  the  substitution  of  electromagnets  for  permanent 
magnets,  the  principle  is  still  applied  in  some  disc  dynamos. 

It  is  more  common,  however,  instead  of  dragging  a  loop  or 
a  coil  sideways  past  the  polar  face  of  a  magnet,  to  provide  for 
it  an  axis  of  rotation  traversing  the  middle  of  a  nearly  uni- 
form magnetic  field,  at  right  angles  to  the  magnetic  lines  of 
induction  of  that  field ;  and  so  to  fit  up  the  loop,  that  at  two 
points  in  its  rotation  round  that  axis,  it  may  be  so  placed  as  to 
embrace  the  greatest  possible  number  of  lines  of  induction. 
This  may  be  done  in  two  ways :  first,  by  making  the  axis  of 
rotation  lie  across  the  loop  itself,  which  is  spun  in  the  field 
round  its  own  diameter ;  or  second,  by  making  the  axis  of  rota- 
tion lie  outside  the  loop  itself,  and  parallel  to  it,  so  that  the  loop 
is  swung  in  a  circle  in  the  field.  In  both  these  cases,  the  loop, 
at  one  part  of  its  rotation,  will  embrace  a  maximum  number  of 
magnetic  lines,  to  which  its  own  axis  is  then  parallel ;  when  it 
has  turned  round  through  90°,  the  axis  of  the  loop  is  at  right 
angles  to  these  lines,  and  the  loop  itself  embraces  none  of  them. 
When  it  is  in  this  latter  position,  it  is  most  rapidly  altering  the 
number  of  lines  which  pass  through  it,  and  the  induced  current 
in  that  loop  is  then  a  maximum. 

The  number  of  lines  of  induction  embraced  by  it  when  its  axis  is  parallel 
to  these  lines  is  B  =  its  area  A  x  b ;  when  it  is  at  an  angle  6  ( =  STrnt)  to 
that  position,  the  induction  through  it  is  reduced  to  B  •  cos  0  =  B0.  But 
the  current-strength  ie  in  any  position  is,  omitting  self-induction,  —&B9/r8t, 
r  being  the  resistance ;  and  this,  on  giving  B0  its  value,  and  differentiating, 
leads  to  the  result  that  i$  =  (B/r)  .  2irn  -  sin  0 ;  and  the  maximum  value 
of  this  is  i  =  (B/r)  •  27rn,  when  6  =  90° ;  that  is,  when  the  axis  of  the 
coil  is  at  right  angles  to  the  lines  of  induction.  Hence  the  maximum  value 


xvi.]  DYNAMO-ELECTRIC   MACHINES.  729 

of  e  =  27rnB ;  and  at  any  position  0,  e&  —  e  •  sin  0  =  e  -  sin  27rnt;  the  condition 
required  for  finding  the  Lag  and  the  Impedance  (p.  722),  when  self-induc- 
tion is  taken  into  account.  The  average  value  of  e  is  2/7r  x  the  maximum 
value  (p.  85),  and  is  therefore,  for  a  single  loop,  equal  to  4nB ;  n  being  the 
number  of  complete  revolutions  of  the  loop,  through  360°. 

In  the  latter  of  the  two  methods  of  rotation  referred  to,  the 
whole  loop,  and  in  the  former,  any  part  of  it,  is  carried  round 
in  the  course  of  its  rotation  from  a  positive  into  a  negative  part 
of  the  field,  and  the  same  operation,  with  negative  sign,  is 
repeated  there ;  the  current  now  produced  is,  as  regards  the 
loop  or  the  part  of  it  referred  to,  now  in  the  opposite  sense, 
and  passes  through  a  maximum  in  the  same  way.  In  any  given 
part  of  the  loop,  therefore,  the  current  produced  passes  through 
alternating  positive  and  negative  maximal  values ;  and  its  varia- 
tion between  these  extremes  is  simple-harmonic,  one  complete 
alternation  to  each  complete  revolution  of  the  loop. 

The  extremities  of  the  loop  may  be  connected  with  two 
separate  rings,  which  rotate  along  with  the  loop  round  the  axis 
of  rotation  ;  and  if  an  external  circuit  terminate  in  flexible 
metallic  "  brushes,"  which  rest  upon  these  collecting  rings,  the 
alternating  currents  developed  in  the  loop  will  be  propagated 
round  that  circuit. 

A  single  loop  is,  however,  an  illustrative  rather  than  a 
practical  apparatus.  A  coil  would  produce  a  greater  current, 
for  each  turn  in  it  would  be  acted  upon,  practically,  as  if  it  were 
a  loop ;  and  in  Siemens'  Inductor  a  coil,  with  a  soft-iron  core, 
was  rotated  round  an  axis  passing  through  its  own  centre,  and 
was  so  shaped  as  to  lie  as  close  to  the  magnetic  pole-faces  as 
possible,  and  at  the  same  time  to  have  a  minimum  moment  of 
inertia.  But  in  the  course  of  each  revolution  there  is  a  period 
during  which  such  a  coil  or  loop  is  very  nearly  idle:  that  is, 
when  its  own  plane  is  at  right  angles  to  the  lines  of  induction : 
and  since  there  is  only  one  alternation  per  revolution,  the  speed 
for  rapidly-alternating  currents  would  have  to  be  excessive.  It 
is  therefore  the  practice  in  alternating-current  machines,  or 
Alternators,  to  multiply  the  opposed  magnetic  fields  through 
which  the  coil  has  to  travel;  and  this  is  done  by  multiplying 
the  opposite  pole-faces  past  which  the  coil  is  driven.  Such 
machines  are  said  to  be  multipolar:  and  the  frequency  of 
alternation  is  correspondingly  increased  by  this  device,  though 
the  alternations  are  now,  initially,  not  so  nearly  simple-harmonic 
in  their  character  as  when  a  single  loop  or  coil  is  employed  in 


730  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

a  single  uniform  field.  But  further,  instead  of  a  single  coil 
passing  one  magnet-pole  at  a  time,  a  number  are  put  into  simul- 
taneous action,  all  similarly  situated  with  respect  to  their  several 
magnetic  fields ;  and  great  ingenuity  has  been  displayed  in  con- 
triving the  machine  so  that  these  may  act  in  concert.  These 
coils  may  move  past  the  magnet-poles,  or  the  magnet-poles  past 
them :  or  again,  both  coils  and  magnets  may  be  stationary,  the 
strength  of  the  magnetic  fields  being  alternately  increased  and 
diminished  by  masses  of  soft  iron  moving  in  or  near  those  fields. 
For  some  purposes  it  is  convenient  so  to  arrange  the  coils  and 
their  metallic  connections  as  to  send  along,  say,  three  wires 
three  equal  currents  differing  in  their  phase  of  alternation  by 
equal  amounts ;  alternators  of  this  kind  are  called  multiphase 
alternators. 

The  congeries  of  coils  is  borne  by  a  drum,  by  a  ring,  or  on 
the  periphery  of  a  disc;  and  each  coil  has  a  laminated  soft-iron 
core,  which  forms  a  part  of  the  magnetic  circuit,  and  greatly 
intensifies  the  inductive  effect  on  the  wire  of  the  coils.  The 
whole  arrangement  of  coils,  with  their  core  or  cores,  is  called 
the  Armature. 

The  magnetic  field  is,  in  all  modern  machines  of  any  size, 
that  of  an  electromagnet;  it  has  to  be  intense,  while  at  the 
same  time  there  must  be  plenty  of  iron  in  the  magnetic  circuit. 
The  electromagnets  are  excited,  when  the  machine  is  at  work, 
by  a  part  of  the  current  from  the  machine  itself ;  but  as  this  is 
alternating,  it  would  not  excite  an  electromagnet  so  as  to 
impart  to  it  a  uniformly-directed  magnetic  polarity,  unless  its 
alternate  phases  had  been  made  to  go,  not  in  opposite,  but  in 
the  same  directions.  This  is  accomplished  by  a  Commutator: 
the  two  "brushes,"  which  take  current  off  for  the  electromag- 
nets, do  not  each  continuously  touch  one  of  the  collecting  rings ; 
but  each  comes  in  contact,  first  with  a  projecting  tooth  of  the 
one,  and  then  with  a  projecting  tooth  of  the  other  collecting 
ring.  Thus,  in  step  with  the  alternations  of  the  current  yielded 
by  the  machine,  there  is  an  alternation  of  the  directions  into 
which  it  is  guided ;  and  the  result  is  a  current  not  uniform  in 
strength,  but  constant  in  direction,  and  useful  for  the  electro- 
magnet. 

Two  alternators  put  in  series  tend  to  assume  opposition  of 
phase  and  to  deliver,  jointly,  no  current;  but  they  will  work 
in  parallel.  They  then  go  into  step,  and  tend  to  keep  step, 
co-phasally. 


xvi.]  DYNAMO-ELECTRIC   MACHINES.  731 

Direct-Current  Dynamos.  —  In  these  the  whole  cur- 
rent of  the  machine  is  led  through  a  Commutator.  The  current 
from  a  single  loop  or  coil  would  vary  from  zero  to  a  maximum  and 
back  to  zero  twice  in  each  revolution  ;  and  as  currents,  merely  in 
different  states  of  variation  of  positive  or  of  negative  value  on 
one  side  only  of  zero,  do  not  tend  to  neutralise  one  another 
when  sent  into  the  same  wire,  but  tend,  by  their  summation,  to 
render  the  aggregate  current  more  uniform  in  character,  the 
direct-current  dynamo  generally  has  its  several  coils  or  groups 
of  coils  so  arranged  that  each  sends  its  own  current  into  the 
general  circuit,  in  whatever  phase  it  may  happen  to  be,  and  is 
cut  out  of  that  circuit  only  during  such  time  as  there  may  be 
danger  of  other  coils  or  groups  of  coils  being  short-circuited 
through  it  during  its  own  comparatively  idle  period.  The  arma- 
ture, by  multiplication  of  loops  or  coils  lying  across  the  field  or 
towards  its  periphery,  usually  takes  the  form  of  a  drum  or  of  a 
ring ;  and  its  soft-iron  core  is  laminated,  to  prevent  the  forma- 
tion of  eddy-currents. 

Both  in  ring  and  drum  armatures,  the  armature-core  tends 
to  become  magnetised  transversely  to  the  main  magnetic  field. 
The  actual  magnetic  field  is  thus  the  resultant  of  the  main  field 
and  a  cross-field.  If  it  had  not  been  for  this,  the  proper  posi- 
tion for  the  brushes  would  have  been  at  right  angles  to  the 
field,  so  as  to  lead  off  the  current  through  those  coils  which,  at 
the  moment,  are  least  engaged  in  the  actual  production  of  cur- 
rent ;  but  the  effect  of  the  resultant  obliquity  of  the  field  is  that 
the  brushes  must  also  lie  obliquely,  to  an  equal  extent;  and 
thus,  as  is  said,  the  brushes  must  be  given  a  certain  lead. 
The  amount  of  this  lead  is,  further,  somewhat  increased  by  the 
necessity,  in  order  to  prevent  sparking  at  the  brushes,  of  letting 
each  loop  or  coil  get  a  little  way  into  the  opposing  field  before 
being  cut-out ;  by  which  means  the  extra  current  is  neutralised. 
This  conduces,  however,  to  demagnetisation  of  the  magnetic  cir- 
cuit as  a  whole. 

The  mean  current  produced  is  proportional  to  the  speed, 
less  a  certain  number'of  revolutions  per  second,  called  the  Dead 
Turns;  and  is  also  proportional  to  the  number  of  coils  in  the 
field  and  to  the  strength  of  that  field.  At  speeds  less  than  the 
so-called  Dead  Turns  the  machine  will  not  deliver  any  current 
at  all. 

As  to  the  mode  of  excitation  of  the  Field  Magnets,  that  is, 
of  the  electromagnets  which  produce  the  magnetic  field  within 


732  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

which  the  armature  rotates :  these  are  feebly  excited  by  an 
ordinary  magnet,  or  there  may  be  sufficient  residual  magnetism 
in  them  to  serve  the  purpose,  or  they  may  be  feebly  magnetic 
under  the  induction  of  the  earth's  magnetic  field;  then,  when 
the  armature  is  set  in  rotation,  an  extremely  feeble  current  is 
generated.  This  feeble  current  is  not  permitted  at  once  to  pass 
away,  but  is  sent,  either  wholly  (in  "Series  "  dynamos)  or  partly 
(in  "  Shunt "  dynamos,  by  means  of  a  shunt  always  kept  in 
action),  round  the  soft-iron  magnet,  and  thereby  increases  its 
magnetisation.  The  soft-iron  electromagnet,  thus  strengthened, 
induces  a  still  stronger  current  in  the  rotating  armature ;  and 
thus,  the  current-intensity  attains  in  a  short  time  a  maximum, 
the  potential  of  which  depends  upon  the  speed  of  rotation  and 
upon  the  product  of  the  intensity  of  the  current  actually  pass- 
ing round  the  field-magnets  into  the  number  of  turns  which  it 
takes  round  them,  as  well  as  upon  the  number  of  turns  within 
the  armature,  effectively  connected  in  Series.  In  "Series-Shunt" 
machines  the  current  is  divided  into  two  parts;  one  part  runs 
in  a  shunt  round  the  electromagnet :  the  other  runs  both  round 
the  electromagnet  and  through  the  external  circuit. 

In  a  Series  machine,  as  the  Amperes  increase,  the  total  voltage  in  the 
circuit  also  increases,  rapidly  at  first,  but  then  more  slowly,  as  the  permea- 
bility of  the  iron  begins  to  fall  off,  and  the  cross  magnetic  field  to  become 
more  intense ;  and  at  extreme  ampereages,  the  total  voltage  even  tends  to 
droop  away  for  these  reasons.  As  the  ampereage  increases,  the  available 
voltage  at  the  terminals  differs  by  a  steadily-increasing  amount  from  the 
total  voltage  in  the  circuit,  because  of  the  increased  voltage  consumed  in 
the  passage  of  a  greater  current  through  the  armature :  hence,  as  the  cur- 
rent increases,  the  available  voltage  at  the  terminals  reaches  a  maximum 
and  then  falls  off.  Within  this  limit,  however,  when  the  resistance  of  the 
circuit  is  increased,  the  electromagnet  is  enfeebled,  and  the  voltage  at  the 
same  time  falls. 

In  a  Shunt  machine,  on  the  other  hand,  as  this  resistance  increases,  the 
tendency  is  for  a  larger  proportion  of  the  total  current  to  pass  through  the 
shunt-winding  round  the  electromagnet,  and  thus  to  strengthen  it  and  raise 
the  voltage.  When  the  voltage  is  so  increased,  the  Amperes  rise  to  a  maxi- 
mum, and  then  fall  off  to  nothing,  while  the  voltage  goes  on  rising  to  a 
maximum,  the  potential-difference  on  open  circuit. 

In  Series-Shunt  machines,  the  electromagnet  is  wound  with  shunt-coils, 
and  the  main  current  is  also  sent  round  it.  When  the  resistance  is  increased, 
the  opposite  variations  of  potential,  due  to  the  shunt  and  to  the  series-wind- 
ing, may  partly  compensate  one  another ;  when  they  'are  so  adjusted  that, 
for  a  particular  speed  of  running,  the  machine  gives  a  constant  voltage 
whatever  be  the  resistance,  the  machine  is  said  to  be  "  Compound-Wound." 

In  another  class  of  machines,  there  is  separate  excitation  of  the 
electromagnet  by  particular  coils  driven  on  the  same  axis  as  the  coils  supply- 
ing the  general  working  circuit,  or  by  a  separate  subsidiary  machine. 


ELECTRICITY  AND   MAGNETISM.  733 


TRANSMISSION  OF  ENERGY  TO  A  DISTANCE. 

All  current  and  electromagnetic  phenomena  are,  as  we  have 
seen,  associated  with  the  transmission  of  Energy  to  a  distance, 
across  the  dielectric. 

We  have  already  considered  the  action  of  Galvanometers ; 
and  also  of  Ballistic  Galvanometers,  in  which  the  throw  of  the 
needle  renders  manifest  the  passage  of  a  very  brief  current ; 
just  as  the  position  of  equilibrium  assumed  by  the  needle,  as  it 
lies  more  or  less  completely  across  the  current,  with  its  axis 
directed  along  the  lines  of  force,  indicates  the  persistence  of  a 
steady  current. 

As  often  as  a  momentary-current  is  sent  round  the  magnet 
of  a  galvanometer,  so  often  will  the  twitch  of  the  suspended 
magnet  be  repeated,  and  at  intervals  of  time  equal  to  those 
between  the  successive  momentary-currents.  This  action  — 
which  is  the  simplest  form  of  transmission  of  energy  to  a  dis- 
tance, for  work  is  done  in  displacing  the  magnet  within  the 
field  —  is  the  basis  of  telegraphic  signalling. 

Longer  and  shorter  currents  produce  longer  throws  and  shorter  twitches 
of  the  galvanometer-needle.  These  form  the  basis  of  a  signal  alphabet  — 
the  Morse  code.  The  following  is  the  alphabet,  the  upper  line,  where  there 
are  two,  being  the  European  or  "  International,"  the  lower  the  American 
form :  — 


A 

B 

c  —  -A— 

D 

B  «  - 

F  -I" 

G 

H  - 

I    " 

j    

K 

L   

M  ' 

N  —  - 

0       I~A~- 

p  

Q    —  — 

R  IT-.'. 

s  •-- 

T 

U 

v  

w  -~ 

x  —  IIIT 

Y  —  A— 

Z  —..TV! 

A   

-      6  —  •  -- 

u  --- 

N  

[CH-  -]  E- 

i-—-ir-_r:-v2v.zi.T7--3: :  :  =  —    4-  -  - 

5_imi^        6 —-.-.-.-.    7=  =  ::-     8=  —  —  •• 

91  -  —  -  0  I 


734  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

Full  stop  I  I  _I_~_I_~_  .  Stroke 

Semicolon (Amer.)  Apostrophe 

Comma  I  ZZ  I  ZZ  ~  ~  Parenthesis  mi  ITJZT  " 

Exclamation  ZZ  ZZ  1I_Z~  Repeat  or  ?  1_1 7Z  ZZ  I  " 

Paragraph (Amer.)  Hyphen 

Italics  "  "  (Amer.)  &  -  A (Amer.) 

In  some  of  the  American  forms  it  will  be  observed  that  a  period  of 
time,  represented  by  A,  intervenes  in  the  midst  of  a  set  of  signals  represent- 
ing one  letter.  The  American  form  (U.S.  and  Canada)  is  the  original,  as 
devised  by  Prof.  Morse  :  the  European  is  an  improved  version  (International 
Congress,  1851). 

In  submarine  telegraphy  the  signals  used  are  not  long  and  short,  but 
right  and  left  deflections  —  that  is,  positive  and  negative  momentary-cur- 
rents. 

By  means  of  differential  galvanometers  two  messages  may  be  sent  along 
the  same  telegraphic  wire  at  the  same  time  (Duplex  Telegraphy). 
Station  A  has  a  single  wire  leading  from  the  positive  pole  of  his  battery : 
the  current  running  in  this  he  divides  into  two  moieties,  which  he  sends  in 
opposite  directions  round  the  needle  of  his  differential  galvanometer :  these 
two  moieties  are  then  sent  on,  the  one  to  the  distant  station  B,  the  other  to 
A's  own  earth-plate.  In  the  course  of  the  one  or  the  other  branch-current 
the  operator  at  A  interposes  resistances  until  the  intensities  of  the  opposed 
currents  round  his  galvanometer-needle  are  equal ;  then,  in  whatever  way 
A  may  make  or  break  circuit,  his  galvanometer-needle  will  remain  steady; 
but  the  needle  at  B  will  respond.  Similarly,  B  sends  signals  to  A,  to  which 
his  own  instrument  is  mute.  The  two  stations  may  thus  signal  simulta- 
neously, two  operators  being  employed  at  each  end,  one  to  transmit,  the 
other  to  receive ;  and  the  variations  of  electric  condition  produced  in  the 
single  connecting-wire  run  through  one  another  in  a  manner  analogous  to 
that  in  which  waves  meeting  on  a  cord  traverse  one  another. 

Bridge- Method  in  Duplex  Telegraphy.  —  Suppose  a  triangle 
ABC ;  the  current  enters  at  A ;  B  is  connected  with  the  distant  station  D ; 
C  is  connected  to  earth  through  a  resistance  equal  to  that  of  the  line  BD ; 
between  B  and  C  is  the  recording  instrument  of  the  home  station.  One 
moiety  of  the  current  which  enters  at  A  will  run  to  earth,  the  other  will 
travel  to  D.  If  the  resistance  in  AB  be  so  adjusted  as  to  be  equal  to  that 
in  AC,  B  and  C  will  be  at  equal  potentials ;  no  current  will  run  through 
BC ;  the  home  instrument  stands  motionless.  At  the  receiving  station  the 
apparatus  may  be  precisely  similar ;  it  will  then  indicate  the  arrival  of  sig- 
nals from  A,  but  will  be  insensible  to  the  movements  of  its  own  key. 

Quadruplex  Telegraphy.  —  A  small  current  always  runs  in  the 
circuit.  There  are  two  transmitting  keys.  The  one  reverses  the  direction 
of  the  current ;  this  causes  a  needle  within  a  magnetic  field  at  the  receiving 
station  to  swing  to  left  or  right ;  an  effect  which  depends  upon  change  of 
direction  of  the  current  within  the  circuit.  The  other  key,  when  depressed, 
introduces  a  new  battery  into  the  circuit;  the  strength  of  the  current  is 
thereby  increased,  and  the  current  is  now  enabled  to  make  a  certain  soft- 


xvi.]  TELEGRAPHY.  735 

iron  electromagnet  move  at  the  receiving  station ;  an  effect  which  depends 
upon  the  strength,  but  not  upon  the  direction  of  the  current  in  the  circuit. 
The  one  receiving  instrument  thus  records  reversals,  the  other  the  enhance- 
ments of  current-intensity.  Two  sets  of  signals  may  thus  be  sent  in  the 
same  direction  at  the  same  time;  and  this  arrangement  when  duplexed, 
preferably  by  the  bridge-method  above  described,  becomes  quadruplex. 
This  is  the  ground-principle  of  Prescott  and  Edison's  system,  which  is 
described  at  length  in  Prescott's  Telephone.  The  practical  details  are 
extremely  ingenious ;  there  may,  for  instance,  be  a  critical  instant  at  which 
the  intensity-receiver  is  liable  to  be  interfered  with  and  to  fail,  through  the 
current  supplied  to  it  fading  away  while  being  reversed  by  the  reversing- 
key;  a  condenser  then  acts  as  a  reservoir,  and  its  discharge  keeps  up  a 
current  which  tides  over  the  critical  instant ;  a  result  which  is  aided  by  a 
subsidiary  local  battery  then  brought  into  action  by  means  of  a  relay. 

In  Multiplex  Telegraphy,  each  operator  gets  the  use  of  the  circuit 
several  times  a  second ;  his  signals  are  like  cyclostyle  writing,  broken  up, 
but  practically  continuous. 

When  at  the  distant  end  of  a  circuit  the  conducting  wire 
is  passed  round  a  soft-iron  core,  that  soft-iron  core  becomes  an 
electromagnet  just  as  often,  and  remains  an  electromagnet  just 
as  long,  as  the  circuit  is  or  remains  completed  by  a  key  at  the 
home  station.  This  electromagnet  may  govern  the  movements 
of  a  neighbouring  mass  of  iron,  and  do  work  upon  it :  and  the 
movements  of  this  second  mass  may  be  utilised  in  an  endless 
variety  of  ways  for  the  repetition  of  movements  similar  to  those 
executed  at  the  home  station  by  the  hand  of  the  operator,  or  by 
any  mechanical  contrivance  adjusted  so  as  to  make  and  break 
contact  in  any  pre-arranged  manner.  The  mass  of  iron  moved 
at  the  distant  station  may  itself,  by  its  movement,  make  and 
break  a  second  electric  circuit,  and  may  thus  control  the  move- 
ment of  metallic  masses  at  still  more  distant  stations,  as  in  the 
case  of  telegraphic  relays. 

Electromagnetic  Interrupter  for  Tuning-Forks.  —  Atuning- 
fork  of  known  pitch  is  set  in  vibration.  As  it  vibrates,  it  alternately  makes 
and  breaks  a  current  which  traverses  the  tuning-fork  itself.  This  current 
is  passed,  in  its  course,  round  a  little  electromagnet,  which  is  alternately 
made  and  unmade.  This  electromagnet  is  so  arranged  as  alternately  to 
attract  and  release  one  of  the  prongs  of  the  tuning-fork,  which  is  thus  kept 
in  continuous  action.  The  intermittent  current  produced  is  sent  round  a 
second  electromagnet,  which  rhythmically  attracts  and  releases  a  second 
tuning-fork;  this  is  thus  kept  vibrating  in  unison  with  the  first,  even 
although  it  be  not  precisely  in  tune  with  it. 

Signalling  by  Alternating  Currents.  —  The  Pho-nophore.  In  this 
there  are  two  wires,  simply  coiled  together  :  their  farther  ends  are  both  free  : 
the  nearer  end  of  one  is  connected  with  the  line-wire.  When  a  brief  current 
is  sent,  the  other  wire  is  acted  upon  by  induction,  and  signal&rmay  be  heard 
in  a  telephone  connected  with  it  and  also  to  earth.  This  instrument  is 


736  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

worked  by  alternating  currents,  which  do  not  affect  the  ordinary  telegraphic 
instruments,  and  will  not  pass  through  their  coils,  being  throttled  by  them. 
These  signals  may  thus  be  made  independently  of  the  ordinary  direct-cur- 
rent signals,  and  both  systems  may  be  duplexed. 

If  a  current  which  has  300  maxima  of  intensity  per  second,  and  another 
of  say  800  maxima  per  second,  be  sent  along  the  same  wire,  the  conjoined 
current  will  present  variations  of  intensity  such  as  might  be  represented 
by  the  curves  of  Fig.  45.  Suppose  a  current  presenting  such  variations  of 
intensity  to  be  passed  round  a  soft-iron  core,  near  the  end  of  which  is  a  steel 
reed  tuned  to  vibrate  300  times  a  second,  and  so  adjusted  as  to  be  attracted 
in  the  sense  of  its  vibrations  when  the  soft-iron  core  attains  its  maximum 
intensity.  The  steel  reed  would,  among  other  impulses  to  which  it  would 
not  respond,  receive  a  set  of  300  maximum  attractions  per  second,  which 
would  set  it  in  vibration.  The  same  current  may  be  also  passed  round  a 
core,  opposite  the  extremity  of  which  is  placed  a  reed  tuned  to  800  vibra- 
tions per  second ;  that  reed  will  pick  out  and  will  respond  to  the  more  rapid 
component  of  variation  of  intensity  of  the  current,  and  will  respond  to  it 
only.  Further,  suppose  that  the  several  components  of  variation  are  each 
not  continuous,  but  interrupted :  the  corresponding  vibrations  of  each  reed 
will  be  similarly  interrupted,  and  one  telegraph  clerk  may  be  occupied  with 
listening  to  each.  A  current  whose  variation  of  intensity  is  as  complex  as 
the  sum  of  eight  distinct  S.H.M.'s  may  be  practically  resolved  by  as  many 
distinct  receiving-reeds  into  distinct  signals ;  and  since  the  duplex  method 
of  working  may  be  applied  to  this  plan,  as  many  as  sixteen  distinct  mes- 
sages may  travel  along  a  single  wire  at  the  same  time.  This  is  the  principle 
of  Mr.  Elisha  Grey's  Harmonic  Telegraph. 

The  Telephone,  in  its  simplest  form,  presents  a  plate  of 
iron,  P,  placed  in  the  magnetic  field  of  a  magnet,  M  :  the  plate  is 
caused,  by  being  spoken  at,  to  enter  into  certain  vibrations  ;  the 
vibrating  plate  P,  by  induction,  acts  upon  the  magnetism  of  the 
F.  257  magnet  M;  the  latter  is  alter- 

nately strengthened  and  weak- 
ened in  accordance  with  the 
varying  position  of  the  vibrating 
plate :  as  M  varies  in  strength  it 
causes  variations  in  the  strength 
of  a  current  passing  through  a 
coil,  C,  wound  round  its  pole, 
or  else,  if  there  be  no  appreciable  current  passing  in  that  wire,  it 
causes  a  current  to  be  formed  in  that  wire  whose  intensity  varies 
continuously  on  either  side  of  zero-value,  being  now  in  the  one 
direction,  now  in  the  other.  This  induced  current  reproduces 
in  the  mode  of  its  variation  the  complex-harmonic  curve  which 
might  have  been  recorded  by  a  delicate  writing-point  attached  to 
the  vibrating  plate.  The  variable  current  thus  produced  passes 
at  the  receiving  station  through  the  coil  of  a  similar  telephone. 
It  there  causes,  by  induction,  variations  in  the  strength  of  the 


xvi.]  TELEPHONE.  737 

magnet,  which  attracts  the  plate  with  varying  degrees  of  force. 
That  plate  is  either  bent  as  a  mass  towards  and  from  the  magnet, 
or  its  molecules  are  disturbed  by  the  varying  induction  :  or  these 
actions  may  be  combined ;  in  any  case,  the  plate  exerts  varying 
pressure  upon  the  surrounding  air  and  produces  in  it  Sound- 
Waves,  which  approximately  reproduce  in  their  complexity  the 
sound-waves  produced  by  the  original  voice. 

There  are  many  causes  of  distortion  of  the  signals  sent,  both  in  the 
instrument  and  in  the  line.  In  the  latter,  the  higher  harmonics  tend  to  thin 
away  more  rapidly  than  the  graver  components,  and  they  are  propagated  at 
different  speeds:  but  the  resulting  distortion  can  be  reduced  to  a  minimum 
by  lowering  the  line-resistance,  and  would  also  be  reduced  through  increas- 
ing the  inductance  L  by  hanging  the  wires  far  apart,  or  through  increasing 
the  leakage  (Heaviside),  though  this  would  weaken  the  current  reaching  the 
receiving  instrument,  or  through  reducing  the  electrostatic  capacity  of  the 
line.  Mr.  Heaviside  has  shown  that  the  distortion  would  be  zero,  if  R/L  = 
D/C,  where  D  is  the  leakage-conductance,  all  per  unit  of  length. 

It  is  a  matter  of  indifference  to  the  receiving  telephone  by  what  means 
the  variations  of  current-intensity  which  it  reveals  have  been  produced. 
These  may  be  due  to  variations  of  electromotive  D.P.  (vibrations  of  one 
of  the  plates  of  an  electrostatic  condenser  or  oscillatory  variations  in  its 
charge,  —  variations  of  the  potential  of  a  mass  of  mercury  vibrating,  while 
in  contact  with  water,  up  and  down  a  conical  capillary  tube),  or  to  varia- 
tions in  the  total  resistance  (length,  cross-section,  conductivity)  of  the  con- 
ducting wire.  The  conductance  of  the  circuit  may  be  caused  to  vary  by 
squeezing  the  wire,  by  causing  a  certain  length  of  it  to  vibrate ;  or  again 
by  interposing  a  certain  length  of  a  conductor  whose  conductivity  varies  with 
varying  pressure  (microphone)  or  with  varying  illumination  (photophone). 

According  to  Prof.  Tait,  the  variations  of  current  in  an  ordinary  tele- 
phone are  equivalent  to  actual  currents  whose  intensity  is  one-thousand 
millionth  part  of  the  current  ordinarily  used  in  telegraphic  work.  This 
telegraphic  current  may,  on  long  lines,  be  stated  to  be  about  one-sixtieth 
Ampere. 

Page-Effect.  —  A  telephone  will  work  feebly  even  without  any  plate 
P;  the  varying  constraint  of  the  particles  of  the  magnet  M  causes  them  to 
exert  varying  pressure  upon  the  air.  If  a  plate  of  any  substance  be  con- 
nected with  the  extremity  of  M,  that  plate  will  act  as  a  sounding-board, 
and  will  enhance  the  sound  produced. 

Reversed  Action.  —  A  reverse  current  of  high  potential 
sent  through  a  frictional  machine  may  maintain  rotation  in  it, 
so  that  a  stronger  machine  in  circuit  with  a  weaker  one  may 
drive  it  backwards.  If  a  dynamo  deliver  all  its  current  in  one 
direction,  an  extraneous  current  sent  through  the  machine  in 
the  same  sense  causes  a  reversed  rotation  of  its  armature.  In 
consequence  of  this,  if  we  couple  two  direct-current  dynamo- 
electric  machines  by  connecting  wires,  so  that  both  dynamos 
(so-called  for  the  sake  of  brevity)  are  on  the  same  metallic  cir- 

3s 


738  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

cuit,  and  if  we  force  the  armature  of  the  one  into  rotation,  the 
armature  of  the  other  rotates  in  a  reverse  sense  (that  is,  against 
its  brushes  unless  these  be  reversed  or  the  armature-connections 
reversed)  as  soon  as  the  current  transmitted  attains  a  certain 
intensity.  The  distant  dynamo,  which  bears  under  such  cir- 
cumstances the  name  of  Electromotor,  may  be  of  any  size,  and 
the  simple  use  of  a  key  or  commutator  arrests  or  reverses  its 
action  at  will.  The  intensity  of  the  current  passing  round  the 
circuit  is  diminished  by  the  reversed  rotation  of  the  electro- 
motor: this  is  equivalent  to  the  production  of  a  reverse  cur- 
rent by  the  electromotor.  The  usefulness  of  the  arrangement, 
the  proportion  of  the  Energy  Absorbed  by  the  electromotor,  in 
rotating  against  resistances,  to  the  Total  Energy  imparted  by 
water-wheel  or  steam-engine  to  the  driving  dynamo,  is  equal  to 
the  ratio  between  the  intensity  of  the  virtual  reverse-current 
produced  by  the  electromotor,  while  running,  and  the  intensity 
of  the  current  produced  by  the  dynamo  when  the  motor  is 
kept  from  rotating.  This  Utility  or  Efficiency  is  not  to  be 
measured  by  the  relative  rapidities  of  rotation  of  the  electro- 
motor and  dynamo,  on  account  of  the  so-called  dead  turns; 
the  rotation  of  the  dynamo  must  exceed  a  certain  speed  before 
any  current  will  be  produced,  and  the  current  produced  must 
exceed  a  certain  strength  before  the  electromotor  will  turn. 
The  Activity  of  the  arrangement  (i.e.  the  rate  at  which  the 
electromotor  can  do  external  work,  the  amount  of  energy  trans- 
mitted per  second)  is  theoretically  greatest  ( Jacobi's  Law)  when 
the  virtual  reverse-current  is  half  that  produced  by  the  dynamo 
when  the  motor  is  stopped —  that  is,  when  the  exterior  circuit, 
including  the  running  motor,  acts  as  if  it  were  a  wire-circuit 
whose  resistance  is  equal  to  that  within  the  dynamo  itself. 

In  practice  it  is  better  to  make  the  whole  resistance  about  *££-  times  the 
internal  resistance. 

Jacobi's  Law  is  arrived  at  thus:  —  During  each  second  the  external 
work  done  is  w  ergs ;  w  also  represents  numerically  the  Activity  in  ques- 
tion ;  the  energy  converted  into  heat  in  the  whole  circuit  is  PR,  and  the 
energy  provided  by  the  dynamo  is  El,  ergs  per  sec.  Then  El  =  I2R  +  w ; 
a  quadratic ;  whence  I  =  (E  ±  \/E2  -  4Rw)/2R.  The  quantity  (E2  -  4Rw) 
cannot  have,  physically,  a  negative  sign,  for  its  square  root  would  then 
become  an  impossible  quantity.  (E2  —  4Rw)  cannot  be  less  than  zero : 
whence  w  cannot  be  greater  than  E2/4R:  when  it  is  equal  to  E2/4R, 
I  =  E/2R,  and  the  intensity  has  been  diminished  from  E/R  to  E/2R,— 
that  is,  to  one  half,  —  while  the  total  resistance  must  have  been  doubled. 

The  Efficiency-relation  may  be  thus  arrived  at:  —  If  the  dynamo  run 
while  the  motor  is  held  fast,  the  E.M.D.P.  and  current-intensity  will  be 


xvi.]  ELECTROMOTORS.  739 

ED  and  ID.  If  the  motor  were  to  run  at  its  actual  speed  while  the  dynamo  was 
held  fast,  the  reverse-current  produced  by  it  would  be  at  EM  and  IM.  When 
the  two  are  coupled,  the  actual  current  is  (ID  —  IM)  :  the  energy  supplied  by 
the  dynamo  during  each  second  is  w0  ergs  ;  that  taken  up  and  transmitted 
by  the  motor  is  M?M  :  the  resistance  of  the  whole  circuit  is  R  :  and  (TD  -  IM)2  R 
is  the  Heat  developed  in  the  whole  circuit  :  then  the  Energy  supplied  by  the 
dynamo  is  WD  =  ED(ID  -  IM)  =  ED(ED  -  EM)  /R  =  {«,„  +  (ID  -  IM)2R)  = 
(wM  +  (ED  -  EM)  /  R}  ;  whence  the  Efficiency  WM/W»  =  EM/  ED  =  IM/  ID. 
The  ratio  EM/ED,  and  therefore  the  Efficiency,  may  be  raised  by  raising  the 
value  of  EM  :  and  this  may  be  done  by  giving  the  motor  a  small  load,  so  that 
it  may  rotate  rapidly,  or  by  making  its  magnetic  field  a  comparatively  strong 
one  ;  so  that  efficiencies  of  86  per  cent  have  been  attained  at  such  distances  as 
600  metres  (f  mile)  ;  44-8  per  cent  at  36  miles  (Creil-Paris),  with  6000  Volts. 

As  to  the  thickness  of  conducting  wire  necessary,  there  is 
no  limit  other  than  that  imposed  by  the  necessity  of  very  good 
insulation.  An  ordinary  telegraph  wire  could  convey  the  whole 
energy  of  Niagara  Falls,  and  convey  it  to  any  distance  ;  but  the 
wire  would  be  at  so  high  a  potential  that  sparks  would  fly  from 
it  into  the  surrounding  air.  In  the  same  way,  if  the  amount  of 
onflow  of  a  fluid  in  a  pipe  were  found  to  vary  directly  as  the 
motive  difference  of  pressure,  any  amount  of  energy  might  be 
transferred  from  one  place  to  another  by  the  smallest  flow  of 
water,  for  any  water  allowed  to  flow  out  of  the  pipe  might  be 
made  to  escape  with  any  assignable  velocity  ;  provided  always 
that  the  tube  were  strong  enough  at  all  points  to  sustain  at  all 
intermediate  points  the  necessary  pressure. 

If  a  dynamo  of  resistance  5  Ohms,  and  producing  a  difference  of  poten- 
tial of  1000  Volts,  be  the  source,  and  a  similar  machine  be  the  electromotor, 
while  the  connecting  wire  offers  a  resistance  of  R  Ohms,  the  intensity  of  the 

current  produced  is  /  -  —  •  j  Amperes.    If  500  such  dynamos  be  coupled 

in  file,  their  joint  E.D.P.  will  be  500,000  Volts,  and  their  resistances  2500 
Ohms  ;  if  the  receiving  electromotors  be  also  multiplied  five-hundredfold, 
their  resistances  will  be  2500  Ohms  ;  if  the  connecting  wire  be  unaltered,  the 

500  000 
intensity  of  the  current  passing  will  be  '         —  —  Amperes  ;   but  if 


the  connecting  wire  be  also  500  times  as  long  as  at  first,  the  intensity  is 

_  500'000  _  =  (    100°   }  Amperes,  the  same  as  in  the  former  case. 
2500  +  2500  +  500R      (lO  +  R  / 

Though  the  intensity  of  the  current  passing  is  the  same,  the  energy  trans- 
mitted per  second  is  not  the  same  :  it  is  500  times  as  great.     In  the  former 

case  it  is  Intensity  x  E.D.P.  =  (   *000  *}  x  1000  Ampere-Volts  or  Watts:  in 

\  10  -4-  J*/ 

the  latter  it  is  -1M-.  Amperes  x  500,000  Volts  =  500'00°Q00  Watts. 
10  +  R  iy  +  J* 

•p 

When  the  total  resistance  is  the  internal  resistance  Rf,  I  =—>  '•  ;  when  it  is 

F  ?f 

(Rf  -j-  Re),  I,  =  —  =3  —    These  two  distinct  sets  of  circumstances  are  linked 


740  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

together  by—  (1)  The  criterion  of  maximum  activity,  Rt-+e  =  -W  Rfl  and  (2) 
The  energy  imparted  to  the  dynamos  is  a  constant,  =  EJ,  =  w  ergs  per  sec- 
ond.  From  these  equations  we  find  I,=  VT%8ff  w/Ri?  and  E;=  Ri+el,  =  (-^YR-O 
(V^w/R^m  V  -Vg0-  RiW  =  VRHew.  We  must  now  choose  numerical  values 
for  Ri+e,  the  total  resistance  internal  and  external,  and  w,  the  energy  im- 
parted to  the  system  per  second.  We  shall  use  C.G.S.  Electrostatic  units. 

Let  the  resistance  be  that  of  4000  kilometres  of  copper  wire  of  1  sq.  cm. 
in  cross-section,  and  that  of  x  dynamos  and  x  electromotors.  The  dynamos 
are  each  supposed  to  generate  an  E.D.P.  of  1000  Volts,  or  3£  C.G.S.  Electro- 
static units.  Their  number  must  be  (E/  -=-  3i). 

Let  the  joint  resistance  of  each  dynamo  and  motor  be  10  Ohms,  or 
9oo.ooo.°oooooo  C.G.S.  Electrostatic  unit  of  resistance.  The  resistance  of 
the  (E,-*-3&)  pairs  of  machines  will  be  {(E, -s- 3J)  x  90o,ooo.°oooooi)}>  or 

L C.G.S.  Electrostatic  units. 

300,000,000000 

The  wire  (4000  kilometres)  will  offer  a  resistance  of  about  648  Ohms,  or 
Electrostatic  C.G.S.  units. 


3oo,ooo.oooooo 


The  total  resistance  Rt+e  =  goo.ooo'.oooooo  *E/  +  216}  C.G.S.E.S.  units. 

Let  w  be  the  energy  of  the  Falls  of  Niagara,  per  second,  also  in  C.G.S. 
units  or  ergs.  About  100,000000,000000  grammes  of  water  fall  per  hour 
through  a  height  of  about  4830  cm.  The  potential  energy  lost  by  the  water 
is  about  132,000,000000,000000  ergs  per  second  =  w. 

The  equation  E,  =  VRf+ew;  is  now 

E,  -  Vairo>000i000000  {£,  +  216}  x  132,000,000000,000000,  a  quadratic  ;  whence 
E,  =  440,020  C.G.S.E.S.  units  or  132,006000  Volts. 

If  iron  telegraphic  wire  4  mm.  in  diar.  were  used,  its  resistance  would 
be  (the  resistance  of  iron  being  fff  that  of  copper)  31680  Ohms;  the  total 
resistance  would  be  ,00.ooo1.oooooo  (E,  +  10560);  and  E,  =  450,320  C.G.S.E.S. 
units,  or  135,096000  Volts. 

No  practicable  insulation  could  be  set  up,  adequate  to  sustain  perma- 
nently so  great  a  stress  ;  and  no  possible  dynamo-electric  machines  could  be 
ranged  in  file  to  the  number  necessary,  for  the  insulation  of  their  coils 
would  be  broken  down  by  sparks  from  the  wire  to  the  outer  air. 

It  is  practicable,  however,  with  ordinary  telegraphic  wires  insulated  in 
the  ordinary  way,  and  with  a  16-horse-power  dynamo,  to  drive  a  6-horse 
power  electromotor  at  a  distance  of  30  miles. 

The  wire  must  also,  by  possessing  sufficient  thickness,  offer  so  little 
resistance  that  it  is  not  so  far  heated  as  to  deteriorate  in  conductivity. 

Alternating  Current  Motors.  —  If  two  alternating  current 
machines  be  coupled  in  circuit,  and  if  they  be  once  in  synchro- 
nous motion,  they  will  tend  to  assume  and  to  maintain  uniform- 
ity of  phase  ;  while  if  the  mechanical  load  on  the  motor  be 
increased,  within  appropriate  limits,  the  motor  will  present  a 
less  complete  opposition  of  phase,  and  a  stronger  current  will 
run,  so  that  the  mechanical  forces  upon  the  motor-armature  will 
be  correspondingly  increased.  The  magnetic  field  of  such  a 
motor  must  be  kept  constant  in  its  direction.  The  practical 
difficulty  connected  with  such  motors  is  that  they  are  not  self- 
starting. 


xvi.]  ELECTROMOTORS.  741 

At  pages  90-91  we  learned  that  two  S.H.M.'s,  differing  by  |  period,  pro- 
duce an  ellipse.  Similarly  if  we  have  two  S.-H.-varying  currents,  each  tend- 
ing to  produce  an  alternating  magnetic  field  in  its  own  particular  direction, 
but  differing  from  one  another  in  phase,  the  result  will  be  a  continuous 
rotation  of  the  direction  of  the  resultant  magnetic  field.  In  such  a  field 
any  mass  of  metal  will  tend  to  rotate ;  and  this  is  the  basis  of  rotary-field 
alternating  current  motors  (Tesla,  Dobrowolski).  In  Schallenberger's  alter- 
nate-current meter,  a  vertical  coil  receives  the  alternating  current  to  be 
measured ;  within  it  is  fixed  another  vertical  coil,  closed,  and  standing  at 
an  adjustable  angle  with  the  preceding.  Inside  the  latter  is  a  soft-iron  disc, 
horizontal,  pivoted  on  a  vertical  axis  of  rotation.  The  outer  coil  tends, 
alternating! y,  to  magnetise  the  disc  along  a  certain  line  ;  the  inner  presents 
induced  currents,  nearly  opposite  in  phase,  which  tend  to  magnetise  the 
disc,  alternatingly,  along  another  line,  which  makes  an  angle  with  the  pre- 
ceding :  the  joint  action  of  the  two  coils  (or  sets  of  coils)  on  the  disc  is  to 
magnetise  it  in  a  direction  which  itself  continuously  rotates;  the  disc  tends 
to  rotate  with  a  velocity  proportional  to  the  square  of  the  current-strength. 
The  tendency  is  for  the  disc  to  maintain  a  position  in  which  the  retarding 
eddy-currents  in  the  iron  are  a  minimum.  In  such  apparatus  the  result  is, 
as  regards  the  intensity  of  the  induced  magnetisation  (which  tends,  if  the 
component  inducing  forces  be  not  equal  and  at  right  angles  to  one  another, 
and  at  phases  differing  by  exactly  ^  period,  to  present  maxima  and  minima), 
more  uniform  if  three  or  more  S.H.  variations  be  simultaneously  induced 
in  directions  making  equal  angles  with  each  other,  as  in  Brown's  three- 
phase  alternating-current  motor.  Multiple-phase  motors  are  self -starting, 
and  gain  in  speed  until  synchronism  is  attained. 

The  Lauffen-Frankfurt  experiments  of  1891  gave  an  efficiency  of  72  per 
cent  in  the  transmission  of  108  horse-power  over  110  miles,  at  30,000  Volts. 


OSCILLATORY  ELECTROMAGNETIC  DISTURBANCES  IN  FREE 

ETHER. 

Herz's  Experiments. — Two  metallic  plates,  each  say  16  cm. 
square,  are  suspended  in  the  same  plane,  and  are  connected  each 
with  one  terminal  of  an  induction-coil.  They  are  also  almost  con- 
nected with  one  another  by  means  of  wires  terminated  at  their 
free  extremities  by  polished  knobs,  between  which  there  is  a 
small  air-gap.  When  the  induction-coil  is  at  work,  a  stream  of 
sparks  runs  across  this  air-gap,  from  knob  to  knob.  The  electric 
displacements  in  the  region  of  the  spark  are  of  an  oscillatory 
character,  and  are  parallel  to  the  length  of  the  air-gap,  from 
knob  to  knob.  The  lines  of  force  are  accordingly  parallel  to  the 
length  of  that  gap.  This  apparatus  is  called  Herz's  Vibrator. 
Next  we  have  his  Resonator,  which  is  a  single  circle  of  wire, 
broken  by  an  air-gap  between  two  knobs.  This  resonator  has 
its  own  fundamental  period  of  electric  oscillation.  The  vibrator 
sends  out  a  mixture  of  oscillatory  ether-disturbances'of  various 


742  ELECTRICITY  AND   MAGNETISM.  [CHAP. 

periods.  Now  assume  that  the  vibrator  air-gap,  in  its  length, 
runs  East  and  West,  horizontally.  Lay  the  resonator  hori- 
zontally, with  its  air-gap  also  East  and  West,  and  facing  the 
vibrator  centrally.  Then  sparks  pass  in  the  resonator  air-gap. 
Sparks  will  continue  to  pass  in  this  air-gap,  though  with  a 
diminished  striking  distance,  when  the  resonator  is  turned 
round  in  its  own  plane  into  any  position,  so  long  as  it  is 
kept  horizontal.  Now  turn  the  resonator  into  a  vertical  plane, 
which  plane  lies  parallel  to  the  length  of  the  vibrator  air-gap, 
that  is,  East  and  West;  some  sparks  will  pass  should  the 
resonator  air-gap  be  also  parallel  to  the  vibrator  air-gap,  but 
when  it  is  at  right  angles  to  the  same  no  sparks  will  pass. 
Again,  turn  the  resonator  round  so  that  its  vertical  plane  lies 
North  and  South,  or  at  right  angles  to  the  length  of  the  vibrator 
air-gap;  no  sparks  will  pass  at  all,  whatever  be  the  direction 
of  its  air-gap.  It  is  (J.  J.  Thomson)  as  if  the  lines  of  electric 
force  in  the  Leyden-jar  discharge  through  the  vibrator  air-gap 
were  parallel  to  the  length  of  that  gap ;  and  as  if  when  travel- 
ling broadside-on,  outwards  from  that  gap,  they  produced  some 
sparks  when  they  struck  the  resonator  air-gap  in  such  a  way 
that  their  length  coincided  with  its  length,  and  produced  a 
maximum  effect  when  they  struck  the  wire  of  the  resonator 
longitudinally,  so  that  their  lengths  coincide  with  its  length, 
while  at  the  same  time  the  magnetic  induction  was  directed 
along  the  axis  of  the  resonator.  In  the  last  case  there  would 
be  reflexion,  from  end  to  end  of  the  resonator,  of  these  lines  of 
force ;  and  these  lines  of  force  would  oscillate  to-and-f ro  along 
that  resonator,  with  a  result  analogous  to  Resonance  in  Acous- 
tics. Those  disturbances,  radiated  from  the  vibrator,  which  had 
been  in  tune  with  the  resonator  would  be  taken  up  and  piled  up 
by  it,  until  sparks  passed  in  the  resonator  air-gap,  or  until  a 
Geissler-tube  held  in  or  near  that  gap  would  light  up. 

By  this  means  the  Resonator  can  be  used  to  detect  the 
existence  and  the  direction  of  electric  oscillatory  disturbances 
in  the  Ether,  such  as  have  periods  corresponding  to  the  rate  of 
propagation  of  electric  disturbance  along  the  wire  of  the  reso- 
nator, and  to  the  length  of  its  wire  ;  but  the  resonator  will 
respond  to  disturbances  of  a  considerable  range  of  frequencies 
above  and  below  this  limit. 

The  waves  produced  by  this  electric  method  traverse  brick 
walls  with  ease,  but  they  are  reflected  by  metallic  mirrors.  If 
the  resonator  be  used  to  explore  the  path  of  the  reflected  waves, 


xvi.]  HERZ'S  EXPERIMENTS.  743 

it  is  found  that  there  is  interference  between  the  direct  and  the 
reflected  waves,  exactly  as  in  the  case  of  Sound  or  of  Light; 
the  resonator  gives  maxima  of  sparking  or  of  illumination  at 
distances  equal  to  half  a  wave-length  from  one  another ;  and  the 
wave-length,  thus  determined,  is  consistent  with  the  velocity  of 
Light,  together  with  the  frequency  of  vibration  as  calculated 
from  the  dimensions  of  the  resonator.  If  a  large  prism  of  pitch 
be  employed,  it  is  found  that  the  waves  are  refracted  by  the 
pitch ;  and  a  large  lens  of  pitch  acts  in  an  analogous  way.  If 
a  vibrator-gap  be  adjusted  in  the  focus  of  a  mirror  which  consists 
of  a  sheet  of  metal  bent  into  a  parabolic  form,  then  a  suitable 
resonator-gap,  placed  in  the  focus  of  a  similar  mirror  opposite  to 
the  first,  may  give  sparks.  These  waves,  like  Light  polarised  at 
right  angles  to  the  plane  of  incidence  and  reflexion,  fail  at  a 
particular  angle  of  incidence  to  be  reflected  from  a  metal  mirror, 
provided  that  the  vibrator  air-gap  be  in  the  plane  of  incidence. 
If  therefore  these  electromagnetic  waves  are  like  waves  of  Light 
or  Radiant  Heat,  or  Actinic  Radiation,  and  differ  from  these 
in  wave-length  only,  the  electric  oscillations,  as  distinguished 
from  the  magnetic  inductions,  are  at  right  angles  to  the  "  Plane 
of  Polarisation."  The  leading  phenomena  of  Light  —  including 
Reflexion,  Refraction,  the  Angle  of  Polarisation,  and  Polarisa- 
tion itself,  Scattering  by  Haze,  and,  to  a  large  extent,  Metallic 
Reflexion  and  the  change  of  phase  on  transmission  through 
thin  films  of  metal,  along  with  Newton's  Rings  and  the  black 
region  of  a  thin  soap-bubble,  as  well  as  Diffraction  —  have 
been  imitated  on  the  large  scale  by  means  of  these  electro- 
magnetic waves ;  while  the  effect  of  the  Magnetic  Field  on 
Light  can  be  largely  explained  if  it  be  admitted  that  Light  con- 
sists of  such  Electromagnetic  Waves  of  small  wave-length,  in 
which  the  electric  oscillations,  at  right  angles  to  the  direction 
of  propagation,  are  also  at  right  angles  to  the  plane  of  polari- 
sation, while  the  magnetic  inductions  are  in  that  plane. 

The  field  in  the  immediate  neighbourhood  of  the  vibrator  is  in  a  singular 
condition  of  alternate  withdrawal  and  emergence  of  Lines  of  Force :  there 
are  various  peculiarities  in  the  amount  of  the  electric  force  and  in  the 
velocity  of  propagation  at  that  place ;  but  the  upshot  is,  that  after  inter- 
ference between  lines  passing  outward  and  lines  re-absorbed  has  had  full 
swing,  the  ether-waves  emerge  as  if  from  wave-centres  about  half  a  wave- 
length from  the  vibrator-gap,  and  the  electric  forces  are  thereafter  exactly 
at  right  angles  to  the  direction  of  propagation,  and  vary  inversely  as  the 
square  of  the  distance. 

If  an  electrostatically-charged  body  could  be  whirled  roun^a  magnetic 
needle  at  the  rate  of  30,000,000000  cm.  per  second,  it  ought  to  act  upon  it 


744  ELECTRICITY   AND   MAGNETISM.  [CHAP. 

much  in  the  same  way  as  a  circulating  electric  current.  At  very  high 
speeds,  such  as  are  physically  within  our  reach,  such  an  effect  should  be 
observed  in  small  degree;  and  Prof.  Rowland  of  Baltimore  has  succeeded  in 
making  it  manifest. 

Maxwell's  Theory  of  Light.  —  These  results,  obtained  by 
Herz  and  others,  furnish  a  verification  of  Clerk  Maxwell's 
Theory  of  Light. 

According  to  this  theory,  the  Electric  Displacements,  parallel  to  the 
•wave-front,  and  at  right  angles  to  the  plane  of  polarisation,  are  the  cause 
of  Optical  Phenomena.  The  Magnetic  Inductions  or  Displacements,  at 
right  angles  to  the  preceding,  and  parallel  to  the  plane  of  polarisation,  but 
also  parallel  to  the  wave-front,  produce  no  effect  on  the  eye.  The  electric 
displacements  will  be  propagated,  in  an  seolotropic  medium,  in  the  same 
way  and  with  the  same  velocity  as  a  light-wave;  and  the  magnetic  dis- 
turbance, whose  energy  is  equal  to  that  of  the  electric,  is  propagated  with 
the  same  velocity.  The  intensity  of  Light,  its  average  energy  per  cub.  cm., 
is  27r/xo-2v2,  where  v  is  the  velocity  of  propagation,  cr  the  maximum  electric 
displacement  per  sq.  cm.  at  the  ends  of  the  lines  of  electric  force,  and  p 
the  permeability  of  the  medium.  In  a  doubly-refracting  medium,  the 
electric  displacement  and  the  magnetic  induction  are  propagated  according 
to  Fresnel's  wave-surface.  There  is  no  dilatational  wave  possible  :  and  this 
removes  a  difficulty  in  optical  theories. 

Since  in  non-conductors  an  electric  Displacement  produces  an  Electric 
Restitution-Force  which  varies  as  the  Displacement — a  criterion  of  vibratory 
movement  propagated  with  a  definite  velocity ;  but  in  conductors  no  such 
force  is  manifest,  and  the  energy  of  electric  disturbance  is  continuously 
dissipated  by  transformation  into  Heat :  then  Light-vibrations  ought  not 
to  be  possible  in  Conductors,  which  should  be  always  opaque,  while  non- 
conductors ought,  if  homogeneous,  to  be  transparent.  With  few  exceptions 
this  is  the  rule. 

The  velocity  of  propagation  of  an  electromagnetic  disturb- 
ance is  the  same  as  the  ratio  of  the  electromagnetic  to  the 
electrostatic  unit  of  current-intensity  or  quantity.  This  ratio 
is  experimentally  found  to  be,  on  the  C.G.S.  system,  in  round 
numbers,  equal  to  30,000,000000 ;  which,  in  cm.  per  second, 
coincides  sufficiently  with  the  velocity  of  light. 

On  comparing  the  formulae  for  the  transverse  variations  of  an  elastic 
solid  with  those  worked  out  to  represent  the  stresses  in  an  Ether  concerned 
in  electromagnetic  phenomena,  it  is  found  (Clerk  Maxwell,  Elect,  and  Magn., 
vol.  ii.,  chap,  xx.)  that  in  the  former  a  term  V,  the  velocity  of  propagation 
of  transverse  disturbances,  occupies  the  same  place  as  1/VK  in  the  latter. 
By  our  electrostatic  convention,  in  vacuo  K  =  1  and  /x  =  l/V2;  .-.  V  = 
(K/x)"*;  but  electromagnetically,  K  =  1/V2  and  /A  =  1 ;  whence  V  =  K~* 
=  V;  i.e.  the  velocity  in  vacuo  is  equal  to  the  ratio-number  V.,  With  other 
media  than  air,  (K/z)~5  has  other  values ;  but  /A  is  nearly  unity  in  most  non- 
conductors; roughly,  V  =  Vl/K;  but  in  Light,  v  =  V,  and  varies  inversely 
as  the  Refractive  Index;  whence  the  Specific  Inductive  Capacity  K  of  a 
dielectric  ought  (Maxwell)  to  be  equal  to  /32,  where  fi  is  the  refractive  index 


xvi.]  MAXWELL'S   THEORY  OF  LIGHT.  745 

for  waves  of  infinite  length.*  In  some  substances  this  is  the  case;  it  is 
so  in  sulphur  (Romich,  Nowak,  Boltzmann),  and  in  turpentine,  petroleum, 
and  benzol  (Silow)  ;  but  in  vegetable  and  animal  oils  (Hopkinson)  and  in 
glass,  Iceland  spar,  fluor  spar,  and  quartz  (Romich  and  Nowak),  and 
generally  in  substances  not  simple  in  chemical  constitution  or  homogeneous 
in  structure,  the  sp.  ind.  cap.  is  too  great.  We  ought  not,  however,  to  ex- 
pect more  than  a  general  agreement ;  even  the  oscillations  produced  by  a 
Leyden-jar  discharge  are  millions  of  times  less  frequent  than  those  electro- 
magnetic alternations  which  we  call  Light ;  and  it  appears  that  the  sp.  ind. 
caps,  for  these  already  approach  the  values  required  by  the  theory. 

The  theory  explains  most  optical  phenomena  as  a  part  of  electrical 
science ;  but  it  is  still  weak  in  its  treatment  of  Dispersion,  both  ordinary 
and  anomalous,  of  Metallic  Reflexion,  and  of  the  rotation  of  the  plane  of 
polarisation  by  magnets.  It  does  not,  however,  profess  yet  to  explain  the 
interaction  of  Ether  and  ordinary  Matter. 


THE  ETHER. 

The  properties  of  the  Ether  which  are  involved  in  the  phe- 
nomena of  Electricity,  Magnetism,  Electromagnetism,  Light, 
Radiant  Heat,  and  Actinic  Radiation  have  been  referred  to  under 
these  several  heads,  where  required.  We  can  now  merely  note  a 
few  remaining  points.  In  terms  of  Maxwell's  Theory,  it  is  sup- 
posed that  there  is  some  kind  of  Rotation  round  each  Line  of 
Force,  upon  which  rotation  the  Elasticity  of  the  Ether  may 
depend :  and  the  facts  of  Electrolysis  or  of  the  Galvanic  Cell, 
in  which  the  charge  that  can  be  liberated  upon  an  electrode  or 
plate  is  limited  to  a  definite  quantity  per  free  atom,  seem  to  show 
that  the  Lines  of  Force  are  not  indefinite  but  definite  in 
number,  so  that  we  may  perhaps  have  (J.  J.  Thomson)  one 
line  of  force  between  each  pair  of  free  atoms.  But  these  lines 
of  force,  on  this  view,  need  not  all  have  free  ends  situated  upon 
matter:  there  may  be,  in  the  Ether,  closed  lines,  like  vortex- 
rings  :  and  these  may,  in  a  Magnetic  Field,  be  so  directed  as  to 
take  up  a  position  parallel  to  one  another. 

Again,  if  Ether  be  a  form  of  Matter,  it  ought  to  have  Iner- 
tia. In  Self-induction  it  appears  to  have  inertia ;  but  in  a 
Steady  Current  it  appears  to  have  absolutely  none.  The  only 
way,  and  mathematically  an  easy  way,  out  of  the  difficulty  seems 
to  be  to  assume  that  the  Ether  is  really  double  in  its  consti- 
tution, so  that  when  we  say,  for  example,  that  positive  lines  of 

*If  j3,  j3,,  be  the  refractive  indices  corresponding  to  the  respective  wave-lengths, 
X  and  X,,  we  know  that  to  a  rough  approximation  ]8  =  A  +  B/X,  where  A  and  B  are 
constants,  found  by  experiment.  From  these,  knowing  the  numerical  values  of 
j8,  j8(,  X,  X,,  we  can  find  that  of  A,  which  is  the  approximate  value  of  j3  when  X  =  « . 


746  ELECTRICITY  AND   MAGNETISM.  [CHAP,  xvi.] 

force  move  in  a  certain  direction,  we  ought  to  add  that  an  equal 
number  of  negative  lines  move  in  an  opposite  direction,  shearing 
past  the  former:  the  apparent  inertia  would  then  be  zero  in  a 
steady  current.  This  would  make  a  steady  current  really  a 
double  current,  positive  one  way  and  negative  the  other ;  but  in 
Electrolysis  we  actually  have  a  double  transfer  of  atoms,  to 
which  this  would  precisely  correspond ;  and  the  phenomena  of 
the  electric  spark  point  in  the  same  direction.  The  sp.  ind. 
cap.  of  a  dielectric  would,  on  this  view  (Lodge),  correspond  to 
a  Shearability,  while  the  permeability  p  at  the  same  time 
represented  an  Inertia  or  Density  of  the  Ether. 

Mr.  W.  Williams  has  shown  (Proc.  Phys.  Soc.  Lond.,  xi.  357)  that  we 
are  practically  restricted  either  to  this  view  or  to  the  opposite,  that  K  is  a 
Density,  or  inertia  per  unit  volume,  and  /x  a  Shearability.  On  the  former 
view,  the  magnetic  energy  of  the  field  is  kinetic  :  and  if  we  take  care  to  keep 
in  view  the  proper  Direction  of  each  element  of  Length  which  enters  into 
the  Dimensional  Equations,  these  equations  themselves  bring  out  the  closest 
analogies  between  magnetic  and  electric  phenomena  and  those  of  vortex- 
motion  and  transverse  stresses  in  an  incompressible  fluid.  The  Dimensions 
of  fj.  in  this  view  would  be  M/L8,  where  the  three  L's  are  at  right  angles  to 
one  another,  so  that  their  product  truly  represents  a  volume  ;  and  that  of  K 
would  be  T2L/M  or  more  properly  T2L2/LM;  where  the  three  directions 
have  again  to  be  distinguished.  Such  an  expression  as  [h]  —  [Mi/LsT/n*] 
then  becomes  [h]  =  [L/T],  and  h  is  a  Linear  Velocity  along  the  Lines  of 
Force ;  and  so  forth.  This  tends  to  elucidate  the  purely  mechanical  aspect 
of  magnetic  and  electrical  phenomena  in  the  Ether :  and  the  paper  should 
be  consulted. 

But  the  subject  is  still  more  obscure  when  we  consider  the 
relation  of  Ether  to  ordinary  Matter.  Why  the  magnetic  in- 
duction under  a  given  magnetising  force  should  be  300  times  as 
great  in  a  particular  sample  of  iron  as  in  a  corresponding  amount 
of  air  or  of  copper  is  still  a  mystery ;  and  even  if  it  were  estab- 
lished that  the  density  of  the  Ether  was  greater  in  iron,  that 
would  itself  have  to  be  explained. 

Meantime,  it  will  be  kept  in  view  that  all  our  statements  as 
to  positive  and  negative  quantities  and  currents  are  based  upon 
the  purely  arbitrary  convention  that  vitreous  electricity 
is  positive,  and  resinous  electricity  negative.  This  convention 
happens  to  harmonise  with  that  which  regards  the  north- 
seeking  end  of  a  magnet  as  its  positive  pole;  and  thus 
uniformity  of  language  throughout  the  subject-matter  of  this 
chapter  happens  to  have  been  readily  attained. 


APPENDIX. 


Notation. — In  adjusting  the  notation  used  in  this  volume,  the  pur- 
poses kept  in  view  have  been  to  provide,  as  far  as  might  be,  for  the  whole 
subject-matter  of  the  book,  but  to  depart  as  little  as  possible  from  the 
symbols  ordinarily  in  use,  while  the  letters  employed  should  be  distinctive 
and  at  the  same  time  typographically  suitable.  One  guiding  principle  has 
been  to  separate  physical  quantities  in  general  from  the  same  quantities  per 
unit  of  area,  where  such  a  distinction  seemed  needful,  by  using,  in  order  to 
represent  these,  capital  and  small  letters  respectively.  Then  Mr.  Oliver 
Heaviside's  suggestion  as  to  the  use  of  blackf aced  type  for  directed  quantities 
was  found  to  promise  to  work  well,  and  to  enable  useful  distinctions  to  be 
made.  Thus  F  is  a  force  acting,  in  general;  f  is  a  force  per  unit  of  area;  and 
f  is  a  force  acting  per  unit  of  area  in  some  given  direction.  Again,  B  is  a 
total  magnetic  induction,  b  is  a  magnetic  induction  per  sq.  cm.;  and  both 
these,  being  blackfaced,  call  to  mind  the  Lines  or  directions  in  which  the 
induction  acts.  Other  typographical  devices  have  had  to  be  employed  for 
the  sake  mainly  of  obtaining  a  larger  number  of  distinctive  characters;  but 
it  is  hoped  that  none  of  these  will  offend  the  eye,  and  that  the  grouping  is 
reasonably  consistent.  Perfect  symmetry  seems  hardly  attainable,  on 
account  of  the  varying  demand  for  the  different  letters  in  different  parts  of 
the  subject.  Still,  even  with  the  notation  as  it  stands,  it  has  been  interest- 
ing to  the  author  to  note  in  how  many  instances  the  mere  necessity  of 
ascertaining  which  symbol  ought  to  be  employed  has  enabled  him  to  set 
matters  forth  with  greater  definiteness  than  in  the  former  editions. 

In  its  issue  of  Aug.  25,  1894,  the  Electrical  World  of  New  York  has 
published  the  recommendations  of  the  Committee  on  Notation  of  the 
Chamber  of  Delegates  of  the  International  Electrical  Congress,  Chicago, 
1893.  These  recommendations  as  to  notation  are  at  present  the  subject  of  a 
good  deal  of  discussion,  and  it  remains  to  be  seen  to  what  extent  they  will 
be  generally  adopted.  In  the  meantime  they  are  recommendations  only, 
and  will  have  to  ,be  further  considered  when  an  International  Electrical 
Congress  next  takes  place ;  but  the  following  conspectus  shows  their  rela- 
tions to  the  notation  employed  in  this  volume. 

It  is  not  explained  what  distinctions  the  respective  large  and  small 
letters  denote,  if  any;  the  same  Dimensions  apply  to  both  the  large  and 
the  small  letters.  The  measurement  in  the  fourth  and  sixth  columns  is  in 
electromagnetic,  not  in  electrostatic,  units.  The  manuscript  type  in  the 
last  column  is  that  known  as  the  French  Script  of  Messrs.  Damon  &  Peets, 
New  York. 

747 


748 


APPENDIX. 


THIS  VOLUME. 

COMMITTEE. 

THIS  VOLUME. 

COMMITTEE. 

THIS  VOLUME. 

COMMITTEE. 

I 

£,« 

g 

Q,q 

tn 

m 

m 

ir 

e 

E,  e  for  so-called 

h 

5C 

t 

r,< 

E.M.F. 

H 

(P 

A 

#,  s 

J7,M  for  difference 
of  potential 

m 

3E 

SB 

v 

it 

3 

Angle  5,  0,  £ 

0,0 

(C) 
i 

(7,  c 
I,* 

b 

V 

V 

(T>} 

6? 

K 

/c 

H 

0) 

*    ' 

'  ^ 

/* 

A 

(T\*\ 

7 

d 

$ 

'      ' 

L 

z,  z 

F 

F,f 

(R) 

£,  r 

Magnetomotive 

-f-J?    «' 

W 

W 

(R) 

P 

Force 

ff 

Activity 

Power,  P 

W 

Activity 

TF 
Power,  P 

Reluctance  of 
Circuit 

'  (R 

P 

P 

N 

K 

Reluctivity 

"=1/" 

Dimensions.  —  In  the  Dimensions  given  in  this  volume  from  time  to 
time,  it  will  be  observed  that,  for  example,  a  Torque  has  Dimensions 
ML2/T2,  while  those  of  Energy  are  the  same.  There  are  other  instances  of 
the  same  kind.  So  long  as  we  use  the  Dimensions  only  for  checking  our 
equations  numerically,  or  for  translating  from  one  system  of  units  to 
another,  this  identity  of  expression  between  physical  quantities  which,  like 
the  two  in  question,  are  truly  dissimilar,  is  of  no  importance ;  but  if  we  wish 
the  Dimensional  Equations  to  convey  to  us  an  idea  as  to  the  physical  real- 
ity lying  behind  them,  we  must  find  some  means  of  introducing  into  them  a 
representation  of  the  Directions  involved.  Now  the  symbol  V  — 1  signifies 
a  Rotation  through  90°,  for  the  operation  which  it  represents  would,  if 
effected  twice,  convert  a  directed  quantity  x  into  —  x,  that  is,  would  turn  its 
direction  round  through  180° ;  and  it  has  been  proposed  to  distinguish  the 
dimensions  of  a  Torque  from  those  of  Energy  or  Work,  by  introducing  the 
factor  V  — 1  into  it.  The  expressions  would  then  be  ML/T2-  LA/  —  1  and 
ML2/T2  respectively ;  and  the  former  of  these  would  show  that  the  second 
L  was  at  right  angles  to  the  first,  while  in  the  other  case  both  the  L's  are  in 
the  same  direction.  Mr.  Williams,  in  the  paper  referred  to  on  p.  746,  has 
shown  that  this  idea  is  capable  of  great  extension  by  means  of  keeping  the 
three  rectangular  axes  of  direction,  X,  Y,  and  Z,  entirely  separate,  so  that 
the  corresponding  L's  do  not  cancel  on  division  or  multiplication  unless  they 
be  in  the  same  direction.  The  Dimensional  Equations  thus  acquire  a 
deeper  significance  and  an  enhanced  utility. 

4ir.  —  In  the  formulae  of  Chap,  xvi.,  the  factor  4?r,  or  some  multiple  or 
sub-multiple  or  power  thereof,  appears  with  painful  -frequency.  Mr.  Oliver 
Heaviside  has  pointed  out  that  this  is  due  to  the  total  flux  of  force,  or  rather 
of  induction,  I  or  B,  round  a  quantity  Q  or  m,  being  taken  as  equal  to  4?rQ  or 
47rin,  as  the  case  may  be.  This  is  itself  a  necessary  consequence  of  so  choos- 
ing our  units  that  the  force,  F  dynes,  between  two  equal  quantities  Q  (or  m) 
is  Q2/Kd2  (or  m2//^2)-  Mr-  Heaviside  proposes  that  we  should,  while 


APPENDIX.  749 

retaining  the  dyne  as  our  unit  of  force,  so  alter  our  units  of  electrical  and 
magnetic  quantity  as  to  make  the  force  F  =  Q2/47rKd2  or  m2/47r/W2.  In 
other  words,  he  proposes  to  make  the  units  such  that  not  4?rQ  or  47rtn,  but  Q 
or  tn  lines  radiate  from  each  quantity  Q  or  m,  as  measured  in  the  new  units. 
Then  the  new  numerical  jralue  Qr  or  tn^  which  stands  in  the  place  of  the  old 
Q  or  m,  is  equal  to  Q  V4?r  or  to  m  V4^,  as  the  case  may  be.  Whence  the 
new  units,  which  Mr.  Heaviside  calls  rational  units,  of  quantity  ai*e 
smaller  than  the  present  air-units  in  the  ratio  of  1  to  V^TT.  If  this  change 
were  effected,  most  of  the  units  used  in  electricity  and  magnetism  would 
have  to  be  changed  at  the  same  time.  If  we  make  the  suffix  r  signify  that 
the  number  indicated  by  the  letter  is  now  to  be  a  number  expressing  the 
same  physical  quantity  in  terms  of  the  new  units,  we  find  that  Qr/Q  =  ov/<r 
=  Tr/I  =  E/Er  =  <|>/«j,r  =  V4TT  ;  uir/m  -  h/hr  =  b/br  =  Sr/«  =  G/a  =  Vi^  ; 
and  further,  R/Rr  -  Cr/C  =  L/Lr  =  M/Mr  =  4?r;  while  K  and  /x  remain 
the  same.  The  fundamental  electrostatic  equations  would  then  become 
<|>r  =  orr,  ir  =  K<|>r,  /=  o>2/2K,  <l>r  =  Er/d;  or,  dropping  the  suffixes  and  con- 
fining ourselves  to  air,  $  =  o-  =  i  ;  /=  cr2/2  ;  <|>  =  E/rf;  whence  V,  —  Vy/  =  E 


/=  E2/2</2  =  <|>2/2  =  o-2/2  =  i2/2  =  4«r/2  =  icr/2  =  <|>i/2  =  Energy  of  Field 
per  cub.  cm.  ;  and  C  =  KA  /d,  or,  for  a  unit  cube  condenser,  C  =  K,  while 
the  Dielectric  Elasticity  =  1/K.  In  the  magnetic  and  electromagnetic  parts 
of  the  subject  there  would  be  corresponding  simplifications.  The  factor  4?r 
would,  however,  make  its  appearance  in  other  formulae  where  it  does  not  at 
present  occur,  but  only  where  the  conditions  of  the  problem  are  truly  spheri- 
cal and  not  merely  superficial  or  linear  ;  for  example,  instead  of  b  =  tn/r2  per 
sq.  cm.  at  a  distance  r  from  a  central  pole  m,  we  would  have  br  —  nv/47rr2 
=  Br/area,  which  is  a  better  representation  of  the  fact.  This  change  of  the 
fundamental  units  of  electrical  and  magnetic  measurement  would  involve 
the  study  of  an  additional  system  of  units.  But  it  is  not  clear  that, 
admitting  the  reasoning,  the  proposal  goes  far  enough.  The  relation  be- 
tween the  Unit  of  Mass  and  the  Unit  of  Force  is  equally  based  upon  neglect  of 
the  central  nature  of  gravitational  forces,  and  of  the  Field  of  Force,  with  its 
Lines  of  Force  and  Equipotential  Surfaces,  round  an  attracting  mass.  If 
the  same  reasoning  were  here  applied,  either  the  unit  of  mass  or  the  unit  of 
force,  or  both,  would  have  to  be  changed.  If  the  gramme  were  retained  as 
our  unit  of  mass,  and  if  the  Force  of  Gravitation  between  two  equal  masses, 
m  grammes  each,  at  distance  d,  were  written  Gs  =  ra2/47rfZ2,  the  new  unit  of 
force  would  be  equal  to  4?ry  dynes.  Then,  between  two  equal  electrical  quan- 
tities Q,  as  measured  in  the  present  C.G.S.  units,  the  force  would  be 
FR  (  =  F  /  47ry)  =  QB2/  4fl-Krf2  ;  and  QR2  =  FR  •  4arKd*  =  F/  4-jry  x  4rrK</2  = 
FKd2/y  =  QY  y  5  so  tna*  tne  effect  would  be  to  set  up  still  another  system 
of  electric  and  magnetic  measure,  in  which  the  units  of  electric  and  mag- 
netic quantity  would  be  equal  to  Vy  x  the  present  C.G.S.  units;  or  else,  if 
y  were  left  in  the  formula  for  G,  with  its  present  value  unaltered  (GR= 
y  •  w  2/47rd2),  we  would  have  our  unit  of  force  equal  to  4?r  dynes,  and  our 
unit  of  mass  equal  to  {one  gramme  -*•  Vy},  while  we  would  at  the  same  time 
restore  the  present  C.G.S.  units  of  electric  and  magnetic  quantity,  etc.  If 
we  took  the  "  astronomical"  unit  of  mass,  1/y  gramme,  as  our  standard,  our 
unit  offeree  would  be  47r/y  dynes,  and  the  electrical  andjnagnetic  units  of 
quantity  would  be  equal  to  the  present  C.G.S.  units  •*•  Vy.  If  we  adhered 
to  the  dyne  as  our  unit  of  force,  although  we  based  it  on'the  equation 
F  =  ma,  we  would  have  to  make  our  standard  unit  of  mass  equal  to  (one 


750  APPENDIX. 

gramme  -s-  V4?ry}  and  our  unit  of  acceleration  equal  to  V4?ry  cm.-per-sec.  per 
second ;  and  the  electrical  and  magnetic  units  of  quantity  would  then  bear 
to  the  present  C.G.S.  units  the  ratio  of  1  :  V47r,  as  in  Mr.  Heaviside's  pro- 
posed system. 

It  seems  at  any  rate  clear  that  if  the  principle  were  thoroughly  applied 
throughout  the  whole  of  Physics,  then,  though  the  electrical  formulae  were 
simplified,  the  4-n-  would  be  transferred  from  the  more  remote  parts  of  the 
subject  to  its  very  threshold ;  and  it  is  doubtful  whether  this  would  not 
cause  more  inconvenience  than  it  would  remedy. 


BIBLIOGRAPHY. 


THE  following  representative  list  may  perhaps  be  found  helpful  to  the 
student  in  the  course  of  his  further  reading :  — 

History  of  Physics.  —  Poggendorff's  "  Geschichte  der  Physik."  —  Heller's 
"Geschichte  der  Physik"  (2  vols.).  —  Rosenberger's  "Geschichte  der 
Physik"  (3  vols.).  —  Mach,  "The  Science  of  Mechanics"  (London: 
Watts  &  Co. ;  Chicago  :  The  Open  Court  Co.,  1893).  —  Montucla,  "  Histoire 
des  Mathematiques  "  (for  Optics  and  Mechanics).  — Hoppe,  "  Geschichte 
der  Elektricitat "  (Leipzig:  Earth,  1884).  —  Weber,  "Geschichte  der 
Optik."  —  Marie,  "Histoire  des  Sciences  math6matiques  et  physiques" 
(12  vols.).  — Mendenhall,  "A  Century  of  Electricity." 

Tables  of  Constants.  —  Clarke's  "Constants  of  Nature"  (in  course  of  publica- 
tion by  the  Smithsonian  Institute  of  Washington,  D.C.,  U.S.A.  —  Car- 
nelley,  "  Melting  and  Boiling  Points "  (2  vols.)  (London  :  Harrison  & 
Sons).  —  Everett's  "Units  and  Physical  Constants."  —  Landolt  und 
Bornstein,  " Physikalisch-chemische  Tabellen"  (Berlin:  Springer). — 
Lupton,  "Numerical  Tables  and  Constants"  (Macmillan).  —  Bieder- 
mann,  "Chemiker  Kalender." — Nippoldt  und  Uppenborn,  "Kalender 
f  iir  Elektrotechniker  "  (Munchen:  Oldenbourg). 

Text  Books. — Jamin  et  Bouty,  "Cours  de  Physique"  (2  vols.).  —  Miiller- 
Pouillet-Pfaundler,  "Lehrbuch  der  Physik  "  (3  vols.).  —  Verdet,  CEuvres 
(with  excellent  bibliographies). — Violle,  "Cours  de  Physique."  —  Fer- 
net, "  Cours  de  Physique."  —  Reis,  "Lehrbuch  der  Physik." — Wullner, 
**  Lehrbuch  "  and  "  Kompendium."  —  Atkinson's  Ganot's  "  Physics."  — 
Everett's  Deschanel.  —  Miller,  "Chemical  Physics."  —  Kirchhoff,  "Vor- 
lesungen  iiber  d.  mathem.  Physik."  —  Winkelmann,  "  Handbuch  der 
Physik"  (vols.  31-33  of  Trewendt's  Encyclopedic  der  Naturwissen- 
schaften,  Breslau).  — Anthony's  "Physics." — Barker's  "Physics." 

Fick,  "  Die  medicinische  Physik."  —  Wundt,  "  La  Physique  niSdicale  " 
(French  translation  by  Monoyer).  —  Grehant,  "Manuel  de  Physique 
medicale."  —  Desplats  et  Gariel;  "  Nouveaux  Elements  de  Physique 
medicale."— Hoh,  '.'Die  Physik  in  der  Medicin." —Hermann,  "Physi- 
ologie"  (5  vols.). — Milne-Edwards,  "Physiologic"  (14  vols.).  —  Mac- 
Gregor-Robertson,  "  Physiological  Physics."  —  Draper,  "  Medical  Physics." 

Works  of  Reference.  —  Watt's  "Dictionary  of  Chemistry,"  with  supplements. 
—  The  physical  articles  in  the  "  Encyclopaedia  Britannica,"  9th  edition. — 
"Jahresberichte  der  Fortschritte  der  Physik"  (yearly  classified  abstracts 
published  by  the  Berlin  Physical  Society,  generally  about  five  years  in 
arrears,  but  otherwise  of  extreme  value).  —  Berghaus,  '^Physikalischer 
Atlas." 

751 


752  BIBLIOGRAPHY. 

Tolhausen,  "  technologisclies  Worterbuch,  deutsch-franzosisch-eng- 
lisch."  —  Werslioven,  "  technologisches  Worterbuch,  deutsch-franzosisch 
mid  deutsch-englisch." —  Eger  und  Brandes,  "technol.  Worterb.,  in 
englischer  und  deutscher  Sprache"  (Braunschweig:  Vieweg). — Rohrig 
and  Kermarsch,  do.,  Eng.-Fr.-Germ.  —  Offinger,  do.,  Eng.-Fr.-Ital.-Germ. 
(4  vols.,  2|-  mk.  each,  Stuttgart,  1890).  —  Andreeff,  "Diet,  technol.," 
Fr.-Russ.-Germ.-Eng.  (St.  Petersburg,  1884). 

Periodicals.  —  Current  Summaries  and  News.  —  "Nature."  —  "English  Me- 
chanic." —  "  Scientific  American."  —  "  The  Electrical  Review."  —  "  The 
Electrician"  of  London.  —  Ditto  of  New  York.  —  "Electrical  World," 
New  York.  —  "  Centralblatt  fur  Elektrotechnik."  —  "  Elektrotechnische 
Zeitschrift."  —  "  Journal  of  the  Chemical  Society  "  (for  Chemical  Physics). 

—  Silliman's  "  American  Journal  of  Science  and  Arts."  — Exner's  "  Reper- 
torium  der  Physik."  —  "  Zeitschrift  fur  Instrumentenkunde." 

Original  Papers.  —  "Philosophical  Transactions  of  the  Royal  Society  of 
London."  —  "  Comptes  rendus  de  1'Academie  des  Sciences."  —  "  Abhand- 
lungen,  physikal.,  d.  Berliner  Akademie."  —  Poggendorff's  and  Wiede- 
mann's  "Annalender  Physik  und  d.  Chemie"  and  "Beiblatter"  thereto. 

—  "  Annales  de  Chimie  et  de  Physique."  —  "London,  Edinburgh,  and 
Dublin  Philosophical  Magazine."  —  "Journal  of  the  Physical  Society  of 
London."  —  "The  Physical  Review."  —  "Archives  des  Sciences  physiques 
et  naturelles."  —  "Bulletin  des  Sciences  physiques."  —  "Journal  de  la 
Physique  theorique  et  appliquee." —  "Proceedings  of  the  Royal  Society 
of  London."  —  "Proceedings  and  Transactions  of  the  Royal  Society  of 
Edinburgh."—"  Collection  de  m^moires  rel.  a  la  Physique,"  (5  vols.)  (Soc. 
Franc_.  de  Physique  :  Paris,  Gauthier-Villars). 

See  Royal  Society  Catalogue  of  Scientific  Papers,  for  lists  of  original 
papers  under  Authors'  names:  vols.  i.-ix.,  to  end  of  1883  (Camb.  Univ. 
Press). 

Dynamics,  etc.  —  Sir  Isaac  Newton,  "Philosophise  Naturalis  Principia  Mathe- 
matical' — Thomson  arid  Tait,  "Treatise  on  Natural  Philosophy,"  vol.  i., 
2  parts,  new  edition.  —  Clerk  Maxwell,  "Matter  and  Motion."  —  Clifford, 
"Elements  of  Dynamic"  (unfinished). — J.  J.  Thomson,  "Application 
of  Dynamics  to  Physics  and  Chemistry."  —  Minchin,  "  Uniplanar  Kine- 
matics."—  Minchin,  "Statics,  with  application  to  Physics." — Besant, 
" Dynamics."  —Ball,  "  Mechanics."— Goodeve,  " Mechanics."  — Barthol. 
Price,  "  Analytical  Mechanics "  (2  vols.). — Routh,  "Analytical  Statics," 
vol.  i.  —  Routh,  "Dynamics  of  a  System  of  Rigid  Bodies"  (elementary 
part). — Airy,  " Undulatory  Theory  of  Optics."  —  Airy  on  "Tides  and 
Waves."  —  Weber,  "  Wellenlehre."  —  Wernicke,  "  Mechanik."  —  Streintz, 
"  Grundlage  der  Mechanik."  —  George  Green,  "  The  Application  of  Mathe- 
matical Analysis  to  the  Theories  of  Electricity  and  Magnetism"  (Notting- 
ham, 1828;  facsimile,  Berlin,  1889) .  —  Clausius, "  Die  Potentialf unction  u.  d. 
Potential."— Rankine,  "Applied  Mechanics."  —  Cotterill's  "Applied 
Mechanics"  (bibliographical  references)  and  "Lessons"  in  do. — Jamie- 
son,  "Elementary  Applied  Mechanics."  —  Selby,  "Elementary  Mechanics 
of  Solids  and  Fluids."  —  Goodeve,  "  Mechanism."  — -.Kennedy,  "Mechanics 
of  Machinery."  —  Twisden,  "Practical  Mechanics." — Alexander,  "Ele- 
mentary Applied  Mechanics."  —  Chalmers,  "  Graphical  Determination  of 
Forces." —Chambers,  "  Astronomy."  —Robert  H.  Smith,  "Graphics." 

—  Cremona,  "Graphical  Statics."  —  Clarke,  "Graphic  Statics."  — Gray 
and   Lowson,    "Graphical    Arithmetic    and    Statics"    (elementary). — 


BIBLIOGRAPHY.  753 

Wormell,    "  Plotting."  —  Newcomb,    "  Astronomy."  —  Langley,    "  The 
New  Astronomy." 

Tait  and  Steele,  "Dynamics  of  a  Particle."  —  Hicks,  "Elementary 
Dynamics  of  Particles  and  Solids."  —  Somoff,  "  Theoretische  Mechanik." 

—  Lagrange,  "  Mecanique  Analytique."  — Laplace,  "  Mecanique  CSleste." 

—  Poisson,  "  Mecanique." — Poinsot,  "Rotation." — Worthington,  "Dy- 
namics of  Rotation."  — Perry,  "  Spinning- Tops."  —  Cayley,  "  Reports  on 
Theoretical  Dynamics,  British  Association  Reports,  1846,  1857,  1862."  — 
Jacobi's  "Gesammelte  Werke."  —  "Archimedis  opera,"  Leipzig  (3  vols., 
12mo.).  — Abel,  "CEuvres"  (2 vols.)  (Christiania) .  — Fourier,  "CEuvres" 
(2  vols.). — Galileo,  "Opere,"  Florence  (16  vols.). — Jacobi,  "Gesam- 
melte Werke"  (5  vols.  and  supplement). — Lagrange,  "CEuvres"  (14 
vols.),  1867-92.  — Laplace,  "CEuvres"  (13  vols.,  seven  published) . 

Properties  of  Matter.  —  Watson,  "Kinetic  Theory  of  Gases."  — O.  E.  Meyer, 
"  Die  kinetisctie  Theorie  der  Gase."  —  Graham,  "  Chemical  and  Physical 
Researches."  —  Frankland,  "Experimental  Researches." — J.  F.  Daniell, 
"Chemical  Philosophy."  —  Wurtz,  "The  Atomic  Theory."— Abney, 
"Treatise  on  Photography"  (molecular  chemistry). — Van't  Hoff, 
"  Etudes  de  Dynamique  chimique  "  and  his  "  St6reochimie,"  1892.  —  Leh- 
mann,  "  Molekular-Physik "  (2  vols.). — Lord  Kelvin,  "Popular  Lect- 
ures," vol.  i. — Boys,  "Soap-Bubbles."  —  Ostwald,  "Solutions,"  and 
"Lehrb.  d.  allgem.  Chemie." — Ibbetson,  "Elasticity." — Todhunter, 
"  History  of  the  Theory  of  Elasticity,"  with  volume  on  Saint-Venant's 
researches.  —  Love,  "Mathematical  Theory  of  Elasticity." — Neumann, 
"Elasticitat  d.  festen  Korper  u.  d.  Lichtathers." —Klein,  "Elasticitat." 

—  Klimpert,  "  Elasticitat."  —  Poincare,  "  Elasticity."  —  Weyrauch,  "  Auf- 
gaben  zur  Theorie  elastischen  Korper."  — Tait,  "  Properties  of  Matter." 

Dynamics  of  Fluids.  —  Plateau,  "Statique  des  Liquides." — Besant,  "Hydro- 
dynamics" (2  vols.).  —  Basset,  "Elementary  Treatise  on  Hydrodynamics 
and  Sound."  —  Minchin,  "  Hydrostatics  and  Elementary  Hydrokinetics." 

—  Greenhill,  "  Hydrostatics." —Lamb,  "Motion  of  Fluids." —  Stanley 
on  the  "  Motion  of  Fluids  "  (many  new  experimental  facts).  —  Ruhlmann, 
"Hydromechanik." — Stokes,  "Mathematical  and  Physical  Papers."  — 
von    Helmholtz,    "  Wissenschaftliche    Abhandlungen."  —  Hicks,    "Br. 
Assoc.  Rep.  Hydrodynamics,  1881,  1882."  —  L.  D'A.  Jackson,  "  Hydraulic 
Manual."  —  Graeff,  "  Trait6  d'Hydraulique  "  (3  vols.).  —  Bodmer,  "  Hy- 
draulic  Motors."  —  Robinson,    "Hydraulic  Power." — J.   J.    Thomson, 
"  On  Vortex-Rings."  —  Buchan,  "  Meteorology." 

Sound.  —  Stone,  "Lessons  on  Sound."  —  Sedley  Taylor,  " Sound  and  Music."  — 
Blaserna,  "  Theory  of  Sound."  —  Tyndall,  "  Sound."  —  Donkin,  "  Acous- 
tics," part  1  (unfinished).  —  Chladni,  "  Acoustique."  —  Herschel,  Encyl. 
Metrop.,  "Sound."— von  Helmholtz,  "The  Sensations  of  Tone,"  Ellis's 
English  translation",  with  additions.  —  Gavarret,  "  La  Phonation  et  1' Audi- 
tion."—Lord  Rayleigh,  "Theory  of  Sound. "  — Koenig,  "  Quelques  Ex- 
pediences de  P  Acoustique,"  1882.  — Melde's  "  Akustik." 

Heat. Clerk  Maxwell's  "Theory  of  Heat." — Baynes's  "Thermodynamics." 

Dixon   on  "Heat." — Balfour  Stewart  on  "Heat."  —  Him,  "De  la 

Chaleur."  —  Madan,  "  Heat."  —  Devillez,  "  TraitS  de  la  Chaleur."  — 
Thurston,  "  Heat  a  Form  of  Energy." —Preston,  "Theory  of  Heat."  — 
Count  P.  de  Saint-Robert,  "  Principes  de  Thermodynamique."  —  Briot, 

3c 


754  BIBLIOGRAPHY. 

"  ThSorie  mScanique  de  la  Chaleur."  — Zeuner,  "  Grundzuge  der  mechan- 
ischen  Warmetheorie."  —  Clausius  on  "  Heat."  —  Clausius,  "  Die  mechan- 
ischen  Warmetheorie"  (3  vols.),  1879-91.  —Him,  "Thermodynamique." 

—  Berthelot,    "Mecanique   Chimique."  —  Thomsen,    "  Thermocmhiische 
Untersuchungen." — Perry    on    "Steam."  —  Rankine,    "The    Steam- 
Engine. "  —  Cotterill,  "The  Steam-Engine  as  a  Thermodynamic  Engine." 

—  Peabody,  "Thermodynamics  of  the  Steam-Engine,  etc." — Thurston, 
"History  of  the  Steam-Engine."  —  D.  K.  Clark,  "The  Steam-Engine" 
(Blackie).— Bryan  Donkin,  "Gas,  Oil,  and  Air  Engines." —Rankine' s 
Papers. — Willard   Gibbs,   "Thermodynamic   Papers." — Lord  Kelvin's 
Papers. — Fourier's  "Analytical   Theory  of   Heat."  —  Tait's  "Thermo- 
dynamics" and  "Heat."  —  Day,  "Numerical  Examples  in  Heat." 

Light  and  Radiant  Heat.  —  Provost,  "Du  Calorique  rayonnant."  —  (Euvres  de 
Fresnel,  de  Cauchy,  d'Arago. — Stokes's  "Collected  Papers." — Kirch- 
hoff' s  "  Collected  Papers."  —Beer,  "  Hohere  Optik."  —  Engel  und  Schell- 
bach,  "Optik."— Mascart,  "  Traite"  d'Optique"  (3  vols.,  two  published) 

—  Neumann,  "Vorles.  tib.  theor.  Optik." — Aldis,  "Geometric  Optics." 
— Larmor,  "Geometric  Optics." — Pendlebury,  "Lenses  and  Systems  of 
Lenses." — Steinheil  und  Voit,  "  Handbuch  d.  angewandten  Optik,"  vol.  i- 

—  Preston,  "Theory  of  Light."  —  Basset,  "Physical  Optics." — Glaze- 
brook,  "Physical  Optics."  —  Lloyd's  "Undulatory  Theory  of  Light."  — 
Tumlirz,    "  Elektromagnetische  Theorie  des  Lichtes "   (1884). —Tait's 
"Light."  —  Stokes,    "Burnett  Lectures  on   Light."  —  Becquerel,    "La 
Luraiere :  ses  Causes  et  ses  Effets." — Moigno,  "Repertoire  d'Optique." 

—  Baden  Powell,  "View  of  the  Undulatory  Theory  as  applied  to  the  Dis- 
persion of  Light." — Landholt,  "The  Polariscope,"  and  "Das  optische 
Drehungsvermogen." — L.    Fletcher,    "The   Optical  Indicatrix"    (Oxf. 
Univ.  Press). — Lommel,   "Optics  and   Light."  —  Lockyer,  "Spectrum 
Analysis."  —  Roscoe,    "  Spectrum    Analysis."  —  Schellen,    "  Spectrum 
Analysis."  —  Watts,  "  Index  of   Spectra"   (Manchester:    Heywood). — 
Macmunn,  "The  Spectroscope  in  Medicine."  — Rosenberg,  "The  Use  of 
the  Spectroscope  in  its  application  to  Scientific  Medicine  "  (New  York : 
Putnam).  —  Tidy,  "Optics  applied  to  Medicine"  (Cantor  Lectures,  1872- 
73).— Leconte,    "Sight"   (New  York:    Appleton).  —  Rood,    "Modern 
Chromatics."  —  Church,    "  Colour."  —  Whitmell,    "  Colour  "    (Cardiff  : 
Lewis).  —  Abney,  "Colour  Measurement  and  Mixture."  —  von  Helmholtz, 
" Physiologische  Optik"  (French  translation  by  Javal  and  Klein,  with 
extensive    bibliographies  ;    2d    German    edition,   materially  revised,    in 
progress).  —  Traill  Taylor's  "Optics  of  Photography." — Van  Heurck, 
etc.,  "The  Microscope"  (Crosby  Lockwood,  1893). 

Electricity  and  Magnetism.  —  Clerk  Maxwell,  "  Elementary  Treatise."  —  Clerk 
Maxwell,  "Treatise,"  in  2  vols. — J.  J.  Thomson,  "Recent  Researches 
in  Electricity  and  Magnetism,"  1893.  —  Mascart,  ^Traite"  d'Electricite 
Statique." — Mascart  and  Joubert,  "Legons  sur  I'Electricite1  "  (2  vols., 
translated  by  Atkinson).  —  Wiedemann's  "Die  Lehre  von  der  Elek- 
tricitat"  (4  vols.,  a  most  admirable  storehouse  of  experimental  informa- 
tion).—  G6rard,  "Electricity." — Lodge,  "Modern  Views  of  Electricity." 

—  Eintage,  "Introduction  to   Mathematical  Theory  of   Electricity  and 
Magnetism." — Frohlich,   "Handbuch  d.  Elektricitat  u.  Magnetismus," 
1887.  — Duhem,  "Legons  sur  I'Electricite  et  le  Magnetisme,"  1892.— 
Jamieson,    "Electricity    and    Magnetism/' — Neumann,     "  Elektrische 
Strome."  —  Keaviside,  "  Electromagnetic  Theory."  — Ewin0,  "  Magnetic 


BIBLIOGEAPHY.  755 

Induction."—  Poincare,  "Electricity  et  Optique"  (2  vols.).— Boltzmann, 
"Clerk  Maxwell's  Theory"  (2  vols.,  Leipzig:  Earth). —Gordon,  "A 
Physical  Treatise  on  ^  Electricity  and  Magnetism"  (2  vols.).  —  Gariel, 
"Traite  pratique  d'Electricite"." —  Fleeming  Jenkin,  "Electricity  and 
Magnetism."  —  Silvanus  Thompson,  "  Elementary  Lessons  in  Electricity 
and  Magnetism." — Noad  and  Preece,  "Student's  Text  Book." — Fer- 
guson's "Electricity,"  Blyth's  edition. —Frederick  Guthrie,  "Magnetism 
and  Electricity."  — A.  C.  Becquerel,  "Traite"  d'Electricite"."  —  Riess, 
"die  Lehre  vori  der  Reibungselektricitat. "  —  Neumann,  "Theorie  der 
Magnetismus." —Beer's  "  Einleitung  in  der  Elektrostatik."  —  Airy, 
"  Magnetism."  —  Be  La  Rive,  "  Electricity."  —  Humphrey  Lloyd,  "  Mag- 
netism." —  Lament,  "  Erdniagnetismus." — Plante",  "  Recherches  sur 
PElectricite."  — Cumming,  "Theory  of  Electricity." — Serpieri,  "II  Poten- 
ziale  Elettrico  nelP  insegnamento  elernentare  della  Elettrostatica. "  — 
Tumlirz,  "  Das  Potential." 

Cavendish's  "Electrical  Researches." — Faraday,  "Researches  in 
Electricity."  —  Lord  Kelvin,  Reprinted  Papers  on  "Electrostatics  and 
Magnetism,"  and,  in  part,  his  other  reprinted  papers.  —  Clerk  Maxwell, 
Scientific  Papers  (2  vols.). — Henry,  Papers  (2  vols.)  (Smithsonian). — 
Heaviside,  Electrical  Papers  (2  vols.)  (Macmillan). — J.  Hopkinson, 
"Papers  on  Dynamo  Machinery  and  allied  subjects"  (London:  Whit- 
aker).  —  Clausius,  "  Mechanische  Behandlung  der  Elektricitat." — Am- 
pere, "The"orie  des  Phenomenes  electrodynamiques. "  —  Hertz,  "Unter- 
suchungen  lib.  d.  Ausbreitung  cl.  Elektrischen  Kraft"  (English  translation 
by  D.  E.  Jones,  Macmillan,  1893). —Gauss,  "  Gesammelte  Werke."  — 
von  Helmholtz,  " Wissenschaftliche  Abhandlungen."  —  G.  S.  Ohm,  "Die 
Galvanische  Kette  mathem.  gearbeitet,"  1827  (reprint,  Vienna,  1887). 

Du  Moncel,  "Les  Applications  de  PElectricite"  (5  vols.),  1878. — 
Hospitalier,  "  Les  Applications  de  PElectricite." —British  Patent  Office 
Catalogues  of  Specifications.  —  "Applications  of  Electricity,"  by  Instit. 
Civ.  Engineers. — Vaschy,  "  Traite"  d'Electricite"  et  de  Magnetisme; 
theorie  et  applications,"  1890. — Delahaye,  "L'Annee  electrique"  (a 
yearly  summary). — Paget  Hicks,  "Electric  Transmission  of  Power."  — 
Kapp,  "Electric  Transmission  of  Energy."  —  Prescott,  " Electricity  and 
the  Electric  Telegraph."  —  Prescott,  "The  Telephone,  the  Electric  Light, 
and  other  recent  inventions."  —  "Engineering"  Summary  of  Electrical 
Patents.  —  Silvanus  P.  Thompson,  "Dynamo-electric  Machinery,"  and 
"  The  Electromagnet."  —  D.  C.  Jackson,  "  Electromagnetism  and  Motors." 

—  Hawkins  and  Wallis,  "The   Dynamo."  —  Schellen,  "Die  magneto- 
und  dynamoelectrischen   Maschinen." — De   Cew,   same  title    (Hartle- 
ben's  "  elektrotechnische  Bibliothek  ").  — Martin  and  Wetzler,  "  Electric 
Motor."  —  Ternant,  "  Telegraphic  "  (popular).  — Du  Moncel,  "  The  Tele- 
phone" ;  "L'Eclairage  electrique"  (popular).  —  Schwartze,  "Telephon, 
Mikrophon  und   Radiophon "    (Hartleben's  Bibliothek).  —  Preece  and 
Maier,  " The  Telephone." — Reynier,  "Les  Accumulateurs  e"lectriqiies " 
(Paris,  1884).  —  Fermi,  "  Technologie  der  Elektricitat  und  des  Magnetis- 
mus." — Zetzsche,  "  Handbuch  der  elektrischen  Telegraphic"  (4  vols.). 

—  Siemens  und  Halske  in  Berlin,  "  Kataloge."  —  Gratz,  "  Die  Elektricitat 
und  ihre  Anwendungen  "  (Stuttgart,  1883). —  Gore's  "Electrometallurgy." 

Report  of  the  Brit.  Assoc.  Committee  on  Electrical  Standards,  edited 
by  Prof.  Fleeming  Jenkin  (London  :  Spon,  1873).  —  Weber,  "Elektrody- 
namische  Maassbestimmtmgen."  —  H.  F.  Weber,  "  Absolute  elektromag- 
netische  und  calorimetrische  Messungen "  (Zurich,  1875)  —  Latimer 


756  BIBLIOGRAPHY. 

Clark,  "Electric  Measurements."  —  Wilke,  "Die  Elektrischen  Mess- 
und  Pracisions-Instrumente  "  (Hartleben's  Bibliothek). — Andrew  Gray, 
"Absolute  Measurements  in  Electricity  and  Magnetism."  —  Day,  "  Exer- 
cises in  Electric  and  Magnetic  Measurement."  —  Culley's  "Handbook 
of  Telegraphy."  —  Williams,  "  Manual  of  Telegraphy."  —  Schwendler's 
" Instructions  for  Testing  Telegraphic  Lines." — Kempe,  "Handbook  of 
Electrical  Testing."  —  Hoskier,  "Guide  to  Electrical  Testing."— Preece 
and  Sivewright,  "  Telegraphy."  — Latimer  Clark  and  Sabine,  "  Electrical 
Tables  and  Formulae."  —  Munro  and  Jamieson,  "Pocket-Book  of  Elec- 
trical, Rules  and  Formulae."  — Aime  Witz,  "  Problemes  pratiques  d'Elec- 
tricite"  (fully  worked  out). — Hospitaller,  "Formulaire  de  1'Electricien." 

—  Zech,  "  Electrisches  Formelbuch  "  (Hartleben's  Bibliothek). 
DuBois  Raymond,  "  Thierische  Elektricitat."  —  De  Watteville,  "  Medi- 
cal Electricity."  — Smith's  "Medical  Electricity."— Althaus,  "Medical 
Electricity.1' — Bardet,    "  Traite  d'Electricite  me"dicale"   (Paris,  1884). 

—  Rosenthal  und  Bernhardt,  "  Elektricitatslehre  fur  Mediziner  "  (Berlin, 
1883). 

Laboratory  Work,  etc.  — Balfour  Stewart  and  Gee,  "  Lessons  in  Practical  Phys- 
ics."—  Worthington,  "First  Course  of  Physical  Laboratory  Practice." 

—  Weinhold,    "  Experimental    Physics."  —  Weinhold,    "  Physikalische 
Demonstrationen."  —  Rechnagel,    "Experimen.    Physik."  — Pickering, 
"Physical  Manipulation."  —  Kohlrausch,  "Physical  Measurements."  — 
Mousson,  "Physik  auf  Grundlage  der  Erf ahrung "  (3  vols.).  — Shelley's 
"Workshop  Appliances." — Marey,   "La  Methode  graphique." — Airy, 
"Errors  of  Observation." — Aime"    Witz,    "Cours   de  Manipulation  de 
Physique"    (Paris:    Gauthier-Villars,   1883).  —  Glazebrook  and  Shaw, 
"Practical  Physics."— Ayrton,  "Practical  Electricity."— Walker,  "The- 
ory and  Use  of  a  Physical  Balance." 


INDEX. 

The  Darker  Figures  serve  as  a  Guide  to  the  more  Explanatory  References. 


ABERRATION  of  Light,  508,  511 

Aberration,  Chromatic,  541,  571,  573  ; 
Spherical,  of  lenses,  537  ;  of  mirrors, 
525  ;  in  the  eye,  57 1 

Abnormal  Dispersion,  532 

Abscissa,  11 

Absolute  Potential,  192,  584 

Absolute  Temperature,  288,  364,  397, 
399 

Absolute  or  C.G.S.  Units,  13;  see 
Units 

Absolute  Zero,  364,  400,  489 

Absorbents,  491,  497 

Absorption  of  Actinic  Rays,  486,  497, 
502 ;  of  Energy  during  Change  of 
State,  355  ;  of  Energy  from  Source, 
384,  396,  615,  652  ;  of  Ether-waves 
by  the  Atmosphere,  486  ;  of  Gases 
by  Solids,  328  ;  of  Gases  by  Licjuids, 
328  ;  coefft.  of  do.,  328  ;  of  Radiant 
Heat,  497  ;  of  Light,  497  ;  — ,  Selec- 
tive, 498 

Absorption-bands,  499 

Acceleration,  18,  166  ;  mean  — ,  19  ; 
Angular,  75,  159,  166  ;  —  in  Circular 
Motion,  79,  165,  166;  in  a  Curved 
path,  79  ;  —due  to  Gravity,  22,  205  ; 
negative  —  in  Kinetic  Friction,  180  ; 
The  Parallelogram,  etc.,  of  — s,  68; 
Positive  or  negative,  69;  in  S.H.M., 
83  ;  Unit  of  — ,  18 

Accelerated  Motion,  69,  152 ;  com- 
pounded with  uniform,  71 ;  under 
gravity,  205 

Accommodation  of  the  Eye,  570 

Accumulator,  (1)  Electrostatic  con- 
denser, 596,  624;  (2)  secondary 
cell,  665 

Achromatism,  531 ;  achromatic  prisms, 
531  ;  Lenses,  541 

Actinic  Rays,  481,  502,  503,  505,  623 

Actio  Agentis,  59 

Action  =  Force,  20 

Action  (Maupertuis  and  Hamilton),  59 

Action  and  Reaction,  6,  23 

Activity,  42,  59,  166;  —  in  belting, 
183 ;  —  of  steady  current,  647;  of 
alternating,  723  ;  of  dynamo-circuit, 
738.  740  ;  of  galvanic  cell,  000 


Adhesion,  37,  256,  339;  measurement 
of,  37 

Adiabatic  Compression  or  Expansion 
of  Gases,  324,  358,  373 ;  —  Equa- 
tion, 324,  395  ;  —Lines,  395,  398,  400 

Adiathermancy,  497,  498,  502. 

Aelotropic,  687,  744 

Affinity,  Capillary,  282  ;  Chemical,  52, 
256,  32S,  356,  614,  615 

Air,  density  of,  224,  324  ;  air  dissolved 
in  water,  329  ;  —  in  electric  arc,  654  ; 
electrification  of  — ,  629,  630 ;  fric- 
tion in,  335 ;  air-flames,  723 ;  air- 
gaps  in  Herz's  apparatus,  741 ;  in 
Magnetic  Circuit,  691  ;  —  as  an  insu- 

.  lator,  589  ;  air-gun,  45 ;  air-pump, 
326  ;  air  liquefied,  389  ;  mercury  air- 
pump,  343  ;  Potential  of—,  607,  611, 
629  ;  Sound  in  — ,  459  ;  Sp.  heat  of 
— ,  370;  Sprengel's,  326;  air-ther- 
mometer, 401 

Air  as  a  Standard  Medium  in  Electro- 
statics, 578,  598,  604,  709 ;  in  Mag- 
netism, 675,  707,  709 

Ajutages,  305 

Alcoholometer,  222 

Alloys,  636 

Alternating  Currents,  721-733,  735, 
740;  in  Electrodynamometer,  717; 
in  Electromotors,  740 

Alternators,  729  ;  multiphase  — ,  730  ; 
multipolar,  729 

Altitude,  12 ;  — s  indicated  by  the 
Barometer,  348 

Amalgamated  Zinc,  356,  618,  664 

Ammeters,  717,  723 

Ammonia,  355,  358,  389 

Ampere,  353,  638,  647,  660,  711,  715; 
Ampere-turns,  711  ;  Ampere-volts 
(=  "Watts"),  42,  647,  711 

Ampere-currents,  693,  701 ;  their  di- 
rection, 693 

Ampere's  Formula,  670  ;  his  Theory  of 
Magnetism,  674,  692 

Amplitude  in  S.H.M.,  82  ;  of  Vibration, 
414 

Analyses,  556 

Analysis  of  a  complex  harmonic  mo- 
tion (Fourier's  theorem),  102;  of  a 


757 


758 


INDEX. 


Colour  by  a  Prism,  486  ;  of  a  Sound, 
429  ;  of  a  Vowel,  475  ;  Chemical  — , 
217  ;  Spectrum  — ,  217,  494 

Andrew's  Critical  Point,  233,  325,  375 

Aneroid,  297,  347 

Aneurism,  290 

Angle  of  Capillarity,  276 ;  Critical  — 
in  Kinetic  Friction,  180 ;  of  Inci- 
dence, 120,  124,  518;  of  Lag,  722; 
Limiting  — ,  177  ;  of  Overturning, 
209 ;  of  complete  Polarisation,  521, 
743  ;  of  Precession,  75  ;  of  Reflexion, 
120,  518 ;  of  Refraction,  126,  518, 

528  ;  of  Repose,  178 

Angular  Acceleration,  75,  159,  166 ;  — 
Displacement,  166  ;  —  Momentum, 
163,  166  ;  —  Velocity,  75,  166  ;  inini- 
num  do.,  164;  do.  in  S.H.M.,  83; 
Dimensions,  75 

Aorta,  315,  322 

Aperture,  Wave  traversing  an,  116, 
117,  131,  140 

Aplanatic,  571 

Approximate  Foci,  129,  525 

Aqueous  Vapour  in  the  Air,  392,  493 ; 
— particles,  579 

Arc-lamps,  636,  723 

Archimedes'  Principle,  294,  333 

Area,  12 

Areometer :  Fahrenheit's,  223  ;  Nichol- 
son's, 223 

Armature :  of  a  magnet,  692,  707  ;  of 
a  dynamo,  730,  737 

Arrival-Curve,  697 

Arterial  Tension,  322 

Aspiration,  339,  340 

Astatic,  691 

Astronomical  Time,  9  ;  —  Units,  202 

Atheroma,  312,  323 

Atmosphere,  336 ;  homogeneous  — , 
347 

Atmospheric  Electricity,  629 

Atmospheric  Pressure,  229,  293,  336- 
349,  388,  452  ;  Standard,  349 

Atom,  239,  745  ;  Gramme ,  367 

Atomic  Heat,  366,  367  ;  —  Refractions, 

529  ;  —  Theory,  239  ;  —  Oscillations, 
351  ;  —Weights,  217 

Atomicity,  244 

Atomistic  Chemical  Formulae,  240 

Atoms,  238,   245;   chemical,  238-245; 

physical,  245-253 ;    size  and  nature 

of,  245,  246 
Attenuation    of    higher    components, 

477,  722 
Attraction,    187 ;    in  particular  cases, 

188 ;    conventionally  negative,    189  ; 

Capillary  — ,  277  ;  Electric,  577,  579, 

601  ;  Magnetic,  676 
Attwood's  Machine,  36,  206 
Audiphone,  467 
Auditory  meatus,  465 
Auditory  Nerve,  469 


Aurora  Borealis,  679,  680 

Availability  of  Energy,  50 

Avvogadro,  242  ;  A.'s  Law  (chemical), 
242  ;  (physical),  249 

Axes:  Crystalline  Axes,  551,  552,  556, 
623;  Magnetic  Axis,  674,  679;  Op- 
tic Axis,  557,  599  ;  —  of  Reference, 
11,  66,  67  ;  —  of  Rotation,  74,  76, 
162 ;  —  of  Spontaneous  Rotation, 
165 

Axioms,  Newton's,  5 

Axle-friction,  184 

Azimuth,  12 

BABINET'S  Compensator,  564 
Balance,   35,    37,   608  ;    Spring  — ,  37  ; 

Langley's  Thermic  — ,  542,  551,  717  ; 

Coulomb's  Torsion  — ,  607  ;    Ctore's 

Voltaic  —,617 
Ballistic  Galvanometer,  687,  713 ;   — 

Pendulum,  214,  713 
Band  Spectrum,  496 
Banjo,  449 

Bar,  Conduction  of  Heat  in  a,  408 
Barads  (dynes  per  sq.  cm.),  25 
Barlow's  Formula,  336 
Bar-magnet,  675,  677 
Barometer,  147,  342  ;  aneroid  — ,  297, 

347  ; vernier,  28 

Barometric   Column,  341  ;    —  Height, 

342  ;  — Pressure,  see  Atmospheric  Air 
Basilar  Membrane,  469 
Basis  of  Support,  208 
Bassoon,  451 
Battery,  Galvanic,  616,  618  ;  +  and  — 

terminals  of,  617  ;   Leyden  — ,  600  ; 

Secondary   — ,    665,    667;    Thermo- 
electric — , 
Bathing,  165 
Beats,  100,  458,  472 
Becquerel's     Phosphorescence  -  Effect, 

505,   542;  his  Thermo-electric  Pile, 

628 

Beetz's  dry  Cell,  621 
Bell-Crank,  175  ;  Oscillation  and  Swing- 
ing of  a  Bell,  444,  445  ;  Vibration  of 

a  Bell,  412,  417,  445 
Belting,  182  ;  Activity  in  — ,  183 
Benzene,  500  ;  —  vapour  in  Space,  252, 

500 

Bichromate  Cell,  619 
Bifflar  Suspension,  215,  606 
Binaxial  Crystals,  556 
Biology,  1 
Biprism,  542 
Biquartz,  568 

Black,  576  ;  Chevreul's,  576 
Blackburn's  Pendulum,  211 
Bladder,  291,  332 
Blue  colour  of  opalescent  bodies,  503, 

523 

Blue-hot,  488 
Body  suspended,  164,  213 


INDEX. 


759 


Bohnenberger's  Electroscope,  605 
Boiling,   386;    boiling-point,  237,  284, 

388,  402;    b.-pt.   at  different  pres- 
sures, 237,  388 
Bole,  19 
Bolometer  (Thermic Balance),  542,  551, 

717 

Bombardon,  453 

Bound  charge,  591,  595,  601,  630 
Bourdon's  steam-gauge,  297 
Bow  (and  arrow),  45  ;  of  Violin,  435, 

441 
Boyle's  Law,  230,  250,  253,  254,  325, 

338,370,  372,  377,  382,  394,  395 ;  in 

the  Kinetic  Theory,  250 ;  in  case  of 

Vapours,  390 
Boys  on  the  Gravitation  Constant,  202  ; 

Torsion  Experiments,  -263 
Brakes.  180 
Brass,  245 

Breaking  Weight,  167,  261 
Bridge,  Wheatstone's,  645,  706 
Bridge-method :    De   Sauty's,   719 ;   in 

Duplex  Telegraphy,  734 
Briot's  Law  of  Refraction,  509 
British   Units  of  Measurement  (Foot, 

Pound,  Second),  13 
Brittle,  262,  268 
Broadside-Method,  681 
Brushes,  729,  730  ;  Lead  of  —  ,  731 
Bubbles,  273,  279 ;  —  in  boiling,  387, 

388  ;  Soap  —  electrified,  582 
Bugle,  453 
Bullet  crushed,  148 
Bumping,  388 
Bunsen's  Cell,  621. 
Buoyancy,  295 

CABLE,  Submarine,  600,  696 

Callipers,  27,  30 

Calorescence,  506 

Caloric,  351 

Calorie,  404  ;  calorie,  353,  404 

Calorimeters,  404 

Calorimetric  Coefficient  of  Thermal 
Conductivity,  407 

Calorimetry,  404,  513 

Camera  obscura,  548 

Camphor,  274,  567 

Candle,  Decimal,  513 

Caoutchouc,  contraction  on  heating, 
360,  379 

Capacity :  Electrostatic,  592,  597,  599, 
603,  604,  612,  624,  664,  696,  710  ;  of 
a  sphere,  593,  711  ;  of  polarized  elec- 
trodes, 664  ;  Comparison  of  — s,  719  ; 
Standards  of,  609,  718 ;  Specific  In- 
ductive — ,  597,  602-604,  612,  692, 
709,  744,  746  ;  Thermal  Capacities, 
365,  367,  372. 

Capillarity,  255,  276,  294  ;  angle  of, 
276 

Capillary,  Affinity,  282 ;  —  Blood  ves- 


sels, 315 ;  —  Tubes,  Flow  through, 
315-317 

Capstan,  172 

Carbon  monoxide,  238,  240,  327 

Carbonic  anhydride,  232,  238,  254,  325, 
327,  364,  370,  375,  389 

Carnot's  Cycle,  395  ;  C.'s  ideal  Engine, 
384,  397  ;  C.'s  Function,  397  ;  C.'s 
Principle,  397,  399. 

Cascade  (Leyden  jars),  600 

Cast-iron,  255,  327. 

Cathetometer,  28 

Causality,  Law  of,  3 

Cause  and  Effect,  3 ;  Simultaneous 
Causes,  4 

Caustic  by  Reflexion,  123,  525  ;  by  Re- 
fraction, 129,  130,  537 

Cavendish's  Experiment,  202r  263 

Cells,  Galvanic,  616 ;  various  forms, 
618-622,  664  ;  cells  coupled  in  sur- 
face,, in  series,  617,  618>  640;  Ar- 
rangement of,  640 ;  Discharge  of,, 
632  ,-  E.M.D.P.  of  —  measured,  646  ; 
internal  Resistance  of  —  measured, 
647  ;  standard  — ,  622,  646 

Centigrade  Thermometer,  402 

Centimetre,  11 

Centre  of  Figure  or  Mass,  73,  146,  158, 
165,  206, 446  ;  —  of  Gravity,  74,  206  ; 

—  of  Inertia,  146 ;   Optical  — ,  534, 
541;  of  Oscillation,  164,  214;  —  of 
Percussion,  165,  214 ;  —  of  Suspen- 
sion, 165,  214 

"Centrifugal  Force,"  165-169 

C.G.S.  (Centimetre-Gramme-Second)  or 
Absolute  System,  13  et  passim 

Change  of  Direction,  18,  68 

Change  of  Motion,  6 

Change  of  State,  235,  354-358 

Change  of  Velocity,  18,  68 

"Character"  of  Sound,  415 

Charge,  52,  578,  591  ;  Division  of,  592  ; 
''free"  and  "bound,"  591,  595,  601, 
630  ;  Static  Charge  of  Conductor,  579, 
600  ;  on  Ellipsoid,  580  ;  on  point,  580  ; 
do.  in  motion,  743. 

Charging  a  Condenser,  648 

Charles's  Law,  251,  253 

Chemical  Action  as  a  source  of  Elec- 
trical Energy,  611,  614;  —Affinity, 
52,  256,  328,  354,  356,  358,  614,  615  ; 

—  Analysis, 21 7  ;  —Atoms, 238-245  ; 

—  Combination,  236  ;  —  Decomposi- 
tion by  Light,  481,  482,  486,  488,  503 
569 ;  by  Radiant  Heat,  482  ;  —  Effect 
of  Electric  Current,  657-667  ;  —  Ele- 
ment, 217,  219,  494  ;  —Energy,  356  ; 

—  Equivalence,  220,  239  ;  —  Equiv- 
alents,   52,    239  ;    -  -   Forces,    355 ; 

—  Formulae,  239,  240,  244  ;  —  Rays, 
481  ;  _  Work,  355,  359 

Chemistry,  1,  217,  219,  24* 
Cheval-vapeur,  42,  647 


760 


INDEX. 


Chevreul's  Black,  576 

Chimney  roaring,  477 

Chirp  of  insects,  414 

Chlorophane,  507 

Chlorophyll,   49,   482,    499,    504,    505, 

569. 

Chromatic  Aberration,  541,  571,  573. 
Cigar  smoke,  503 
Circle,  Osculating,  79 
Circle  of  Reference  in  S.H.M.,  81,  82, 

185 
Circuit,    Electric,    613-616,   639,   643; 

open,  613,  614,  617  ;  closed,  613,  643  ; 

Positive  side  of,  672 
Circuit,  Magnetic,  691,  725,  730,  731 
Circuits,  Primary  and  Secondary,  700, 

724 

Circuit,  Thermo-electric,  624,  651 
Circular   Motion,    8,  57,  78,  161-169; 

=  2  S.H.M.'s,  87,  101 ;   Friction  in, 

185 
Circular   Polarisation,   514,   560,   562, 

564 ;  detected,  563 
Clamond's  Thermo-electric  Pile,  629 
Clarionet,  451 
Clepsydra,  34 

Clerk  Maxwell ;  see  Maxwell 
Clock,  8  ;   Energy  in,  45  ;   Pendulum, 

9,  34,  212,  380  ;  Wheelwork  in,  34 
Closed    Galvanic    Circuit,    613,    643; 

closed  Magnetic  Circuit,  691 
Cloud,  203 
Cobbler's  Wax,  226 
Cochlea,  468 
Coefficient  of  Absorption  of  Gases,  328  ; 

of  Cubical  Compressibility,  259,  301  ; 

of  Linear  do.,  262  ;  of  Electric  Con- 
ductivity, 634;  — s  of  Thermal  do., 

407  ;    —  of  Thermometric  do.,  407, 

408  ;   of  Diffusibility,  283 ;   of  Elas- 
ticity, 264, 322  ;  of  Elastic  Restitution, 
264  ;  of  Electrostatic  Induction,  600, 
604  ;  of  Expansion  by  Heat,  378,  380  ; 
of  Extensibility,   260 ;    of    Induced 
Magnetisations,  686,  693;  of  Inertia, 
147,   166;  of  Kinetic  Friction,  179; 
of  Statical  Friction,  176  ;  of  Magnetic 
Induction,  685,  692,  694  ;  of  Mutual 
Induction  of  Charged  Bodies,  600,  of 
Currents,  658,  704  ;  of  Resistance  to 
Compression,  259,  to  Extension,  261, 
to  Shear,  260,  to  Twist,  263  ;  of  Res- 
titution, in  Elasticity,  264,  in  Impact, 
151 ;  of  Rigidity,  226,  260 ;  of  Self- 
induction,  705,  710;    of   Solubility, 
280,  328  ;  of  Transmission  of  Light, 
etc.,  499 ;  of  Transpiration,  331  ;  of 
Viscosity,    227,    307,    316;     Kine- 
matical  — -  of  do.,  228 

Coercitive  Force,  684,  690,  691 
Cohesion,   256  ;     solids,   257 ;    liquids, 

254,  270,  345 
Cohesion  figures,  279 


Coil:  Induction ,  646,656,706,  725; 

Resistance-,  634,  646 

Collecting  Rings,  729,  730 

Collodion  Films,  273 

Colloids,  284 

Colour,  281,  483-488,  501;  by  trans- 
mitted light,  498 ;  by  reflected  light, 
501  ;  produced  by  interposed  doubly- 
refracting  lamina,  559;  Complemen- 
tary — s,  487,  563,  574 

Colour-Blindness,  575 

Coloured  Light :  Simple,  483 ;  Com- 
pound, 486 

Colours :  Analysis  of,  486  ;  Matching 
of,  575  ;  Mixture  of,  574  ;  Primary  — , 
575 

Colours  of  Metals,  502,  503 

Colours  of  Thin  Films,  545 

Combinational  Tones,  473 

Combining  Proportions,  239 

Combustion-equivalent,  356 

Combustion,  Heat  of,  49,  236,  354,  357. 

Comet,  80,  203 

Comma,  423,  424 

Commensurable,  Commensurate,  93, 103 

Common  Light,  515,  563 

Communicating  Vases,  293 

Commutator,  730,  738 

Compensation-Method  of  Comparison 
of  Electrostatic  Capacities,  720 

Compensation-Pendulum,  380 

Compensator  in  Saccharimeter,  568 ; 
Babinet's  Compensator,  564 

Complementary  Colours,  487,  563,  564, 
565,  574,  575 

Complementary  Distribution  of  Elec- 
tricity, 580,  591,  592 

Complex  HarmoniCjMotion ;  see  Fourier- 
motion,  103 

Complex  Sound- Waves,  432 

Components  of  a  Force,  143 ;  of  a 
Fourier-motion,  416  ;  of  a  Velocity, 
60,  63 

Component  Tones  of  a  Note,  417,  429 

Composition  of  Forces,  143;  of  Rota- 
tions, 74;  of  S.H.M.'s,  86-103;  of 
Transversal  Vibrations,  107  ;  of  Uni- 
form with  Accelerated  Motion,  71  ; 
of  two  Velocities,  60  ;  of  more  than 
two,  65 

Compound  Coloured  Light,  486 

Compound  Pendulum,  213 

Compound  Winding,  732 

Compressibility,  Cubical  Solids,  259; 
gases,  325 ;  water,  462 ;  coefficient 
of  — ,  259 

Compressibility,  Linear,  262 ;  coeffi- 
cient, 262 

Compression,  adiabatic,  324,  358,  373 

Concert  Pitch,  421 

Concertina,  412,  442,  451 

Condensation  of  gases,  231,  375,  377; 
condensation-temperature,  392 


INDEX. 


761 


Condenser,  596, 630  ;  charging  a  — ,  648 ; 
discharge  of  a  — ,  632  ;  Sliding  — , 
600 ;  Standard  — s,  609,  718  ;  — s  in 
Submarine  Telegraphy,  698  ;  vibra- 
tion of,  737 

Condenser  and  Source:  in  Galvanic 
Cell,  615,  648  ;  in  Heat-engine,  384, 
396  ;  in  Thermodynamic  Circuit,  652 

Conductance,  633,  634,  638,644;  "  — 
a  Velocity,"  638 

Conduction-Current,  586 

Conduction  of  Electricity,  588, 603,  692  ; 

Surface ,  696,  721 ;  Electrolytic- 

— ,  588,  622 

Conduction  of  Heat,  406-410  ;  602,  692, 
710,  740 ;  in  crystals,  409 ;  in  gases, 
236,  251,  409 

Conductivity,  Electrical^  281,  634,  635, 
638;  of  electrolytes,  646,  658,  722; 
Variable,  636,  737  ;  Magnetic,  686 

Conductivity,  Thermal,  407,  602,  636  ; 

three  coefficients  of ,  497  ;  Ther- 

mometric  — ,  407,  408 

Conductor  of  Frictional  Electric  Ma- 
chine, 610 

Conductors  of  Electricity,  588,  603  ; 
kinds  of,  590 ;  currents  in  homoge- 
neous — ,  638  ;  in  heterogeneous  — , 
639 ;  in  wide  — ,  643  ;  Charge  of  — , 
579,  600.  Conductors  of  Heat,  406 

Conical  Pendulum,  80 ;  its  Energy,  142  ; 
in  Viscous  Medium,  185 

Conical  Kefraction,  557 

Conjugate  points,  130,  526,  535 

Conservation  of  Electricity,  581 

Conservation  of  Energy,  7,  8,  47,  52, 
127,  144,  230,  517  et  passim 

"  Conservation  of  Force,"  52,  144, 
230,  292 

Conservation  of  Vires  vivce,  517 

Conservative  System,  44,  186,  255 

Constancy  of  Nature,  2 

Constitution  of  Matter,  238  et  seq. 

Contact-Effect,  true,  612,  616,  624,627, 
649 

Contact  of  Metals  (electricity),  611  ;  of 
Non-conductors,  610 

Contact-Breaker,  707,  725 

Contact  in  Friction,  179 

Continuity  between  Liquids  and  Gases, 
232,  254,  375,  496 

Continuity,  Law  of,  300,  334,  602 

Continuous  Spectrum,  495    . 

Contraction  on  Heating,  379  ;  on  Pull- 
ing, 260 

Convection  of  Heat,  410  ;  Convection- 
currents,  410 ;  Electric  Convection, 
651 

Convections :  as  to  Attraction  —  and  Re- 
pulsion +  ,  189  ;  Standard  Pitch,  421  ; 
as  to  Electric  and  Magnetic  Formulae 
in  Air,  578,  598,  604,  675,  707  ;  as  to 
Gravitation-Potential,  192  ;  North- 


seeking  Pole  of  Magnet,  Positive, 
675,  746  ;  as  to  Plane  of  Polarisation, 
521  ;  Vitreous  Electricity  Positive, 
579,  746 

Convergent  Lenses,  533,  535 

Convergent  Rays  or  Beam,  117 

Cooling,  during  evaporation,  389  ;  — 
during  expansion,  358  ;  —  in  still  air, 
410 ;  Dulong  and  Petit's  Law  of,  410  ; 
Newton's  Law  of,  410 

Copper  electroneagtive  to  zinc,  positive 
in  the  battery,  611,  615,  617,  643;  — 
cast  under  water,  364  ;  colour  of  — , 
498,  502  ;  repulsion  of,  724 

Cord,  Vibrations  of,  134,  413 

Core,  soft-iron,  687,  728,  731 

Cornet,  453;  cornet  of  guttapercha, 
454 

Cornopean,  453 

Corona,  551 

Corresponding  Points  of  Eyes,  576 

Corti,  469 

Coulomb,  638,  711,  715 

Coulomb's  Torsion-Balance,  607 

Counterpoising  of  Forces,  37 

Couple,  158  ;  equilibrium  of  — s,  160 ; 
examples  of,  160 ;  Moment  of,  159, 
166,  677  ;  Magnetic,  676,  681,  693 

Crank,  89 

Creeping,  380 

Crest  and  Trough,  104 

Critical  Angle  in  Kinetic  Friction,  180  ; 
—  Density,  376  ;  —  Pressure  (Car- 
nelley's),  236  ;  —  Pressure  (Van  der 
Waals's),  233,  376  ;  —  State  of  Mat- 
ter, 232,  376  ;  —  Temperature,  232, 
237,  255,  376,  390,  (magnetic)  689 ; 
Villari's  —  Value,  690  ;  —  Volume, 
233,  376 

Crookes's  Layer,  363 

Crooks  of  Trumpet,  453 

Crossed  (Nicol's)  prisms,  559 

Cross-field  (dynamo),  731 

Crushing,  262 

Crust  of  Earth,  228 

Cryophorus,  389 

Crystalline  form,  255 

Crystalloids,  284 

Crystals,  Axes  of,  551,  556,  623 ;  Bi- 
naxial,  556  ;  Conduction  of  Heat  in, 
409  ;  Expansion  in,  378  ;  Positive  and 
Negative  doubly-refracting  crystals, 
555 ;  electrified  on  heating,  624, 
on  pressure,  623 ;  Principal  Section, 
551 

Cubical  Compressibility,  259,  325,  462 

Currents,  586,  603,  611,  632-674  ;  Con- 
duction — ,  586  ;  Displacement,  586, 
611,  648,  695  ;  Eddy-  — s,  703  ;  En- 
ergy of  —  49,  615,  647,  651;  Oscillat- 
ing — ,  721-733,  735,  740  ;  Steady  — 
632-674,  745  ;  Direction  of  — ,  615  ; 
Mutual  Action  of  — ,  670,  702,  724  ; 


762 


INDEX. 


Secondary  — ,  687,  700  ;  Simultane- 
ous — s,  644  ;  —  in  Telephones,  737 

Current-Density,  633,  649 

Current-Induction,  699-707 

Current-Intensity  or  Strength,  633,  637, 
638,  659,  684,  694,  703,  707,  709 

Current-Sheet,  693 

Curvature,  79 ;  Radius  of,  79,  80,  165 

Curves:  see  Adiabatic,  Arrival,  Gau- 
gain,  Harmonic,  Isentropic,  Isother- 
mal, Periodic,  Sines 

Curved  path,  57  ;  Acceleration  in,  79 ; 
Velocity  in,  59 

Curved  Surface,  Refraction  at,  130 

Curved  Wave-Front :  Reflexion  of,  120  ; 
Refraction  of,  128 

Cyanite,  674 

Cycle,  394 ;  Carnot's,  395 

Cyclical  order,  63,  66,  645 

DALTON'S  Atomic  Theory,  239.;  D.'s 
Law  of  Gaseous  Pressures,  250,  253 

Damp  air  as  an  insulator,  589 

Damping,  186,  299,  3U7,  701,  703,  716 

Daniell's  Cell,  620,  638,  661,  663,  704  ; 
its  various  forms,  620;  its  C.M.D.P. 
computed,  661 

Dark-Heat  Waves,  481 

Dark  Lines  in  Spectrum,  494  ;  in  Heat- 
Spectrum,  495,  717 

Dead-Beat,  703,  716 

Dead  Points  of  a  Crank,  89 

Dead  Turns  in  Dynamo,  731,  738 

Decay,  357,  507 

Decimal  Candle,  513 

Declination,  678 

Decomposition,  Chemical,  236 ;  of 
Water  by  Electrolysis,  658 ;  by  a 
Grove  Cell,  not  by  a  Daniell,  663  ;  by 
Light,  481,  482,  486,  488,  503,  569; 
by  Radiant  Heat,  482.  See  Dissocia- 
tion 

Deferred  Restitution-force,  266 

Deflection-methods,  680 

Deformation,  Resistance  to,  259 

Degradation  of  Energy,  399 

Degrees  of  Freedom  of  a  Particle,  72, 
357  ;  of  a  Rigid  Body,  76 

Demagnetisation,  692,  725,  731 

Demon,  Clerk  Maxwell's,  52,  398 

Denser  Medium,  Wave  entering,  124; 
wave  leaving,  125 

Densimeter,  223 

Densities  of  gases,  241,  250,  324 

Density,  measurement  of,  221-224,294, 
295,  382  ;  measurement  of  Vapour- 
density,  391 

Density  of  Air,  324  ;  of  the  Earth,  202  ; 
of  the  Ether,  235,  746  ;  of  Hydrogen, 
224,  324  ;  of  Matter,  220  ;  of  Solu- 
tions, 281  ;  of  Water,  13,  220,  224 ; 
Maximum  Density  of  Water,  13,  359 

Density,  Critical,  376  ;  Optical  — ,  509  ; 


Specific,  221 ;  Superficial,  of  Matter, 
188  ;  —  Electrical,  579,  583,603,  604  ; 
Magnetic,  683,  686,  692,  693  ;  Vol- 
ume-density, 188 

Density  of  Current,  633,  649 

Derived  Currents  :  steady,  644,  653  ; 
oscillatory,  723 

Dermis,  287 

De  Sauty's  Comparison  of  Capacities, 
719 

Deviation  :  minimum,  529  ;  —  without 
dispersion,  531;  dispersion  without 
deviation,  532 

Dew,  393  ;  dewpoint,  392 

Dextro-rotatory,  567 

Dialysis,  286 

Diagram,  Indicator,  55,  593  ;  Thermo- 
electric, 626,  651 

Diamagnetic,  685,  (oscillating  currents) 
724 

Diaphragm  in  Lenses,  537 

Diathermancy,  497 

Diatonic  Scale,  422 

Dichroism,  Dichromatism,  499 

Dicrotic  Pulse,  322 

Dielectric,  587,  589,  590,  591,  593,  595, 
597,  602,  603,  611,  648,  695,  744  ;  - 
in  Oscillating  Currents,  721  ;  —  Elas- 
ticity, 603,  604 ;  Energy  of  the  — , 
602,  603,  604,  695 

Difference  of  Potential,  193,  584,  593. 
597,  603,  604,  609,  612,  623,  624 
709;  Actual  D.P.  in  closed  circuit, 
643;  Electromotive  I).  P.,  587,  608, 
609,  616,  625,  646  ;  do.,  measured 
608,  646 ;  observed,  604  ;  produced, 
609 

Differential  Equations,  83,  185,  186, 
395,  705,  718,  722,  725,  728 

Differential  Galvanometer,  713,  734 

Differential  Tones,  473 

Diffraction,  139;  —  of  Sound- Waves, 
458 ;  of  Light,  548  ;  of  e.-m.  waves,  743 

Diffraction-grating,  141,  486,  549 

Divisibility,  Coefficient  of,  283 

Diffusion  of  gases,  247,  251,  330;  of 
liquids,  247,  283,  288  ;  of  solids,  257  ; 
of  gases  through  membranes,  332 

Diffusivity,  Thermal,  407 

Dilatancy,  288 

Dimensions,  Theory  of,  15,  746 ;  —  of 
Physical  Quantities,  59,  75,  166,  224, 
263,  308,  598,  603,  604,  638,  693,  694, 
708,  709,  714 

Dimensions  of  Space,  9,  10 

Dip,  678 

Direct  Extra-Current,  705 

Direct-Vision  Spectroscope,  532 

Direction,  57  ;  Change  of  — ,  18,  68  ;  — 
of  Current,  615  ;  —  of  Electromag- 
netic Lines  of  Force,  668  ;  —  of  Mag- 
netic Lines,  676  ;  of  Movement,  191  ; 
—  of  Sound,  471 


INDEX. 


763 


Discharge,  580,  599,  632,  696,  725,  742, 
744  ;  residual  — ,  599 ;  —  in  a  vacu- 
um, 656,  662 

Discharging  Electroscope,  605 

Discord,  471 

Discs,  vibration  of,  137,  443,  479 

Dispersion,  245,  509,  530,  745  ;  Abnor- 
mal, 532  ;  Irrationality  of,  531  ;  De- 
viation without  — ,  531  ;  —  without 
Deviation,  532 

Displacement,  41,  82,  166  ;  angular  — , 
166  ;  electric,  602,  603,  611 

Displacement-Current,  586,  611,  648, 
695 

Disruptive  Tension,  582 

Dissipation  of  Energy  ;  see  Availability 
(50)  and  Degradation  (399)  of  En- 
ergy 

Dissipation  of  Sound  in  Air,  460,  463 

Dissociation,  241,  243,  248,  355,  367, 
615,  662;  —in  Solutions,  248,  280, 
386,  590,  614 

Dissonance,  471 

Distance,  Action  at  a,  577 

Distorted  Image  (lenses),  538 

Distortion  of  Air-waves,  477  ;  of  elec- 
tric waves,  722,  737 

Divergent  Lenses,  533-537  ;  —  Rays, 
117 

Dividing  Engine,  30 

Divisibility  of  Matter,  220, 238/245,  246 

Division  of  Charge,  592 

Doppler's  Principle,  465,  484 

Double  Bass,  450 

Double  Refraction,  228,  551-566,  575, 
599,  744 

Double-Refracting  Lamina,  559;  — 
Power,  detection  of,  565 ;  Double- 
Refraction  Dynamometer,  565 

Drum,  412,  444,448;  of  Ear  (mem- 
brana  tympani),  457,  466,  473 

Dry  bulb,  392 

Dry  pile,  605,  622 

Ductility,  259 

Dulong's  Water  Calorimeter,  405 

Dulong  and  Petit's  Law  of  Atomic 
Heat,  366,  367  ;  of  Cooling,  410 

Duplex  Telegraphy,  733 

Dust,  583  ;  dust-haze,  503,  522,  523 

Dynamical  Coefficient  of  Thermal  Con- 
ductivity, 407 

Dynamo-Electric  Machines  ("dyna- 
mos"), 353,  637,  692,  725,  737; 
Direct-Current  — ,  731  ;  'series,  732  ; 
shunt,  732 ;  series-shunt,  732 ;  sep- 
arately excited,  732  ;  Activity,  000  ; 
Efficiency,  ooO  5  Alternators,  729-730 ; 
Disc-dynamos,  728  ;  drum  — ,  730  ; 
ring  — ,  730 

Dynamometer  (Force),  38;  (Energy), 
53;  Double  -  Refraction  — ,  665; 
Friction  — ,  184 

Dyne,  21 


EAR,  the  Human,  448,  458,  465-471 ; 
the  Ear-trumpet,  428,  461 

Earth :  Attraction  by,  6,  201 ;  Crust 
of,  228  ;  Currents,  644  ;  Density,  202  ; 
— 's  Drag  on  the  Ether,  510  ; In- 
ductor, 687,  718 ;  Loss  of  Heat  by 
— ,  407  ;  Mass,  202 ;  as  a  Magnet, 
679,  693  ;  — 's  Permeability  to  the 
Ether,  570 ;  — 's  Positive  Pole  the 
Southern,  693;  its  Potential  Zero, 
588  ;  Pressure  of  Sunlight  on  — ,  570  ; 
Putting  to  — ,  698,  719,  720;  Radii 
of,  205 ;  Rotation  of,  164,  168,  205 ; 

—  in  Telegraphy,  644  ;  Variations  of 
g  over  — ,  22,  41,  205 

Ebullition,  386 

Echo,  460 

Ecliptic,  81 

Eddies,  306,  313 

Eddy  Currents,  703,  724,  725 

Effect,  Cause  and,  3 

Effective  Mean  Intensity,  723 

Efficiency    of    Electromotor,    738 ;    of 

Heat-engine,  397;   of  Carnot's  ideal 

do.,  397 

Effusion  of  gases,  330 
Egg,  spinning  of,  76,  164 
Elastic  Bodies,  their  Impact,  117,  151, 

251,  266;   E.  intermediary,  56,  268; 

—  Restitution,   264 ;   —  Toughness, 
265 ;   —  Tubes,  Flow  through,  320- 
323 

Elasticity  :  coefficient  of  — ,  264,  322  ; 
in  Solids,  256,  263-267  ;  in  Liquids, 
000  5  in  Gases,  229,  267,  324  ;  Perfect 
and  Imperfect  —  in  Solids,  265,  267  ; 
Fatigue  of,  267  ;  physiological  exam- 
ples, 268;  Limits  of,  256,  265; 
Mechanical  advantages,  56,  268 ; 
Vibrations  due  to  E.,  151,  251, 
266 ;  Electric  —  of  Dielectric,  603, 
604 ;  E.  of  the  Ether,  235,  745 ;  E. 
of  Volume,  229,  259 

Elastivity,  603,  604 

Electric  Arc,  654 ;  Attraction,  234,  577, 
581,  602,  721  ;  —  capacity,  592,  597, 
599,  603,  604  ;  —  Charge,  578  ;  free, 
591  ;  bound,  591 ;  —Circuit,  613-616, 
672  ;  —  Conduction,  588,  603,  692  ; 

—  Conductivity,  634, 658  ;  —  Convec- 
tion of  Heat,  651 ;  —  Cautery,  653, 
666  ;  —  Currents,  586,  603,  611,  632- 
674,  721-733,  735,  740 ;  —  Density, 
579  ;  —  Displacement  of  Dielectric, 

603,  611,  744  ;  —  Elasticity  of  Dielec- 
tric, 603  ;  —  Energy,  49,  52,  593,  603, 

604,  648  ;  —  Equipotential  Surfaces, 
582-588,  592,  595,  602,  648,  672 ;  — 
Force,  581, 583, 587,  599,  603, 604, 609, 
648,  710  ;  lines  of  Force,  583,  591-595, 
604,  648,  667,  671,  692,  699,  700,  742, 
745  ;  Motion  of  these  lines,  591,  648, 
699  ;  in  oscillating  field",  721 ;  — Fur- 


764 


INDEX. 


nace,  655  ;  —  Fuses,  653  ;  —  Light, 
484,  636,  653,  723,  727  ;  —  Machines 
(frictional),  610,  656,  737  ;  Matter 
(imaginary),  578,  68.1,  583  ;  —  Pres- 
sure, 587 ;  —  Quantity,  578,  594, 
604,  638  ;  —  Screen,  601 ;  —  Storage 
of  Energy,  666  ;  —  Stress,  234,  577, 
693,  599,  603,  744;  —Tension,  582, 
603  ;  —  Welding,  653 ;  — al  Wind, 
580,  602 

Electricity,  7  ;  not  a  form  of  Energy, 
577, 594 ;  Conduction  of,  588, 603, 692  ; 
electrolytic,  588,  622 ;  Conservation 
of  — ,  581 ;  Separation  of  Electrici- 
ties, 602,  609 ;  —  in  the  Universe 
=  0,  581 

Electrocapillarity,  624 

Electrochemical  Equivalence,  659;  — 
Equivalent,  661 

Electrodes,  657,  745  ;  Capacity  of,  664  ; 
Polarization  of,  664 

Electrodynamic  Units,  670 

Electrodynamometer,  716,  723 

Electrokinetic,  234,  591 

Electrolysis,  49,  327,  388,  590,  657-667, 
723,  727,  745,  746  ;  —  in  Gases,  662  ; 
of  Mixtures,  663  ;  —  under  Oscillat- 
ing Currents,  723 

Electrolytes,  386,  590,  615,  657,  722, 
664  (glass)  ;  conductivity  of,  646, 
658  ;  in  oscillating  currents,  722 

Electrolytic  Conduction,  588,  622  ;  — 
Field  of  Force,  615,  657 

Electromagnet,  38,  689,  692,  707,  (al- 
ternate current)  724 

Electromagnetic  Field  of  Force,  667, 
683,  689,  695,  701  ;  —  Current-Induc- 
tion, 699-707 ;  —  Interrupter  for 
Tuning  Forks,  267, 447,  735  ;  —Lines 
of  Force,  667,  671,  699,  702  ;  their 
Direction,  668  ;  —  Measure,  625, 634, 
670,  684,  703, 707,  714  ;  —  Units,  see 
Magnetic  Units ;  —  Unit  of  Heat 
(=  Joule),  41,  353,  647;  —Waves, 
479-481,  498,  741 

Electrometer,  245,  604,  606 

Electromotive  Force,  587,  588,  633,  706 

Electromotive  Intensity,  587,  603 

Electromotor,  738 ;  Alternating  Cur- 
rent — ,  740;  triphase  do.,  741; 
Efficiency  in  — ,  QOO 

Electronegative,  611,  617,  643 

Electrophorus,  630 

Electroplating,  663 

Electropositive,  611,  617,  643 

Electroscopes,  579,  604,  612 

Electrostatic,  234,  591;  —  Capacity, 
592,  597,  599,  603,  612,  624,  664,  696, 
710;  —Energy,  593,  603,  604,648; 

Electromagnetic  Ratio  V,  708, 

709,  744  ;  —  Field,  Intensity  of,  583  ; 
—  Retardation,  696 ;  —  Units,  see 
Units 


Element,  Chemical,  217,  219,  494 ;  — s 
of  Charge,  602;  Galvanic  — ,  616; 
Geometrical,  58 

Elliptical  Motion  =  2  S.H.M.'s,  90 
Elliptically  Polarized  Light,  515,  560 ; 

detected,  563 
Elongation  in  S.H.M.,  82 
Elongation  under  Traction,  260 
E.M.D.P.,  Electromotive  Difference  of 
Potential    (  =  "E.M.F.,  Electromo- 
tive Force  "),  587,  608,  609,  616,  625, 
646,  709  ;  Measured,  608,  645 
End-on  Method,  681 
Endosmotic  Equivalent,  287 
Energy,  2,  42,  354 ;   Potential  — ,  43, 
45,  190,  354,  684,  699,  702  ;  Kinetic 
— ,  46  ;  Availability  of  — ,  50  ;  Con- 
servation of,  7,  47,  52,  125  et  passim  ; 
Degradation  of,  399  ;   Flow  of,  648, 
684,    695  ;    Fluctuation   of  — ,   164  ; 
Graphic  representation  of —  (areas}, 
52,  53-56,  393  ;  Indestructibility  of, 
7,    47  ;    Measurement  of  — ,    5,    3  ; 

—  Slope,     42,    60,    603;     Storage 
of  — ,  in  the  Ether,  586,   593,  603, 
648,  721 ;  Transformations  of,  47  et 
passim 

Energy  absorbed  or  evolved  during 
Change  of  State,  237,  356 ;  do.  do.  in 
Galvanic  Cell,  614,  615  ;  —  in  Charg- 
ing a  Condenser,  648  ;  Chemical  — , 
356  ;  —  of  Conical  Pendulum,  142  ; 

—  of  Dielectric,    603,    604,  695  ;    of 
Electrified   Body,    593  ;    of    Electric 
Current,  49, 615,  647, 651 ;  —  in  Elec- 
tric Field  of  Force,  603,  604,  648  ; 
Flow  of  —  in  Current-Field,  648,  684, 
695  ;  —  in  Impact,  151,  152  ;  Intrin- 
sic — ,  48,  247,  356  ;  —  of  Jet,  303  ; 

—  of  Magnetic  Field,  686  ;  —  of  Mole- 
cules, 351 ;  —  of  Niagara  F.alls,  739  ; 

—  in  case  of  Repulsion,  190  ;  —  of 
Rotation  of  a  Particle,  162  ;  of  a  Mass, 
162  ;  —  in  Secondary  Batteries,  666  ; 

—  of  S.  H.M.,  141  ;  of  Sound-waves, 
142,  414,  476  ;    of  Sun,   50,  478 ;  of 
Sunlight,    479 ;     —    in    Superposed 
Waves,  458  ;  Transmission  of  — ,  182, 
383,  230;    by  Steady  Current,  632, 
648  ;  —  in  Vision,  573  ;  —  of  Wave- 
motion,  142 

Engine :  doing  and  not  doing  Work,  49, 
352  ;  Carnot's  — ,  397  ;  Dividing  — , 
30  ;  Harmonic  — ,  476  ;  Marine  — , 
89  ;  Perfect  — ,  374  ;  Railway,  181  ; 
Reciprocating,  397 

Engineer's  Unit  of  Force,  23 

Entropy,  395,  398,  399 

Epoch  in  S.H.M.,  83,  103 

Equal  Temperament,  425 

Equalisation  of  a  Current,  667 

Equation  to  a  Curve,  72 

Equator,  Magnetic,  679 


INDEX. 


765 


Equilibrium,  187  ;  stable,  unstable,  and 
neutral,  210  ;  E.  of  Couples,  160  ; 
Electrostatic  — ,  590,  613,  615,  657  ; 

—  of  Forces,   145  ;    —  of    Liquids, 
289 

"  Equilibrium-position,"  39,  263 

Equipotential  Sucfa££S^i03-199,  301, 
409,  5&>r&8%  585,  588,  592,  595,  599, 
602,  648,  667,  672,  676,  682,  683 

Equivalence,  Chemical,  220, 239;  equiv- 
alents, 52,  239 

Equivalence  of  Forces,  3 

Equivalence  of  Shell  and  Circuit,  683, 
707 

Equivalent,  Electro-chemical,  661 ;  En- 

dosmotic  — ,  287  ;  Gramme ,  367  ; 

Water ,  405 

Equivalent  Lens,  538 

Ereinacausis,  507 

Erg,  41 ;  Ergten,  41 

Ericsson's  Sun-motor,  491 

Essential  Properties  of  Matter,  216 

Ether,  the,  234,  361,  411,  478,  479,  480, 
481,  488,  497,  504,  508,  509,  510,  511, 
512,  513,  514,  521,  522,  570,  577,  582, 
583,  586,  589,  591,  593,  598,  602,  612, 
648,  669,  671,  679,  694,  700,  704,  721, 
727,  742-746  ;  its  Constitution,  745  ; 
its  Density,  235,  513  ;  its  Elasticity, 
235,  513,  593,  612,  745 ;  Energy  stored 
in,  586,  593,  603,  648;  Transverse 
Vibrations  of,  479,  509 

Ether-Shear,  746 

Ether-Stress,  234,  577,  582,  586,  599, 
602,  603 

Ether- Waves,  234,  350,  478,  508,  741 ; 
their  Length  measured,  542,  544,  550  ; 
their  Pressure,  570  ;  their  Velocity, 

"  480,  510,  512,  637,  743 

Ethylene,  389 

Euphonium,  453 

Evaporation,  237,  386 ;  Cooling  during, 

389  ;  Electricity  in  — ,  623 ;  —  of  Ice, 

390  ;  Latent  Heat  of  — ,  390 
Evolution  or  absorption  of  Energy  dur- 
ing Change  of  State,  237,  356 

Exchange  of  Radiations,  489 
Excitation  of  Field  Magnets,  731 
Expansion  of  Gases,  indefinite,  229, 250, 

326,  338  ;  explained,  250 
Expansion  by  Heat,  Coefficient  of,  378  ; 

—  in  Crystals,  378  ;  Measurement  of, 
380-382  ;  examples  of  — ,  379 ;  —  in 
hollow  bodies,  379 

Experimentation,  5 

Exploration  of  air-potential,  629  ;  —  of 

Fluid  Pressure,  296 
Explosives,  4,  44,  148 
Exposure  in  Photography,  482 
Extensibility,  260  ;  Coefficient  of,  260 
Extension  of  Matter  in  Three  Dimen- 
sions, 218 
Exterior  Work  done  by  Heat,  359,  370 


External  Conical  Refraction,  658 
Extra-Current,  704,  705,  731  ;  Direct, 

705 ;  Reverse,  705 
Extraordinary  Ray,  553 
Eye,  536,  570 

FAHRENHEIT  —  Areometer,  223 ;  — 
Thermoneter,  364,  402  ;  F.  Zero,  386, 
402 

Fall  of  Electric  Potential,  638,  641,  650, 
654,  656 

Fall  under  Gravitation,  21,  202 ;  — 
down  inclined  plane,  173 

Farad,  715 

Faraday's  Laws  of  Electrolysis,  659 

Fatigue  of  Elasticity,  267 

Faure's  Accumulators,  665 

Federmanometer,  297 

Format's  Law,  133 

Ferromagnetic,  685 

Field  of  Force,  198  ;  Electrostatic,  582- 
585,  588,  590,  591,  593,  595-599,  605, 
609,  612,  615,  632,  685  ;  discharge  of, 
632  ;  intensity  of,  583,  676  ;  Electro- 
lytic, 615,  657  ;  Electromagnetic,  667, 

683,  689,  695,  701,  708;    Magnetic, 
669,  676,  682,  686,  701,  745  ;  Rota- 
tory, 741 

Field  Magnets,  731 

Fife,  453 

Figure,  Centre  of,  73,  146,  158,  165, 
206,  446 

Films,  Colours  of,  545 

Filtration,  340 

Fish,  air-bladder  of,  325  ;  luminosity 
of,  507 

Fixity  of  proportions  in  chemical  com- 
pounds, 238 

Fizeau's  method,  512 

Flageolet,  453 

Flame,  355,  506,  576,  723  ;  as  a  Reflec- 
tor, 459  ;  Singing  and  Sensitive,  454 

Flexibility,  262 

Flexible  Lenses,  537  ;  —  Mirrors,  527 

Flexure  of  a  Rod,  263 

Flow  of  Electrical  Charge,  602-603 

Flow  of  Energy  in  Current-Field,  648, 

684,  695 

Flow  of  Gas,  334,  336;  measurement, 
336 

Flow  of  Heat,  407,  409,  602  ;  in  a  Bar, 
408 

Flow  of  Liquids,  288,  299-323  ;  Forces 
producing  — ,  300 ;  in  Suspended 
Loops,  301 ;  —  through  rigid  Tubes, 
309  ;  —  in  Capillary  Tubes,  315  ;  — 
in  Elastic  Tubes,  320 ;  Lines  of  — , 
300;  Measured,  317;  Steadiness  of 
— ,  300 

Flow  of  Magnetism,  692 

Flow  of  Temperature,  409 

Fluctuation  of  Energy,  164 

Fluid,  218,  225 ;  —  Friction,  184 ;  — 


766 


INDEX. 


Pressure,  296 ;  Pressure  in  Vibrat- 
ing — ,  335 

Fluorescence,  484,  486,  504 

Flute,  453 

Flux,  Magnetic,  685,  686,  691 

Fluxion-notation,  59 

Fly-wheel,  89,  164,  168,  169,  667 

Focal  Distance  in  Eefraction,  130 ; 
Principal ,  130,  539 

Focal  Length  of  Minor  ( = principal  focal 
distance),  525  ;  of  Lens,  534,  535,  536 

Focus,  117  ;  Approximate  — ,  129,  525  ; 

—  of  Mirror,  525  ;  Principal  — ,  525  ; 
Conjugate,  526 ;    Virtual,    525,  527  ; 

—  of  Lens,  533  ;  real,  533  ;    virtual, 
533;  principal,  533,  534;   Conjugate, 
535;   Heat  F.  and  Photographic  F., 
541,  542  ;  F.  of  Eye,  538 

Fogg,  411 ;  Clearing  — ,  583 ;  Reflexion 
of  Light  from,  503 ;  Sound  through, 
460 

Foot,  10 

Foot-pound,  40,  41 ;  Foot-poundal,  41 

Force  (1)  Any  Cause  of  Motion,  4,  20  ; 
(2)  Action  =  ma,  20,  166  ;  (3)  Time- 
rate  of  Change  of  Momentum,  20,  60, 
248 ;  (4)  Space-rate  of  Change  of 
Energy,  Energy- Slope,  42,  60,  603; 
(5)  Rate  of  Variation  of  Change  of 
Configuration,  47  ;  —  said  to  Act 
upon  bodies,  20,  40  ;  to  do  Work,  40, 
or  to  have  Work  done  against  it,  40  ; 
"  Centrifugal  — ,"  165-169 ;  Chemical 
— ,  355 ;  Components  of  a  — ,  143 ; 
Composition  of  — s,  143  ;  Conserva- 
tion of  — ,  52,  144,  230,  292  ;  Elec- 
tromotive — ,  587,  588,  633,  706; 
Equilibrium  of  — s,  145 ;  Equiva- 
lence of  — s,  3;  Field  of  — ,  198; 
electrostatic,  582-585,  587,  588,  590, 
591,  593,  595-599,  603-605,  609,  611, 
612,  615,  632,  648,  685,  710,  742; 
electromagnetic,  667-674,  683,  689, 
695,  701  ;  magnetic,  618,  667, 676,  682, 
683,  701 ;  electrolytic,  615,  657  ;  uni- 
form, 198,  583,  608,  673  ;  —  produc- 
ing Flow,  300;  Graphic  representation 
of  —  (lines),  52  ;  Lines  of  — ,  196, 
582,  591,  592,  595,  604,  648,  667,  692, 
745 ;  electromagnetic,  667,  672,  699, 
702,  direction  of,  668 ;  magnetic,  676, 
their  direction,  676  ;  Magnetic,  — , 
676,  693  ;  Magnetomotive  — ,  691  ; 
Molecular  — s,  253-256,  measurement 
of,  271  ;  Measurement  —,20,  35-39  ; 
Moment  of  — ,  155,  166;  Parallelo- 
pipedon  of  — s,  145 ;  Parallelogram 
of  — s,  143 ;  Polygon  of  — s,  145  ; 
Resolution  of  — s,  143  ;  Resultant 
Electric  — ,  581,  58'3,  603  ;  Simul- 
taneous — s,  143,  145;  Triangle  of 
— s,  145  ;  Tubes  of  — ,  197,  583,  602  ; 
Unit  of  — ,  21  ;  Unit  Tubes  of  — ,  197 


Forced  Vibrations,  447-449,  479 

Force-pump,  345,  587 

Forceps,  170 

Form,  of  Matter,  218;  Perception  of 
—,576 

Formulas,  Chemical,  239,  244  ;  Graphic, 
244 ;  Mathematical,  15 

Foucault's  Principle,  512  ;  his  Prism, 
556 

Fourier-motion  (a  periodic  motion  com- 
pounded of  commensurate  S.H.M.'s), 
(Fourier's  Theorem),  103,  266,  412, 
414,  416,  417,  434 

Fourier's  Theorem,  103 ;  applied  to 
vibrating  strings,  135 

Fragility,  258 

Fraunhofer  lines,  484,  485,  494,  550 

Free  Charge,  591,  595,  597,  630 

Free  Path  of  Molecules,  251,  252,  253, 
464 

Free  Vibration,  434-445,  479 

Freedom,  Degrees  of,  72,  76,  351 

Freezing  Mixture,  235;  —Point,  281, 
402,  lowered  by  molecules  in  solu- 
tion, 281,  386 

French  Horn,  453 

Frequency,  83,  112,  412,  414,  435,  479  ; 
of  S.H.M.,  83;  of  Alternating  Cur- 
rents, 725  ;  Ether-Waves,  480 

Fresnel,  379,  517,  519,  521,  543  ;  Fres- 
nel's  Rhomb,  562 

Friction,    7,  176,  203,   352,    393,   436  ; 

—  at  Axles,  184  ;  Coefficient  of  Stati- 
cal —  between  Solids,    176  ;  Coeffi- 
cient of  Kinetical  —  between  Solids, 
179 ;    variations   therein,    179,    184  ; 

—  in  special  cases,  179-184  ;   —  in 
air,    335  ;    —   -Dynamometer,    184  ; 

factor,  184  ;   —  between   Fluids 

and   Solids,  184  ;   —  within   Fluids, 
306  ;   —  in  the  Mechanical  Powers, 
180;  —a  Resistance,  176,  180;  Roll- 
ing — ,  182  ;  —  in  S.H.M.,  185  ;  in 
Violin,  435-436  ;  —  -Wheels,  182 

Frictional  Electric  Machines,  610,  656, 
737  ;  vapour-friction  do.  do.,  623 

Fringes,  140 

Frothing,  278 

Function,  15, 82  ;  Carnot's-,  397  ;  Periodic, 
82  ;  Thermodynamic,  395 

Furnace,  Electric,  655 

Fuses,  653 

Fusing  point  affected  by  Pressure,  237, 
384 

Fusion,  360,  384 

GALILEO'S  Principle,  4  ;  his  Doublet, 
572  ;  —  and  Pendulum,  34,  202 

Gallon,  13 

Galvanic  Cell,  353;  —Battery,  Cell, 
Pile,  608,  616,  745  ;  coupled  up,  617, 
640  ;  discharge  of,  632  ;  economical 
use  of,  641 ;  and  —  ends,  615,  617  ; 


INDEX. 


767 


—  Circuit,  616,  639,  643  ;  Constants 
in,  measured,  646 

Galvanometer,  602,  637,  643,  645,  647, 
695,  710,  712,  733  ;  Ballistic  — ,  687, 

713  ; Constant,  V&  ;  Dead-beat 

— s,  716  ;  Differential  — ,  716,  733  ; 
von  Helmholtz's  — ,713  ;  Marine  — , 
691  ;  Resistance  in,  647  ;  Sensitive- 
ness of  — ,  713 ;  Sine  — ,  713  ;  Tan- 
gent — ,  637,  710,  712,  718 

Gases,  229,  241,  250,  324-349,  375,  377|; 
perfect,  369-374 

Gas-battery,  664 

Gate  on  its  hinges,  170,  216 

Gaugain's  Curves,  627 

Gauges,  30  ;  Steam  Gauge,  297 

Gauss's  Lens-sy stem-method,  539 

Gausses,  677,  715 

Geissler's  Vacuum-Tubes,  656,  723 

General  Properties  of  Matter,  219 

Gerhardt,  240 

Gilberts,  715 

Glacier-Flow:  (1)  Creeping,  380;  (2) 
Plasticity,  385 ;  (3)  Regelation,  385 

Glow-lamps,  653 

Gradient,  Potential—,  191,  585,  587, 
597,  598,  602,  633,  639,  648,  710; 

Pressure ,  301  ;  Temperature , 

407,  602 

Gramme,  13 ;  gramme-atom,  gramme- 
molecule,  367  ;  gramme-equivalent, 
660 

Graphic  Formulse  (chem.),  244 

Graphic  Representation  of  Energy,  52, 
53-56,  393  ;  of  Force,  52 

Grating,  Diffraction,  141,  486,  549 

Gravitation,    41,    44,    146,   201,   247 ; 

—  Constant,  189,  202  ;  Fall  under  — , 
21, 173,  202  ;  Law  of  — ,  201  ;  —  Po- 
tential, 192,  195,  584 ;   Universal  — , 
203 

Gravity,  acceleration  g  due  to,  22,  202  ; 
Variations  of  g,  22,  41,  205  ;  Meas- 
urement of  g,  36,  206  ;  Centre  of  — , 
74,  206 ;  Effect  of  —  on  flow,  301  ; 
Intensity  of  — ,  22,  713;  Specific  — , 
221 ;  do.  bulbs,  223 

Gravity  Electric  Machine,  631 

Grays,  576  <£: 

Grove's  Cell,  621,  638 

Guebhard,  663 

Guitar,  449 

Gun,  6,  46 

Gunpowder,  44.  580 

Gyration,  Radius  of,  162 

HAEMODROMOGRAPH,  319 
Haemodromometre,  319 
Haemotachymetre,  318 
Hall's  Experiment,  694 
Halo,  530 
Hammock,  149 
Hardness,  258 


Harmonic  Curve,  85,  07,  106,  111,  736  ; 

—  Engine,  476  ;  —  Telegraphy,  736 
Harmonic  Motion,  80  ;  Acceleration  in, 

proportional  to  Displacement,  83, 
434;  Amplitude  in,  82,  xv?,  114; 
Angular  Velocity  in,  82  ;  Circle  of 
Reference  in,  81,  185 ;  Composition 
of,  86-103  ;  H.M.'s  compounded  by 
Blackburn's  Pendulum,  211,  by  vi- 
brating reeds,  442,  by  vibrating 
strings,  434  ;  due  to  Elasticity,  266  ; 
Elongation  in,  82  ;  Energy  in  (x  am- 
plitude2), 141  ;  Epoch  in,  83,  103  ; 
Frequency  of,  83  ;  Friction  in,  185 ; 

—  Isochronous,  83  ;  Pendulum-move- 
ment,   86,  211  ;   Period  in,  82,  414, 
680  ;    Phase   in,  82  ;    Projection  of, 
84  ;  Resolution  of,  96, 102  ;  Viscosity 
in,  185 

Harmonic  Variations  of  Potential  and 

Current,  721,  722,  729 
Harmonicon,  442 

Harmonics^  416,  429,.  430,  472,  475 
Harmonium,  412,  425,  442,  451,.  472, 

473 

Harmony,  417,.  471 
Harp,  449 

Harpsichord,  436,  449 
Haze,  503,  522,  523,  743 
Head  of   Liquid,  302,  587  ;  of  a  Gas, 

330,   334;    Pressure ,    310,    334; 

Velocity-  — ,  310,  334 
Heart,  311 ;  Valves  of,  346,  431  ;  Work 

done  by,  320 
Heat,  7,  48,  2o4,  350-411 ;    Atomic  — , 

366  ;  —  of  Combustion,  49,  236,  354, 
357  ;  Conduction  of,  406-410,  in  gases, 
236,    409;    Convection    of  — ,  410; 

—  produced  by  Electric  Current,  649, 
652,  711,  723  ;  —  a  Form  of  Energy, 
351  ; Engine,  49,  352,  397  ;   Ex- 
pansion  bv,    378-382  ;    Flow  of  — , 
407,  409,  602  ;    —  Focus,  541,  542  ; 
Latent  — ,  235,  361,  364;   Material 
Theory  of  Heat,  351  ;  Molecular  — , 

367  ;  —  not  Motion,  351  ;  —  not  by 
Pressure  alone,  351  ;  Radiant  — ,  48, 
234,  350,  410,  481,  541,  743  ;  —  Rays, 
481  ;    Sensible  — ,  350  ;   Specific  — , 
353,  365,  370,  404,  405 ;  —  -Spectrum, 
486,  495,  500,  533,  717  ;  Transference 
of  _,  406  ;  Transport  of,  410 

Heavy,  216  ;  Heavy  liquids,  291 
Helium,  218,  495 
Henries,  711,  715 
Henry's  Law,  329 

Herz's  Experiments,  741  ;  his  Resona- 
tor, 741  ;  his  Vibrator,  741 
Heterogeneous  Conductors,  639 
Hollow  Body,  Expansion  of,  379  , 
Homogeneous  Atmosphere,  347 
Homogeneous  Conductors^485,  638 
Homogeneous  Light,  485 


768 


INDEX. 


Homogeneous  Strain,  78 

Hooke's  Law,  256,  265,  473 

Hopkinson,  599,  689 

Horizontal  Component  of  Terrestrial 
Magnetic  Force,  679,  680,  712 

Horn-bands,  Russian,  449 

Horsepower,  42,  647, 653, 655  ;  effective, 
42  ;  nominal,  42 

Humidity,  392,  463 

Huyghens,  34 

Hydraulic  Press,  21,  52,  290,  333 ;  — 
Ram,  148,  312,  704 ;  —  Tourniquet, 
306 

Hydrogen  ;  a  metal,  327  ;  alloy  with 
palladium,  327,  664;  Critical  Tem- 
perature, etc.,  376 ;  Density  of,  224, 
324  ;  Flame  of  — ,  496  ;  Heat  evolved 
on  Combustion  of,  354,  356,  358; 
Molecular  Velocity  of,  250,  465  ;  Self- 
repulsion  of,  253,  375,  377 ;  Sound- 
waves in,  413,  428  ;  —  in  Organ-pipe, 
452 ;  Specific  Heat  of,  366,  369 ; 
Thermal  capacity  of,  366 

Hydrometer,  222,  277 

Hydrostatic  Paradox,  292  ;  —  Pendu- 
lum, 318,  336 ;  —  Pressure,  25,  290, 
301,  310,  325,  602;  —Stress,  225, 
257,  300 

Hydrostatics,  287-299 

Hygrometer,  392 

Hypertrophy  of  Heart,  311,  323 

Hypothesis,  8 

Hysteresis,  690,  725 

ICE  :  Evaporation  of,  390  ;  Fusion  of, 
351,  360,  384,  402  ;  affected  by  Pres- 
sure, 384  ;  —  in  hot  Crucible,  363  ; 
Latent  Heat  of  —,361,  390 ;  Plas- 
ticity of  — ,  385 ;  Spectrum  of  — , 
495 

Iceland  Spar,  551,  553,  555 ;  as  a  Di- 
electric, 599 

Image  :  of  Lens,  real,  533,  535  ;  virtual, 
533,  535  ;  distorted,  538 ;  —  in  Mir- 
ror, 526 ;  real,  526  ;  virtual,  527 

Imaginary  Electric  Matter,  57 8, 581, 583 

Imaginary  Magnetic  Matter,  675 

Imbibition,  282 

Impact,  117,  150-152,  251  ;  Energy  in, 
151,  152  ;  Vibration  on  — ,  151,  251, 
266 

Impedance,  706,  722,  727,  729 ;  —  in 
Transformers,  725 

Impenetrability  of  Matter,  218 

Imperfect  Elasticity,  265,  267 

Impulse,  20,  166  ;  Moment  of  — ,  166 

Incandescence,  654 

Incandescent  Lamps,  653,  723 

Incidence,  Angle  of,  120,  126,  518  j 
Plane  of  — ,  517,  519,  521 

Inclination  (magnetic),  678 

Inclined  Plane,  172  ;  Fall  down , 

173  ;  Pull  up  an ,  181 


Incus,  467 

Indiarubber,  266,  360,  379,  570,  58  9 

Indestructibility  of  Energy,  7,  47  ;  of 
Matter,  7,  219 

Index  of  Refraction,  127,  509,  518, 
522,  528,  534,  540,  744  ;  (for  X  =  co  ), 
745  ;  measured,  528,  529 

Indicator  Diagram,  55,  393 

Induced  Magnetic  Poles,  685 

Induced  Magnetisation,  Coefficient  of, 
686,  693 

Inductance,  705,  722,  737  ;  Mutual  — , 
704 

Induction  :  —  -Coil,  646,  656,  706,  725  ; 
Electromagnetic  — ,  667,  699-707, 
727-729  ;  Electrostatic  — ,  594,  604  ; 
coefficient  of,  601,  604,  692  ;  lines  of, 
595,  596,  598,  602  ;  total,  596,  598, 604 ; 
per  sq.  cm.,  596,  598,  603,  604  ;  induc- 
tive capacity,  597,  598,  603,  604,  692  ; 
Magnetic  — ,  684,  693 ;  coefficient  of, 
685,  692,  693  ;  lines  of,  667,  671,  685, 
702,  705,  710,  718  ;  general  problem 
of  magnetic  — ,  692  ;  Mutual  — ,  600, 
704,  710  ;  Self-  — ,  646,  704,  710,  722, 
724,  745,  746 

Inductive  Capacity,  Specific,  597,  602, 
603,  604,  612,  692,  709,  744 

Inductivity,  685,  714 

Inductor,  Earth-,  687 

Inertia,  5,  146,  219  ;  Centre  of—,  146  ; 
Coefficient  of,  147,  166  ;  examples  of 
— ,  147  ;  —  in  Ether  discussed,  745  ; 
Moment  of  — ,  162,  166,  263,  680  ; 
Radius  of  — ,  162,  166 

Infinite  (oo  ),  58 

Instantaneous  Axis,  75 ;  —  Current, 
695 

Insulation,  582,  583,  589 

Integration,  162,  188,  395 

Intensity  of  Alternating  Current,  723, 
726  ;  mean,  723 ;  virtual  mean,  723  ; 
—  of  Steady  Current,  633,  637,  638, 
659,  684,  694,  703,  707,  709;  meas- 
ured, 637,  694;  Electromotive  — , 
587,  603;  —of  Electrostatic  Field, 
583  ;  —  of  wave-motion  at  a  Focus, 
117;  —  of  Gravity,  22,  713;  —  of 
Light,  513,  744;  —  of  Magnetic 
Field,  676,  685,  693,  701,  708  ;  —  of 
Magnetisation,  677,  691,  693;  —of 
Pressure,  25  ;  —  of  Radiation,  513  ; 

Slope,  200  ;  —  of  Sound,  414  ;  — 

of  Stress,  24  ;  —  of  Tension,  26 

Intensity  of  S.H.M.  (cc  Energy  or 
oc  Amplitude2),  141,  414 

Interference-Bands,  140,  142 

Interference  of  Waves,  137  ;  of  Ether- 
waves,  542 ;  of  electro-magnetic 
waves,  743;  of  Sound-waves,  461, 
463 

Intermolecular,  359 

Internal  Conical  Refraction,  558 


INDEX. 


769 


Internal  Work,  359,  369-371 ;  in  Air, 

etc.,  345,  347,  375 
Interrupter,  Electromagnetic,  735 
Intervals  (musical),  422 
Intramolecular,  351,  356,  359 
Intrinsic  Energy,  48,  247,  356 
Inverse  Squares,  Law  of,  187,  582,  585, 

675 

Inversion  (thermo-elect.),  628,  651 
Iodine,  498,  499,  500 
Ions,  280,  288,  386,  590,  614,  658,  659 ; 

Direction   of,  664;    Travel  of,   658, 

664  ;  Velocity  of,  658 
Iridescence,  547 
Irradiation,  576 

Irrationality  of  Dispersion,  531 
Isentropic  Curves,  395,  398 
Isochronous    Oscillations     of    Elastic 

Body,  267  ;   of  Pendulum,  34,  212 ; 

—  S.H.M.'s,  83 

Isodynamic  Surfaces,  200 

Isothermal  Lines,   394,  400 ;    —  Sur- 
faces, 408,  602 

Isotropic,  113,  138,  547,  551;  (mag- 
netic) 687 

JACOBI'S  Law,  738 

Jelly,  a  Solid,  226 ;  the  Ether  analo- 
gous to  a  — ,  235 

Jet,  72,  277,  302,  303-305,  579  ;  Energy 
of,  303 

Joule,  41,  48,  352  ;  — 's  Equivalent,  48, 
352,  353  ;  his  experiments  on  Gases, 
372-374  ;  his  Law  of  the  Heat  pro- 
duced by  a  Current,  649 

"Joule"  =  107  ergs,  41,  353,  647 

Jupiter's  Satellites,  81,  511 

Just  Intonation,  422,  475 

KALEIDOSCOPE,  524 

Kepler's  Laws,  203 

Kerr's  Experiment,  695 

Kettle  singing,  477 

Kilogramme,    13 ;    Kilogramme-metre, 

23,41 

Kilowatt,  42 
Kine  (C.G.S.  unit  of  Velocity,  one  cm. 

per  sec.),  14 

Kinematics,  57  ;  Kinetics,  143,  434 
Kinetic  Energy,  46  ;  —  Friction,  179  ; 

—  Theory  of  Gases,  247,  361 ;  applied 
to  Radiometer,  361 ;   to   Spheroidal 
State,  363  ;  to  Velocity  of  Sound,  464 

Kirchhoff's  Laws,  644 
Knee,  175 

Koenig's  Manometric  Capsule,  431, 
444,  452 

LAEVO-ROTATORT,  567 

Lag,  722,  724,  729 ;  Angle  of,  722 
Lampblack,  455,  498 
Lamp  wickholder,  408 
Laryngoscope,  524 


Larynx,  148,  433,  475,  476,  524 

Latent  Heat,  235,  361,  364,  384,  411, 

689  ; of  Expansion,  370,  372, 

377  ; of  Ice,  361,  390 ; 

Methods  of  Calorimetry,  406 

Latimer  Clark's  Cell,  622,  646 

Lattice  girders,  269 

Law  of  Causality,  3  ;  —  of  Continuity, 
300,  334,  602  ;  Law  of  Electric  At- 
traction and  Repulsion,  579,  581  ; 
Law  of  Gravitation.  201  ;  Law  of 
Inverse  Squares,  187,  582,  585, 
675 ;  Law  of  Magnetic  Attraction 
and  Repulsion,  675  ;  Laws  of  Motion, 
5  ;  Laws  of  Thermodynamics,  353, 
398.  See  Avvogadro,  Boyle,  Briot, 
Charles,  Dbppler,  Dulong  and  Petit, 
Earaday,  Eermat,  Eoucault,  Fourier, 
Galileo,  Henry,  Hooke,  Jacobi,  Joule, 
Kepler,  Kirchhoff,Lenz,  Newton,  Ohm, 
Pascal,  Poiseuille,  Prevost,  Raoult, 
Stokes,  Torricelli,  Van  der  Waals 

Laws  of  Nature,  1,  5 

Lead  of  Brushes,  731 

Leakage  :  Electric,  618,  698,  737  ;  Mag- 
netic, 692 

Leclanche'  Cell,  619,  666 

Left  hand  of  the  Current,  668,  689 

Left-handed  Polarised,  515,  517,  560 

Length,  Focal,  130,  525,  534,  535,  536 
(  =  Principal  Focal  Distance) 

Length,  Measurement  of,  27  ;  Units  of 
-10 

Length  Reduced,  of  Conductor,  637, 
641,  642 

Length  of  Simple  Pendulum,  206,  212, 
214  ;  of  Compound,  213 

Lens  :  Equivalent,  538  ;  Power  of,  534  ; 
Reversibility  of,  534 

Lenses,  533,  743 ;  achromatic,  541 ; 
convergent,  534,  535  ;  divergent,  534, 
536  ;  flexible,  537 

Lens-method,  Gauss's,  539 

Lenz's  Law,  701 

Level  of  Water,  294 ;  Potential  analo- 
gous to,  193,  195,  586 

Lever,  169-71 

Leyden  jar,  599,  600,  606  ; dis- 
charge, steady,  726  ;  oscillatory,  721, 
725,  742,  744 

Light,  7,  48,  234,  481,  694,  723,  743  ; 
production  of,  506;  —  by  electric 
current,  653 ;  monochromatic,  485, 
542,  543,  545;  coloured,  483,  486; 
polarised,  514  ;  common  or  natural, 
515,  563  ;  Maxwell's  Theory,  234, 
510,  743,  744 

Light  (adjective),  216 

Lightning  Conductor,  586, 696 ;  —  flash, 
distance  of,  463 

Limiting  Angle,  in  statical  friction,  177 

Limits  of  Elasticity,  256V265  ;  of  Hear- 
ing, 471  ;  of  Magnetisation,  688 


770 


INDEX. 


Line,  9 

Linear  Compressibility,  262 
Linear  Currents,  their  Mutual  Forces, 
669-671 

Linear  Waves,  104  ;  Reflexion  of,  117 

Line-Measurement,  27 

Line,  Neutral,  262  ;  Nodal,  137,  443 

Line  of  No  Pressure  (Indicator  Dia- 
gram), 55 

Line  of  Potential,  639,  641,  696,  720 

Lines,  Adiabatic,  '394,  398,  400 ;  Iso- 
thermal, 394,  400 

Lines  of  Electrostatic  Induction,  595, 
596,  598,  602 

Lines,  Fraunhofer,  484,  485,  495,  550 

Lines  of  Flow,  300,  602  ; of  Heat, 

409,  602  ;  Stream ,  300,  602 

Lines  of  Force,  196,  301,  692  ;  Electric, 
582,  591,  592,  595,  602,  671,  700,  742, 
745  ;  closed,  745  ;  whether  limited  in 
number-,  746$  Electromagnetic,  667, 
671,  699,  702  ;  direction  of,  671 ;  cut 
across,  703  ;  travel  of,  648,  671,  699, 
721  ;  Magnetic,  667-670,  702  ;  direc- 
tion of,  668 

Lines  of  Induction :  Electric,  595,  596, 
598,  602,  603,  604,  671;  Magnetic, 
667,  671,  685,  702,  705,  710,  718 

Lines  of  Propagation  of  Heat,  409,  602 

Lines  of  Slope,  200,  687 

Lines  of  Transmission  of  Energy,  671, 
699 

Line-Spectrum,  495,  496 

Liquefaction,  235,  358,  360,  384  ;  lique- 
fied air,  389  ;  marsh-gas,  224 

Liquids,  254,  271-323;  friction  against, 
184  ;  mobile  and  viscous,  226 

Lissajous's  Figures;  e.g.,  figs.  35-41, 
211,  442 

Litre,  12 

Loading  a  Vibrating  Body,  445 

Locomotive,  6 

Locomotive  Pulse,  313 

Logarithmic  Decrement  (viscosity),  185, 
308 

Logarithmic  Increment  (musical  pitch), 
424,  426 

Logarithmic  Spiral,  185 

Longitudinal  Vibration,  110,  135,  441 ; 
Compression  and  Displacement  in, 
111 

Loops  and  Nodes,  134,  137,  438,  452 

Loops,  suspended,  liquid  in,  293,  301, 
344 

Loss  of  Energy  of  Electrification,  594 

Loss  of  Half  a  Wave-Length,  125,  518, 
545 

Loudness,  413,  415,  425,  471 

Lubricants,  176,  184 

Luminiferous  Ether,  234  (see  Ether) 

Luminosity,  507 

Luminous  Radiations,  489 

Lungs,  330,  337,  339,  340 


Lustre,  576 
Lyre,  449 

MACHINES  :     Electric,  610,   656,    727  ; 

Simple,  169 

Magdeburg  Hemispheres  339 
Magnet,  674,  693  ;  Solenoidal,  675  ;  how 

made,  689  ;  Spherical,  678 
Magnetic  Axis,    674,   679  ;  —  Circuit, 

691,  725,  730,731;  —[Conductivity, 

686  ;  —  Declination,  678  —  Dip,  678 ; 

—  Displacement,   744  ;   —  Equator, 
679  ;  —  Equipotential  Surfaces,  667, 
672  ;  —  Ether-vortices,  234,  745  ;  — 
Field,  669,  676,   682,  683,  701,  708, 
745  ;  Energy  of,  686,  744;  its  [nature, 
745  ;    —   Flux,   685,    686,    691  ;    — 
Force,  648,  667,  676,  682,  683,   693, 
701,    746 ;    lines   of,   667-674,   676 ; 
motion  of  these,  671  ;  in  oscillating 
currents,  721  ;  —  Inclination,  678  ;  — 
Induction,  684,  693,  744;  Coefficient 
of  magnetic  induction,  685,  692,  693; 
lines  of,  667,  671,  685,702,  705,  710, 
718  ;    general  problem   of  magnetic 
induction,    692 ;    —    Intensity,   676 
685,  693,  701  ;  —  Lines  of  Force,  676, 
of   Induction,   667,    671  ;    imaginary 
Matter,  675  ;  —  Measure,  625,  634, 
670,    684,    703,  707;    —  Meridian, 
679,  680  ;  —  Moment,  677,  680,  693  ; 
of  Shell,  682  ;  —  North,  678  ;  —  Par- 
allels, 679 ;  —  Permeability,  671,  676, 
682,  683,  684,  685,  686,  691,  703,  706, 
707,  708,  709,  722,  744,  746  ;  —  Pole 
(terrestrial),   679,    of  magnet  \   668, 
675,  685,  691  ;  —  Potential,  682, 693  ; 

—  Quantity,  675, 693  ;  —  Rotatory  Po- 
larisation, 694,  745  ;  —  Screen,  691  ; 

—  Shell,  682,    707  ;  —  Storms,  679, 
strength  of,  682,  684,  693,    707,  po- 
tential of,  683;  —  Strength   (pole), 
675,  (field)  676,693,  701  ;  —  Superfi- 
cial Density,  683,  686,  692,  693 ;  — 
Susceptibility,  686,  690 ;  —  Tick,  690 ; 

—  Twist,  688 ;    —  Units,  625,  634, 
670,  684,  703,  707,  714 

Magnetisation :  Induced  — ,  684 ;  Coeff t. 
of,  686,  693;  Intensity  of  — ,  677, 
691,  693  ;  Limit  of  — ,  688  ;  Residual 
— ,  684,  690;  Weber's  Theory  of, 
688 

Magnetism,  49,  234,  257,  674-695  ; 
Flow  of,  692  ;  nature  of,  674,  692  ; 
separation  of,  686,  689,  692  ;  terres- 
trial, 679,  691 

Magneto-electric  Machines,  629,  687 

Magnetometry,  680 

Magnetomotive  Force,  691,  714 

Malleability,  259 

Malleus,  466,  467 

Manometer,  47,  296,  325,  333 ;  ma- 
nometre  metallique  inscripteur,  298 ; 


INDEX. 


771 


m.  compensateur,  299;  manometric 
capsule,  431,  444,  452 

Marie-Davy's  Cell,  622 

Marine  Engine,  89 

Marsh-Gas  liquefied,  224 

Mass,  12,  166,  216,  247  ;  Centre  of  — , 
12,  146  ;  Measurement  of  — ,  35 

Massive,  216 

Matching  Colours,  575 

Material  Theory  of  Heat,  351 

Materials,  Strength  of,  255 

Mathematical  Formula?,  15 

Matter,  2,  7,  216,  577,  745 ;  its  Consti- 
tution :  chemical  views,  238  ;  phy- 
sical views,  245  ;  Kinetic  Theory  of 
— ,  247  ;  Essential  Properties  of  — , 
216;  Extension  of — ,218;  Impene- 
trability of  — ,  218  ;  Indestructibility 
of  — ,  7,  219  ;  States  of  — ,  225-237  ; 
General  Properties  of  —  5,  219 ;  Iner- 
tia, 5,  146,  219  ;  Weight,  12,  21,  201, 
219,  333  ;  Divisibility,  220,  238,  245, 
246  ;  Porosity,  220  ;  Density,  220 ; 
Perception  of  Matter,  216,  246 ;  Ra- 
diant — ,  234,  252,  656 

Matter,  Imaginary :  Electric,  578,  581, 
583  ;  Magnetic,  675 

Maximum  Density  of  Water,  13,  359 

Maxwell  on  Coercitive  Force,  690  ;  — 's 
"  Demon,"  52,  398  ;  — 's  Discs,  574  ; 
"Electromotive Intensity,"  587,  603  ; 
Theory  of  Light,234, 510,744  ;  Theory 
of  the  Magnetic  Field,  745 

Mean  Free  Path,  251,  278 

Measure :  Electrodynamic,  670  ;  Elec- 
tromagnetic or  Magnetic,  625,  634, 
770,  684,  703,  707;  Electrostatic, 
638  ;  Numerical,  16 

Measurement,  1,  27  ;  —  of  Adhesion, 
37  ;  Calorimetric  — ,  404,  513  ;  —  of 
Density,  221-224,  294,  295,  382; 

End ,  30  ;  —  of  Energy,  53  ;  —  of 

E.M.D.P.,  608,  646  ;  of  Expansion, 
380-382  ;  —  of  Flow  of  Gases,  336  ; 
of  Flow  of  Heat,  408  ;  of  Flow  of 
Liquids,  317;  of  Force,  20,  35-39; 
of  g,  the  acceleration  of  Gravity,  36, 
206 ;  of  Heat,  404  ;  of  Inductance, 
706  ;  of  Intensity  of  Current,  637, 
694,  707  ;  of  Intensity  of  Magnetic 
Field,  708  ;  —  of  Length,  27  ;  Line- 
_,  27  ;  —  of  Liquid  Pressure,  295- 
299  ;  —  of  Mass,  35 ;  of  Magnetic 
Moment,  680 ;  of  Magnetic  Permea- 
bility, 687  ;  of  Molecular  Forces,  270  ; 
of  n,  the  Atmospheric  Pressure,  347  ; 
of  Pressure  in  a  Stream,  317  ;  of  Re- 
fractive Indices,  528,  529  ;  of  Resis- 
tance of  a  Battery,  647  ;  of  Resistance 
to  Electric  Current,  633,  645,  647  ;  of 
Resistance  of  a  Galvanometer,  647  ; 
of  Surface,  32;  of  Surface-Tension, 
37,  275  ;  Temperature,  400  ;  of  Time, 


34 ;  of  V,  708, 709 ;  of  Vapour- density, 
391  ;  of  Velocity  of  Liquid  Stream, 
317  ;  of  Velocity  of  Sound,  461 ;  of 
Velocity  of  Wave-motion,  136 ;  of 
Volume,  33;  of  Wave-lengths  in 
Ether,  544,  550 ;  of  Weight  (spring- 
balance),  37 

Mechanical  Equivalent  of  Heat,  353 

Mechanical  Powers,  169-176;  with 
Friction,  180 

Medium,  44  ;  air  as  a  standard  — ,  578, 
598,  604,  675,  707 

Megadyne,  21 

Megalerg,  41 

Mega  volt,  711 

Megohm,  711 

Melde's  Experiments,  439 

Melting  Point  of  Ice,  360,  384,  402 

Membrana  Basilaris,  469 ;  membrana 
Tympani,  466,  473 

Membrane,  Diffusion  through,  285,  332  ; 
Vibration  of,  free,  137,  444,  forced, 
448 

Mercury,  Expansion  of,  382  ;  Fusing 
point  of,  384  ;  —  Thermometer,  401 ; 
—  Vapour,  243,  369,  465,  662 

Meridian,  9  ;  Magnetic  — ,  678,  680 

Metallic  Reflexion,  502,  521,  561;  in 
electromagnetic  waves,  743,  745 

Metals,  Colours  of,  502,  503 ;  Conduc- 
tivity of,  636  ;  Contact  of,  611 

Meteoric  Dust,  50  ;  Meteorites,  50 

Method  of  Mixtures,  404 

Method  of  Oscillations,  39,  213 

Method  of  Vibrations,  680 

Metre,  10 

Metric  System,  10 

Mho,  634,  715 

Mica,  564 

Microfarad,  711 

Microhm,  711 

Microphone,  636,  737 

Microscope,  571 

Microvolt,  625,  711 

Minimum  Angular  Velocity,  164 

Minimum  Deviation,  529 

Mirrors,  459,  523 ;  concave,  525 ;  con- 
vex, 527  ;  flexible,  527  ;  rotated,  512, 
525,  607  ;  spherical  — ,  123,  525,  527 

Miscibility  of  Liquids,  282 

Mixture  of  Colours,  574 

Mobile  Liquid,  226 

Modulus  of  Superficial  Tension,  276  ; 
Young's  — ,  261,  321,  441,  443,  461 

Molecule,  240  ;  Gramme ,  367 

Molecules,  218,  240,  242,  244,  246,  272, 
675,  685,  688,  692,  723,  727;  Nature 
of,  246  ;  Number  of,  251 ;  Energy  of, 
351  ;  Free  Path  of,  251,  252,  464  ; 
Size  of,  246,  251  ;  Velocity  of  — , 
250,  464,  610,  613 ;  —  in  Vibration/ 
247,251,  351,479,  500 

Molecular  Asymmetry,  244,  479,  568 ; 


772 


INDEX. 


Distance   of    —  Action,   278,    612; 

—  Forces,  253-256,  measurement  of, 
271 ;  —  Heat,  366 ;  —  Kinetic  Energy, 
361  ;  —  Refractive  Power,  529 

Moment :  of  a  Couple,  159,  166,  677  ; 

—  of  a  Force,  156,  166  ;  of  Impulse, 
166  ;  of  Inertia,  162,  166,  263,  680  ; 
Magnetic    — ,    677,    680,   682,   693; 
measurement  of,  680  ;  —  of  momen- 
tum, 166  ;  Principle  of  Moments,  155  ; 
Twisting  — ,  158,  160,  166,  263,  676, 
681,  691,  716,  723 

Momentum,  6, 14,  19,  20,  149, 150,  166, 
351  ;  Angular  — ,  163,  166 ;  Moment 
of  — ,  166;  —  of  a  System,  149; 

—  in  Impact,  150-152  ;  in  the  Elec- 
tromagnetic Field,  704 

Monochord,  437,  449 

Monochromatic  Light,  485,  542,  543, 
545 

Moon,  6,  76,  146,  201,  204,  679 

Morse-code,  733 

Motion,  14  ;  accelerated  — ,  69,  152  ; 
Change  of  — ,  6 ;  —  in  a  Circle,  8, 
57,  78,  161-169;  in  Curved  paths, 
57  ;  Laws  of  — ,  5  ;  The  Perpetual  — , 
7,  193,  357,  385;  —  parallel  to  or 
across  Equipotential  Surfaces,  192, 
194;  —  of  Lines  of  Force,  699; 
Quantity  of  — ,  19  ;  Reciprocating, 
88,  89;  Simple  Harmonic  — ,  80  et 
seq.  ;  Simultaneous  — s,  4,  60 

"  Motion"  =  Momentum,  6,  14,  19, 
149,  166,  351 

Motivity,  51 

Motors  :  electro  — ;  thermo-magnetic 
— ,  689 

Multiphase,  730 

Multiple  Proportions,  Laws  of,  239 

Multiplex  Telegraphy,  735 

Multipolar,  729 

Muscle  :  Breaking  Weight  of,  261  ;  Dif- 
fraction-Grating, 550  ;  Extensibility 
of,  261  ;  Mechanical  Disadvantage, 
171  ;  _  -Sound,  431,  471 

Musical  Box,  451 ;  Intervals,  422  ;  No- 
tation, 423  ;  Pitch,  420  ;  Scale,  420, 
422 

Mutual  Action  at  a  Distance,  577 

Mutual  Attraction  and  Repulsion  of 
Currents  :  Alternating,  724  ;  Steady, 
670,  702 

Mutual  Inductance,  704 

Mutual  Induction  of  two  Charged 
Bodies,  600,  601 ;  of  two  Currents, 
701,  704 

Myopic,  572 

NATURE,  Constancy  of,  2  ;  Laws  of,  5 

Negative  Crystals,  555 

Negative    Electric  Charge   (resinous), 

579,  602 
Negative  Electrode,  657 


Negative  Pole  of  Magnet  (South-seek- 
ing), 675 

Negative  Pressure  in  Thorax,  340 

Negative  Waves,  reflected,  125,  458, 
463,  518,  542,  545 

Nerve-ends,  350,  469,  575,  576 

Neumann  and  MacCullagh,  522 

Neutral  Line,  262 

Neutral  Point  (thermo-elect.),  626,  651 

Neutral  State  (electric),  579,  580 

Newton's  Laws  of  Motion,  5,  146,  167  ; 
his  Law  of  Cooling,  410  ;  Law  of 
Universal  Gravitation,  203  ;  Newton's 
Rings,  546,  in  electromagnetic  waves, 
743  ;  —  on  Sound,  412  ;  — 's  Law  of 
Velocity  of  Wave-motion,  267,  368 

Niagara  Falls,  352  ;  Energy  of,  739 

Nicol's  Prism,  555,  559,  568 

Nobert,  220 

Nobili's  Rings,  663 

Nodal  Lines,  137,  443;  Nodal  Points, 
108,  134  ;  (Gauss's)  539 

Nodes  and  Loops,  134,  137  ;  —  in  a 
Membrane,  137  ;  —  on  Monochord, 
438  ;  —  in  an  Organ-pipe,  452  ;  on  a 
vibrating  String,  438,  439 

Noe's  Thermo-electric  Pile,  629 

Noise,  416 

Non-commensurable,  93,  100,  103 

Non-conductors,  588,  603;  Contact  of 
— ,  610 

Non-conservative  System,  45 

Non-polarisable  Electrodes,  664 

Normal  to  Wave-front,  113;  Propaga- 
tion along  — ,  113,  117,  131,  139 

Normal  Spectrum,  141,  550 

Note,  417,  429 

Numerical  Measure,  16 

Nutation,  76 


OBOE,  451 

Oersteds,  715 

Ohm,  353,  634,  637,  640,  711,  715,  718 

Ohm's  Law,  633,  638,  640  et  passim 

Oil  in  painting,  501 ;  on  waves,  279 

Opacity,  497,  722,  744 

Opalescence,  503,  523 

Open  Circuit,  613,  614,  617 

Opera  Glass,  572 

Ophicleide,  453 

Ophthalmoscope,  572 

Opposition  of  Phase  in  Resonance,  446 

Optic  Axes,  557,  599 

Optical  Centre,  534,  541  ;  —  Density, 
509  ;  —  measurement  of  Length,  32 

Ordinary  Ray,  553,  555 

Ordinate,  11 

Organ  : pipe,  412,  446  ;  open,  451, 

452,  457,  462  ;  stopped,  454 ;  —  -reed- 
pipe,  429,  451 

Oscillating  Currents,  721-733,  735,  740 

Oscillation,  39,  414 ;  as  a  means  of 
measuring  Force,  39,  213,  608,  680 ; 


INDEX. 


773 


Centre  of  — ,  164,  214  ;  —  of  Elastic 
Body,  267  ;  Frequency,  83,  112,  412, 
414,  435,  479 ;  — s  of  a  mercury- 
column,  147  ;  of  Molecules,  351,  479  : 
of  Pendulum,  34,  21? 

Oscillatory  movements  of  Systems  of 
Particles,  103-142 

Osculating  Circle,  79 

Osmosis,  285  :  Osmotic  Pressure,  281, 
288 

Ounce,  13 

Outflow,  302,  303,  334;  from  elastic 
tubes,  323 

Overcooling,  281 

Overturning,  207-210  ;  Angle  of,  209 

Oxygen,  Condensation  of,  224,  232; 
magnetic  properties,  232,  685 ;  spe- 
cific heat,  369 

Ozone,  Condensation  of,  232  ;  forma- 
tion of,  662 

PAGE-EFFECT,  737 

Palladium,  327,  664 

Parabolic  Mirrors,  122,  525;  —Path, 
72 ;  —  Surface  of  rotating  liquid, 
169,  294 

Paradox,  Hydrostatic,  292 

Parallel  Beam,  117 

Parallel,  Cells  arranged  in,  617  ;  Dyna- 
mos in  — ,  730 

Parallelepipedon  of  Accelerations,  68 ; 
of  Forces,  145  ;  of  Velocities,  67 

Parallelogram  of  Accelerations,  68  ;  — 
of  Forces,  143;  experimental  proof 
of,  144  ;  —  of  Velocities,  61 

Paramagnetic,  685,  688 

Partially-polarised,  516,  521,  522  ;  de- 
tected, 563 

Partials  (i.e.,  Harmonics,  ^.v.) 

Pascal's  Principle,  290,  333 

Peltier's  Effect,  649,  650  ;  — 's  Electro- 
scope, 605 

Pendulum,  9,  34,  36,  86,  149,  206,  210 ; 
Ballistic  — ,  214,  713  ;  Blackburn's, 
211;  Compensation ,  380;  Com- 
pound, 213  ;  Conical,  80,  185  ;  energy 
of,  142  ;  —  Formula,  212,  680  ;  Gali- 
leo and  — ,  34,  202  ;  Hydrostatic  — , 
318,  336;  --  Oscillations  Isochro- 
nous, 34,  212 ;  —  movement  Har- 
monic, 86,  211 ;  Length  of  Simple  — , 
206,  212,  214;  of  Compound,  213; 
Simple  — ,  206,  210-213  ;  Work  done 
in  moving  a  Simple  — ,  213 

Penumbra,  547 

Perception  of  Colour,  573,  575 ;  of 
Form,  576  ;  of  Matter,  216,  246 

Percussion,  Centre  of,  165 

Perfect  Conductors  and  Non-conduc- 
tors, 588 

Perfect  Elasticity,  265,  267 

Perfect  Engine,  397,  399 

Perfect  Gas:    denned,  369;    Internal 


Work  in,  372;  No  Physical  Gas  Per- 
fect, 374 

Perfect  Solid,  225 

Perfectly  Conducting  Molecules,  692 

Period  of  S.H.M.,  82,  414,  680;  of 
Wave-Motion,  112 

Periodic  Curve,  103  ;  —  Function,  82 

Permanent  Magnet,  674,  689,  692 

Permeability,  Magnetic,  671,  676,  682- 
684,  685,  686,  691,  694,  703,  706-709, 
722,  744,  746  ;  measured,  687 

Permittance,  592,  597,  599,  603,  604, 
612,  692,  709,  744,  746 

Permittivity,  597 

Perpetual  Motion,  The,  7, 193, 195,  357, 
385 

Phase  in  S.H.M.,  82;  perceived  by 
Ear,  459 

Phonautograph,  433 

Phonograph,  433,  448 

Phonomotor,  457 

Phonophore,  735 

Phosphorescence,  504,  542,  656 

Phosphoroscope,  505 

Phosphorus,  282  ;  —  Vapour,  243 

Photichthys,  507 

Photographic  Focus,  541 

Photographing  Vibrations,  433 

Photography,  433,  481,  503,  569;  Ex- 
posure in  — ,  482  ;  Pinhole  — ,  548 

Photometry,  513 ;  Unit  in,  513 

Photophone,  637,  737 

Physiology,  1 

Pianoforte,  416,  420,  425,  437,  450 

Piccolo,  453 

Piezometer-tube,  295,  310,  334 

Pigments,  575 

Pile,  Dry,  605,  622;  Galvanic,  608, 
616  ;  Thermo-electric,  542, 628 ;  Vol- 
ta's,  617,  618,  623 

Pint,  13 

Pipette,  343 

Piston,  40,  55,  229,  237,  356,  399 

Pitch  of  a  Screw,  173 

Pitch  of  a  Sound,  415,  483;  musical 
Specification,  420 ;  physical,  418 ; 
standard  — ,  421 ;  —  of  a  Vibrating 
String,  435,  441 ;  Variations  of  —  in 
an  Organ-pipe,  452  ;  Variations  of 
—  with  Temperature,  452 

Pitot's  Tubes,  319,  336 

Plane  in  Space,  Pendulum  Swinging  in 
fixed,  148  ;  Wheel  rotating,  149. 

Plane  of  Incidence,  517,  619,  521 

Plane,  Inclined,  172,  173,  181 

Plane  of  Polarisation,  521,  522,  555, 
559,  566,  694 

Plane-polarised,  514,  521,  555,  558,  560, 
666 ;  detected,  563 

Plane  Wave-Front,  115 ;  reflected,  119 

Plant6,  665 

Plasticity,  259  ;  of  Ice,  585 

Plates,  Vibration  of,  443 


774 


INDEX. 


Pneumatometer,  336 

Pneumothorax,  337 

Points  :  Conjugate,  130,  526,  535  ;  Cor- 
responding, 576  ;  Critical,  233,  325  ; 
Dead,  89 ;  Neutral,  626,  651 ;  Nodal, 
108,  134,  (Jauss')  539 

Poiseuille's  Law,  308,  315 

Poisson's  Ratio,  260 

Polarisation  of  Light,  etc.,  514  ;  Plane, 
514;  Circular,  514,  560,  562-564; 
Elliptical,  515,  560;  Partial,  516; 
Right-  and  Left-Handed,  515,  517, 
560;  Rotatory,  517,  566;  Angle  of 
Complete  — ,  521  ;  Plane  of  — ,  521, 
555,  559,  566,  694;  Magnetic  Rota- 
tory — ,  694 ;  —  in  Electromagnetic 
Waves,  743,  745 

Polarisation  of  Dielectric,  603;  —  of 
Electrodes,  664  ;  —  of  Galvanic  Cell, 
619,  632  ;  —  -Current,  664 

Polarised  Light,  514 ;  to  detect , 

563,  564 

Polariser,  516,  555 

Pole :  Magnetic,  of  Earth,  679 ;  of 
Magnet,  668,  675,  685,  691 ;  Secon- 
dary, 677 

Polygon  of  Accelerations,  68  ;  of  Forces, 
145;  of  Velocities,  66,  68;  Skew- 
polygon,  68,  145 

Porosity,  220 

Position,  Change  of,  14 

Positive  Crystals,  555  ;  —  Direction  up- 
wards or  to  the  right,  11 ;  —  Direc- 
tion of  Electromagnetic  Lines  of 
Force,  668;  of  Magnetic,  676;  — 
Electrical  charge  (vitreous),  579,  602, 
746  ;  —  Electrodes,  657  ;  —  Mutual 
Action  Repulsive,  189 ;  —  Pole  of 
the  Earth  (Antarctic),  693  ;  —  Pole 
of  a  Magnet  (north-seeking),  668, 
675,  746  ;  —  Side  of  a  Circuit,  672  ; 
Thermo-electrically  — ,  625 

Potential,  52,  191,  301,  583,  608,  692  ; 
Analogy  of  Sea-Level,  193,  of  Water- 
Level,  193,  195,  586 ;  Analogy  of 
Temperature,  588  ;  Absolute  — ,  192, 
584 ;  Absolute  Zero  of  — ,  192  ;  Ar- 
bitrary Zero,  193,  588 ;  Continuity 
through  Zero  Value,  193,  588  ;  — 
Difference,  193, 584, 586, 587, 595,  597, 
604,  609,  612,  623,  624,  709  ;  meas- 
ured, 608,  646  ;  produced,  609-  ; 
electromotive  difference  of  — ,  587, 
608, 609, 616, 625, 646  ;  —  of  Air,  607, 
611 ;  —  of  Double  Sheet,  200 ;  Elec- 
trical — ,  583  ;  Fall  of  — ,  638,  641, 

650,  654,  656  ;  Gravitation ,  193, 

195,  492,  584  ;  —  Gradient,  191,  585, 
587,  597,  598,  602,  633,  639,  648,  710 ; 
—  Line,  639,  641,  696,  720,  ,  ; 
Magnetic  — ,  682,  693  ;  —  of  Mag- 
netic Shell,  683;  Mutual  — ,  191; 
Positive  and  Negative  — ,  192;  — 


Slope,  191,  585,  587,  597,  598,  602, 
604,  633,  639,  648,  710  ;  The  — ,  588, 
609  ;  Zero  — ,  192,  193 

Potential  Energy,  43,  45,  354  ;  in  cur- 
rent-field, 684,  699,  702  ;  in  case  of 
Repulsion,  190 

Potentiometer,  646 

Pound,  12 

Poundal,  21 

Powdered  Transparent  Substance,  497 

Power,  -  Activity,  42,  59 

Power,  Horse-,  42,  647,  653,  655 

Power  of  Lens,  534 

Power,  Molecular  Refractive,  579 

Powers,  the  Mechanical,  169-176, 180 

Practical  Electrical  Units,  618,  634,  637, 
711 

Precession,  75  ;  Angle  of,  75 

Pressure,  24 ;  Atmospheric,  229,  293, 
336-349;  Critical  —  (Carnelley's), 
236;  Critical—  (Andrews'), 233, 376; 
Electric  — ,  587  ;  Electricity  on  — , 
623;  —  an  Energy-Slope,  42,  603; 

—  of  Ether-Waves,  570  ;  —  affecting 
Fusing- Point,  237,  384  ;  —  in  a  Gas, 
=    coefft.    of    elast.,    324,    368  ;    in 
Gases,  229,  333,  Dalton's  Law,  250, 
253,  by    Kinetic  Theory,  250;   pres- 
sure and  volume  in  gases,  230  ;    — 
produced  by  heated  solids,  377  ;  di- 
minished in  Fluid  in  Motion,  335  ;  — 

-Gradient,  301 ; Head,  310,  334  ; 

Hydrostatic  — ,  25,  290,  301,  310,  325, 
602  ;    Intensity   of  — ,  25  ;  —  Line, 
311,  334  ;   —  in  Liquids,  289,  602  ; 

—  in  Heavy  Liquids,  291 ;  measure- 
ment, 295-299 ;  Osmotic  —  or  Solu- 
tion  ,  281,  288;    Restitution , 

256,    264,  266  ;  Saturation ,  390  ; 

Slope,  301  ;  —  in  Streams,  309- 

317  ;  —  of  Sunlight,  570  ;  Total  — , 

24 ;  Vapour ,  387,  of  a  Solution, 

281,  387  ;  Vibrating  Fluid,  335 

Prevost's  Law,  491 

Primary  Circuit,  700,  724 

Primary  Colours,  575 

Principal  Focal  Distance  (=  Focal 
Length),  130,  525,  534,  535,  536 

Principal  Focus  of  Mirror,  525  ;  of  Lens, 
534 

Principal  Section  of  a  Crystal,  551 

Principle  of  Moments,  155 

Prism,  485,  493,  528,  743  ;  achromatic 
— ,  531;  Foucault's,  556;  Nicol's, 
555,  559,  568  ;  Rochon's,  556 

Projectile,  Path  of,  203 

Projection,  84  ;  —  of  S.H.M.,  84 

Proof-plane,  607  ;'  proof -sphere,  607 

Propagation  of  Elasticity- Waves,  267  ; 
of  Electromagnetic  Disturbance,  479, 
480,  481,  498  ;  —  of  Groups  of  Waves, 
142  ;  of  Heat,  409,  602  ;  —  of  Sound, 
428,  456-465  ;  —  of  Temperature, 


INDEX. 


775 


409 ;   —  of   Waves  along  Normals, 

113,  116,  131,  139 
Properties  of  Matter :  Contingent,  220  ; 

Essential,  216  ;  General,  219 
Proportion,    fixity    of    (chem.),   238 ; 

Multiple  proportions,  239 
Ptolemy's  Law,  133 
Puissance,  42 
Pull,  23 
Pulleys,  174 
Pulse  :  dicrotic,  322  ;  locomotive,  313  ; 

wave,  321 

Pump,  342,  345,  587 
Putting  to  Earth,  698,  719,  720 
Pyrometry,  404 
Pythagorean  Intervals,  425 

QUADRANT,  711 ;  —  of  Earth,  10 

Quadrant  Electrometer,  606 

Quadruplex  Telegraphy,  734 

Quality  of  Sound,  415,  428 

Quantity :  Cells  arranged  in  — ,  641  ; 
Electric  — ,  578,  594,  603,  638  ;  unit 
of  electric  —  (C.G.S.  electrostatic), 
578,  603,  604  (C.G.S.  electromag- 
netic), 675,  693,  714  (practical),  711, 
714 ;  —  of  Magnetism,  675,  193  ;  - 
of  Matter  =  Mass,  12, 166,  216,  247  ; 
—  of  Motion  (=  Momentum j,  6,  19, 
149,  150,  166,  351 

Quarter-undulation  Plate,  561,  563 

Quartz  as  a  Dielectric,  599  ;  —  Prisms 
and  Lenses,  486,  498,  504;  —  in 
Rotatory  Polarisation,  666 

RACEMIC  Acid,  568 

Radian,  75 

Radiant  Heat,  48,  234,  350,  410,  481, 
541, 743  ;  chemical  decomposition  by, 
482 

Radiant  Matter,  234,  252,  656 

Radiation,  410,  478  ;  —  from  a  hot 
body,  488  ;  —  from  gas  and  vapour, 
495  ;  _  from  liquids  and  solids,  495  ; 
exchange  of  radiations,  489  ;  —  and 
absorption,  491 ;  Intensity  of  — ,  513 

Radii  of  Earth,  205 

Radiometer,  361 

Radiophony,  455 

Radius  of  Curvature,  79,  165  ;  —  of 
Gyration,  162 ;  —  of  Inertia,  162, 
166 

Railway  :  Force  of  Engine,  181 ;  Super- 
elevation, 209 ;  wheels,  182 

Rain,  348 

Rainbow,  530 

Raindrop,  255,  272,  530,  579;  friction 
on  — ,  185 

Ram,  Hydraulic,  148,  312,  704 

Raoult's  Laws  of  Freezing  Point,  386, 
Osmotic  Pressure,  288,  and  Vapour 
Pressure,  387,  in  Solutions 

Rarefied  Air,  Sound  in,  413 


Rate  of  Change  of  Momentum  =  Force, 
19,  154,  248 

Ratio,  Electrostatic-Electromagnetic, 
708,  709 

Ray,  116,  131  ;  Kinds  of  Radiation, 
480  ;  Convergent — s,  117  ;  Divergent, 
117 ;  Ordinary  and  Extraordinary 
Rays,  553 

Reaction,  6  ;  Action  and  — ,  6,  23 

Real  Image,  526,  535,  536 

Reciprocal  or  Reciprocating  Motion, 
88,  89 

Reciprocating  Engine,  397 

Recoil,  306,  334 

Reduced  Length,  637,  641,  642  ;  —  Re- 
sistance, 637,  641,  642 

Reduplication,  Principle  of  (in  Pulleys),. 
174 

Reeds,  Vibrating,  442,  443,  451,.  736- 

Reference  to  Axes,  11,.  66 

Reference,  Circle  ofr  81,  82,  185 

Reflexion,  117-124  ;  Angle  of  — ,  120, 
518  ;  Caustic  by  — ,  123,  525 ;  —  of 
Ether- Waves,  517,  743;  Metallic  — , 
502,  521,  561,  743,  745  ;  of  Negative 
Waves,  125,  458,  463,  518,  542,  545  ; 

—  in  electromagnetic  waves,  727,  743  ; 

—  of  Sound- Waves,  459 ;  Total  — , 
519 

Refraction,  124-131, 217  ;  Angle  of  — , 
126,  518,  528 ;  Caustic  by  — ,  129, 
130, 537  ;  Conical  — ,  557  ;  Double  — , 
228,  551-566,  575,  599  ;  —  of  Elec- 
tromagnetic Waves,  743 ;  —  of  Ether- 
Waves,  517,  527  ;  Index  of  — ,  127, 
509,  518,  522,  528,  534,  540;  of 
liquid,  529,  of  solid,  528  ;  for  X  =  oo, 

—  of  Sound- Waves,  461 ;   Total  — , 
520 

Refractions,  Atomic,  529 

Refractive  Index,  127,509,  518, 522,  528, 
534,  540,  744 

Refractive  Power,  Molecular,  529 

Regelation,  385 

Relaxation,  Time  of,  228 

Relay,  735 

Reluctance,  691,  714 

Reluctivity,  714 

Replenisher,  Lord  Kelvin's,  631 

Repose,  Angle  of,  178 

Repulsion  Conventionally  Positive,  189 ; 
Direction  of,  584,  585  ;  —  in  Field  of 
Force  or  Flow,  602,  721  ;  Potential 
Energy  in  case  of,  190  ;  Work  done 
by,  190 ;  Repulsion  of  Resonator,  431 ; 
Self (elect.),  582 

Research,  8 

Residual  Discharge,  599 

Residual  Magnetisation,  684,  690 

Residual  Restitution,  266 

Resinous  (negative),  579,  746 

Resistance  to  Electric  Current,  615,  616, 
633,  634,  638,  639,  644,^54,  710,  725  ; 


776 


INDEX. 


measured,  633,  645,  647,  718 ;  reduced 
do.,  637,  641,  642  ;  —  Coil,  634,  646 

—  to  Deformation,  259,  264,  Compres- 
sion, 259,  Extension,  261,  Shear,  260 
Twist,  263  ;  —  of  Electrolytes,  646  ; 
to  Flow,  306,  310  ;  Friction  a  — ,  176, 
180 1 ;   —  to  Traction,  181  ;   —  a  Ve- 
locity, 710  ;  Viscosity s,  185 

Resistivity,  634,  638,  689,  710 

Resolution,  of  Forces,  143 ;  —  of 
S.H.M.'s,  96,  102  ;  —  of  Velocities, 
63 ;  —  of  Vibrations,  longitudinal, 
110,  transversal,  107,  108 

Resonance,  430,  445,  742 

Resonators,  417,  430,  431,  432,  476; 
Herz's  — ,  741  ;  Repulsion  of  — ,  431 

Restitution-Pressure,  256,  264 ;  de- 
ferred  ,  266;  electric,  744;  Co- 
efficient of  R.  (impact),  151,  (elas- 
ticity), 264,  of  deferred  R.,  266 

Resultant  Electric  Force,  581,  583,  603  ; 

—  Force,  142  ;  —  Harmonic  Motion', 
102  ;  —  Motion,  60 

Retardation,  Electrostatic,  596 

Retina,  570,  573,  575,  576 

Reversal  in  Thermo-Electricity,  627, 
651 ;  Temperature  of  — ,  627 

Reversed  Action,  611,  737 

Reverse  Current,  650,  664 

Reverse  Extra-Current,  705 

Reverse  E.M.D.P.,  654,  (electromo- 
tor), 738 

Reversibility  of  Carnot's  Ideal  Engine, 
397  ;  of  Lenses,  534 

Rheochord,  Rheostat,  647 

Rhomb,  Fresnel's,  562 

Right-handed  Polarised  Light,  515,  560 

Rigid-  Body,  73  ;  degrees  of  Freedom 
of,  76  ;  —  Solid,  225 

Rigidity,  225,  227,  260,  415  ;  coefficient 
of,  226,  260 ;  —  through  Rotation, 
226  ;.  —  affecting  Vibrations,  415,  435 

Rings,  Newton's,  546,  743  ;  Nobili's, 
663 

Rochon's  Prism,  556 

Rods,  Flexure  of,  263  ;  Vibrations  of, 
135,  441,  443. 

Rolling  down  a  Curve,  173 

Rolling  Friction,  182 

Rope  and  Post,  Friction  between,  178  ; 
wetted  rope,  282 

Rotating  Mirror,  512,  525,  607 

Rotation,  74,  161  ;<  Axis  of  — ,  74,  76, 
162;  Composition  of  — s,  74;  — of 
the  Earth,,  164r ,  205  ;  Energy  of  — 
of  a  particle,  162,  of  a  Mass,  163; 
Force  causing  —  Constant  in  Direc- 
tion, 158  ;  Instantaneous  Axis  of  — , 
75 ;  —  round  Lines  of  Force,  745 ; 

—  of    a    Liquid,    168r   294;   —  of 
Magnet-Poles   round   Currents,  669, 

—  of    Molecules,   351,   465;    —  of 
Plane  of  Polarisation,  244,  517,  566  ; 


of  Plane  of  S.H.M.,  101 ;  Simple  — , 

74  ;  Axis  of  Spontaneous  — ,  165 
Rotatory  Field,  741  ;    —  Polarisation, 

517,  566  ;  Magnetic ,  694,  745 ; 

—  Power,  567  ;  —  Vibration,  443 
Rowland,  352,  550,  744 
Ruhelage,  39,  263 
Rupert's  Drops,  255,  388 

SACCHARIMETER,  567  ;  Soleil's,  568 

Safety-tube,  340 

Salt- radicle,  658,  660 

Sand-blast,  258 

Saponine,  278 

Sassafras,  547 

Saturated  Solution,  280  ;  —  Steam,  371, 
390 ;  —  Vapour,  231,  spec,  heat  of, 
371 

Saturation-Pressure,  390 

Savart's  Wheel,  418 

Scale  (measuring),  27,  (musical),  420, 
422,  (therometric),  364,  402. 

Scattering  by  Haze,  503 ;  in  electro- 
magnetic waves,  743 

Schallenberger's  Alternating  Current 
Meter,  741 

Screen,  Electric,  601  ;  Magnetic,  691, 
to  Alternating  Field,  722  ;  effect  of 
S.  on  Waves,  138,  139 

Screw,  29, 173, 181;  male  and  female,  30 

Sea-Level,  Analogy  in  Potential,  193 

Sealing-wax  a  fluid,  226 

Secohm,  711  ;  Secohm-Meter,  706 

Second,  9 

Secondary  Batteries,  Cells,  665-667  ;  — 
Circuit,  700 ;  Current  (induction), 
687,  700,  724,  (polarisation),  665  ;  — 
electrolytic  reactions,  659  ;  —  Poles, 
679 

Selective  Absorption,  498 

Selenite,  563 

Selenium,  570,  636 

Self-induction,  646,  704,  710,  722,  724, 
727,  745  ;  Coefficient  of,  705,  710 

Self- Repulsion :  electric,  582  ;  of  Hy- 
drogen, 254,  375,  377 

Semitone,    true    (if),    422 ;    so-called 

(Iff).  423 
Sensible  Heat,  350 
Sensitive  Flames,  454 
Sensitiveness  of   Galvanometers,   713 ; 

of  Thermometers,  402 
Separately  Excited  Dynamos,  730. 
Separation  of  Electricities,  602,  609  ;  of 

Magnetisms,  686,  689,  692 
Series,  Cells  coupled  in,  618,  640;  — 

Alternators   in    Series,   730 ;    Series 

Dynamos,  732 
Series-Shunt  Dynamos,  732 
Sextant- Vernier,  28 
Shadow,  140,  547 
S.H.M.  (Simple  Harmonic  Motion)  80. 

See  Harmonic  Motion. 


INDEX. 


777 


Shear,  78,  227,  260 ;  Ether-  — ,  746 

Shearability,  260  ;  —  of  Ether,  746 

Sheet,  Current,  693 

Shell,  Momentum  of  explosive,  149. 

Shell,  Magnetic,  682,  707  ;  Equivalence 
of  —  and  Circuit,  683,  707  ;  Poten- 
tial of,  683  ;  Strength  of,  682,  693 

Shunts,  645,  661 ;  Shunt  Dynamos,  732 

Siderial  Time,  9 

Siemens's  Electrodynamometer,  717 ; 
Governor,  168  ;  Inductor,  729 

Simple  Harmonic  Motion,  80  et  seq. ; 
—  —  Variations  of  Current,  etc., 
721,  722,  729 

Simple  Machines,  169 

Simple  Pendulum,  206,  210-213 

Simple  Rotation,  74 

Simple  Translation,  73 

Simultaneous  Causes,  4;  —  Currents, 
644  ;  —  Forces,  143,  145  ;  —  Motions, 

4,  60 
Sine-Galvanometer,  713 

Sines,  Curve  of,  85,  97,  106,  111 

Singing  Flames,  454 

Siphon,  345 

Skew-polygon,  68,  145 

Sky,  503 

Sliding,  176,  177,  210 

Sliding  Condenser,  600 

Slope:  Energy-Slope,  42,  603;  Inten- 
sity-Slope, 200  ;  Lines  of  Slope,  200, 
687  ;  Potential-Slope,  191,  585,  587, 
597,  598,  602,  604,  633,  639,  648,  710 ; 

5.  of  Potential-Line,  639,  641,  Pres- 
sure-Slope, 301 

Soap-bubble,  37,  330,  582  ;  soap-film, 
37,  245,  273,  330,  498,  545 ;  electro- 
magnetic waves,  743 

Sodium-flame,  483,  493,  502 

Soft  Iron,  684,  689 

Softness,  258 

Soft  Solid,  226 

Solar  Time,  9 

Soleil's  Saccharimeter,  568 

Solenoids,  673,  674,  705,  726;  Sole- 
noidal  Magnet,  675. 

Solid,  12,  225,  256-270;  rigid,  225; 
soft,  226  ;  perfect  — ,  225 

Solubility,  328 ;  coefficient  of,  280,  328 

Solution':  Saturated  — ,  280;  super- 
saturated — ,  281  ; pressure,  281, 

288 ;  Density  of  a  — ,  281  ;  Dissocia- 
tion upon,  248,  280,  386,  590,  614  ; 
— s  as  conductors,  590 

Solution  in  Liquids :  of  Solids,  279, 
358  ;  of  Gases,  328,  358  ;  Coefficient 
of  Solubility,  280,  328. 

Solution  of  Gases  in  Solids,  327  ;  —  in 
gases,  330  ;  of  Solids  in  gases,  330 

Sonorescence,  570 

Sound,  7.  48,  412-477  ;  Analysis  of, 
429  ;  Direction  of,  471 ;  Propagation 
of,  428,  456-465;  Velocity  in  Steel, 


267;  Waves  in  Air,  413,  432,  456, 
464,  737 

Sounding-Board,  413,  737 

Source  and  Condenser,  384,  396,  615, 
648,  652 

Space,  9;  Dimensions  of,  10;  —  tra- 
versed under  Uniform  Acceleration, 
70 

Spark,  580,  582,  589,  590,  612,  706,  741 

Speaking-Trumpet,  414 

Specific  Conductivity,  634  ;  —  Density, 
220  ;  Measurement  of,  221-224,  294, 

295, 382 ;  —  Gravity,  221 ; bulbs, 

223 ;  —  Heat,  353,  365,  370,  404,  405, 
653 ;  at  const,  vol.,  367  ;  at  const, 
pressure,  367  ;  ratio  of  —  Heats,  368 ; 
Differences  in  this  Ratio,  369  ;  — 
inductive  capacity,  597,  602-604,  612, 
692,  709,  744,  746  ;  of  dielectric  =  /32, 
744  ;  —  Resistivity,  634,  638,  689  ;  — 
Thermal  Capacities,  324,  365,  367,  372 

Spectrum,  484,  486,  494,  530;  Abnor- 
mal, 532  ;  Band  — ,  496  ;  Continuous 
— ,  495  ;  Dark  Lines  in,  494,  495 ; 

Diffraction ,  550  ;  Heat-  — ,  486, 

500,  533  ;  —  of  Ice,  495 ;  Line , 

495,  496  ;  Normal  — ,  141,  550 

Spectrum  Analysis,  217,  494 

Speed,  15 

Sphere,  Capacity  of,  593,  599 

Spherical  Aberration  of  Lens,  537  ;  of 
Mirror,  525  ;  —  Form,  254  ;  —  Mag- 
net, 678  ;  —  Mirror,  123,  525,  527  ; 
—  Wave,  128, 130,  514 

Spheroidal  State,  363,  623 

Spherometer,  31 

Sphygmoscope,  298 

Spinning-Top,  75 

Spirometer,  336 

Spontaneous  Rotation,  165 

Spoud,  (C.G.S.  Unit  of  Acceleration, 
one  kine  per  second),  18 

Sprengel-pump,  252,  326 

Spring  Balance,  37 

Squares,  Law  of  Inverse,  187,  582,  585, 
675 

Stability,  215 

Standards:  See  Units;  —  Cell,  622, 
646;  —  Condensers,  609, 718 ;  —  Ohm, 
718 ;  —  Pitch,  421 ;  —  Resistance- 
Coil,  634,  646,  718 

Standard  Atmospheric  Pressure,  348, 
349 

Stapes,  467 

Stars,  Twinkling  of,  550 

State,  Critical,  232,  376 

States  of  Matter,  225;  Change  of  — , 
235,  354-358 

Statical  Friction,  176 

Stationary  Vibrations,  134 

Steady  Currents,  632-674,  721,  745 

Steady  Flow  of  Heat,  407  :  of  Liquids, 
299 


778 


INDEX. 


Steadiness  of  Flow,  300 

Steam  :  Steam-engine,  40,  49,  352,  397, 
399 ;  Steam-gauge,  297 ;  Jet  issuing 
into  air,  373,  374  ;  latent  heat  of,  390, 
411 ;  specific  heat  of,  371  ;  saturated 
— ,  371,  390  ;  thermal  capacity  of  — , 
371 

Stereoscope,  576 

Stethoscope,  428,  456 

Stokes's  Law,  492 

Stopping  Component  Vibrations,  135 

Storage  of  Energy,  666 ;  in  the  Ether, 
586,  593,  603,  648 

Straight  Path,  5,  57 

'Strain,  23,  77;  Homogeneous  — ,  78; 
Strain-Ellipsoid,  78 

Streams,  308-323;  Stream-Lines,  300, 
602  ;  Streams  of  Liquid,  299  ;  pres- 
sure in,  309-317  ;  velocity  of,  317- 
319 ;  viscosity  in,  301,  307  ;  work  done 
in  keeping  up,  319  ;  —  of  Gas,  334 

Strength  of  Current,  633,  637,  638,  659, 
684,  694,  703,  707,  709  ;  —  alternat- 
ing, 728,  726 ;  —  of  Magnet,  675  ;  —  of 
Magnetic  Field,  676,  693,  701  ;  —  of 
Magnetic  Shell,  682,  684,  693,  710  ;  — 
of  Materials, 255;  —of  Structures, 269 

Stress,  23,  43,  145,  187,  579  ;  Intensity 
of  — ,  24  ;  Measurement  of,  37  ;  — , 
Electric,  234,  577,  582,  586,  593,  599, 
602,  603  ;  Hydrostatic,  225,  257, 300  ; 
Effect  of  Eepeated  Variations  of,  267 

Stria?,  656,  662 

Striking  Distance,  580,  589,  590,  613 

String  Organ,  451 

String,  Tension  in  Stretched,  26 

Strings,  Vibration  of,  423,  434-441, 
446 ;  when  bowed,  435 ;  plucked, 
434,  436  ;  struck,  436 

Stroboscopic  Disc,  305 

Stromuhr,  318 

Structures,  Strength  of,  269 

Sublimation,  386 

Submarine  Cable,  600,  696  ;  —  Teleg- 
raphy, 600,  696-699,  734 

Sucker,  339 

Suction,  305,  339 

Sulphurous  Acid,  325,  363,  389 

Summational  Tones,  473,  474 

Sun's  Atmosphere  j  218,  486,  494,  496  ; 
— 's  Attraction,  6,  76,  204 ;  —  En- 
ergy, 50,  478  ;  Sunlight,  50,  478,  486, 
488,  491,  496  ;  Energy  of,  479  ;  Pres- 
sure of,  570 ;  Sun-motor,  491  ;  Sun- 
set, 527  ;  Sun's  disc,  491  ;  Sun's 
magnetic  effect,  679  ;  Sun's  Tempera- 
ture, 489,  491 

Superelevation  of  Rails,  209 

Superficial  Charge  of  Conductor,  579, 
583,  600;  —  Density  (elect.),  579, 
583,  603,  604,  612  (magn.),  683,  686, 
692,  693  ;  —  Film,  254  ;  —  Tension, 
modulus  of,  276  ;  —  Viscosity,  278 


Supersaturated  Solution,  281 

Surf,  279 

Surface,  11 ;  —  Adhesion,  306  ;  —  At- 
traction of  a  Gas,  327  ;  Cells  coupled 

in  — ,  617,  640; Conduction  of 

Currents,  696,  721 ; Density,  188, 

579,  583,  603,  604 ;  Equipotential  — s, 
193-199,  301,  409,  582,  583,  585,  588, 
592,  595,  599,  602,  648,  672,  676,  682, 
683;  —  of  Falling  Bodies,  203;  — 
Friction,  306 ;  in  Gases,  no  Free  — , 
229  ;  Isodynamic  — s,200;  Isothermal 
— s,  408,  602  ;  Ley  den  jars  coupled  in 
— ,  600  ;  —  of  Liquids,  169,  272,  294  ; 

Measurement  of,  32  ; Tension,  26, 

254,  272,  294,  305,  579,  603,  624  ; 
measurement  of,  37,  275 

Susceptibility,  Magnetic,  686,  690 

Suspended  Body,  164,  209 

Suspension,  Bifilar,  215,  606 

Swing-back,  538 

Synthesis  of  Sound,  432  ;  of  Vowel- 
Sound,  476 

Syren,  418 

Syringe,  339 

TANGENT,  57  ;  Tangent-Galvanometer, 
637,  710,  713,  718 ;  —  -Scale,  607  ; 
Screw,  29 

Tangential  Velocity,  59,  165 

Tasimeter,  551,  636 

Teinte  de  passage,  568 

Telegraphic  Code  (Morse)  733  ;  —  Re- 
lays, 735 

Telegraphy,  644,  679,  697,  727,  733; 
Deep  Sea  — ,  600,  696-699,  734  ;  Du- 
plex — ,  734  ;  Harmonic  — ,  736  ;  Mul- 
tiplex, 735 ;  Quadruplex,  734 

Telegraph  Wire,  600,  643,  739,  740 

Telephone,  471,  644,  646,  701,  736 ;  — 
-Currents,  737  ;  Mechanical  Pulsion 
— ,  456  ;  Wire  — ,  456 

Telescope,  549,  572 

Temperament,  Equal,  425 

Temperature,  249,  250,  359,  364,  399, 
588  ;  Absolute  — ,  251,  364,  397,  399  ; 
true  C.G.S.  unit  of  — ,  365  ;  —  of 
Condensation  of  Vapour,  392 ;  Crit- 
ical — ,  232,  237,  253,  376,390,  (mag- 
netic) 689;  Flow  of  — ,  409;  — 
-Gradient,  407,  602  ;  Measurement  of 
— ,  400  ;  Propagation  of  — ,  408,  409  ; 
—  of  Reversal,  627  ;  Waves  of  — ,  409 

T:nacity  of  a  Liquid  Stream,  254,  345 

Tension,  25,  167,  183,  184,  260,  415 ; 

Intensity  of,  26  ;  Electric  Surface , 

582,  603  ;  Cells  coupled  in  — ,  640, 
666  ;  —  of  String  in  circular  motion, 
167  ;  Surface  — ,  26,  252,  272,  294, 
305,  579,  603,  624 

Tenth-metre,  480,  483,  575 

Terrestrial  Magnetism,  679,  691 

Tesla's  Experiments,  723 


INDEX. 


779 


Theorem,  Fourier's,  103,  135,  266,  412, 
414,  416,  417,  434 

Theory,  8 

Theory  of  Dimensions,  16 

Thermal  Capacities,  324,  365,  367,  370, 
372 

Thermal  Conductivity,  Coefficients  of, 
407,  602,  636 

Thermal  Diffusivity,  407 

Thermic  Balance,  542,  551,  717  ] 

Thermodynamic  Constant,  370 

Thermodynamic  Function,  395 

Thermodynamics  :  First  Law,  353 ; 
Second  Law,  398 

Thermo-electricity,  612,  624,  649-652 

Thermo-electric  Circuit,  624,  651  ;  — 
Diagram,  626,  651  ;  —  Effect  in  Elec- 
tric Arc,  654  ;  —  Pile,  542,  628  ;  — 
Power,  625,  689;  —  Eeversal,  627, 
651  ;  —  Series,  625 ;  —  Thermome- 
ter, 404,  628 

Thermo-electrically  positive  and  nega- 
tive, 625 

Thermolysis,  243,  247,  248,  355,  367 

Thermo-magnetic  Motors,  689 

Thermometers :  Air,  401  ;  BrSguet's, 
401  ;  Mercury,  401 ;  special  forms, 
403 ;  sensitiveness  of  — s,  402 ; 
thermo-electric,  404,  628 ;  Bolometer, 
717 
,  Thermometric  Conductivity,  407,  408 

Thermometric  Scales,  364,  402 

Thermopile.     See  Thermo-electric  Pile 

Thermoscope,  401 

Thomson's  Effect,  650 

Thoracic  Duct,  309 

Throttling,  722,  736 

Tide,  6,  204,  205,  220 

Tide-calculating  Machine,  103 

Timbre  of  a  Sound  (Quality  or  Charac- 
ter), 415,  428 

Time,  9  ;  measurement  of,  34  ;  Sidereal 
— ,  9  ;  Solar  — ,  9  ;  unit  of,  9  ;  Time 
of  Relaxation  in  Canada  Balsam,  228 

Tone,  418  ;  Combinational  — s,  473 ; 
Differential,  473  ;  Summational,  473, 
474 

Torque,  158,  160,  166,  263,  676,  681, 
691,  716,  723 

Torricelli's  Law,  302,  330,  334;  — 
Vacuum,  342,  344,  349 

Torsibility,  263 

Torsion,  39,  263,  606,  607,  680 

Torsion  Balance,  607 

Total  Induction :  electric,  596,  598,  604 

Total  Number  of  Lines  of  Force  :  elec- 
tric, 583 ;  magnetic,  685 

Total  Pressure,  24 

Total  Reflexion,  519 

Total  Refraction,  520 

Total  Tension,  26,  167,  260 

Toughners,  262,  265  ;  Elastic  — ,  265 

Tourmaline,  556,  623 


Tourniquet,  Hydraulic,  306 

Traction,  26,  181,  260,  435,  444,  582, 
583;  Electricity  on  — ,  623;  Fric- 
tional  Resistance  to  — ,  181 

Transference  of  Heat,  406 

Transformations  of  Energy,  47  et passim 

Transformers,  724  ;  Impedance  in,  725 

Transition  in  Music,  423 

Translation  of  Molecules,  351 ;  —  of  a 
Particle,  73;  —  of  a  Rigid  Body, 
74,  76 

Translucency,  497 

Transmissibility  of  Fluid  Pressures, 
290,  333 

Transmission  of  Energy,  182,  183,  230  ; 
739;  Lines  of,  671,  699;  by  Steady 
Current,  632,  648,  739 ;  by  Oscillat- 
ing Current,  721,  735  ;  by  Intermit- 
tent Current,  733 

Transmission  of  Light,  Coefficient  of, 
499 

Transmutation  of  Elements,  218,  219 

Transparence,  497,  509,  744 

Transpiration  of  Gases,  331 ;  Coefficient 
of,  331 

Transport  of  Heat,  411 

Transversal  Vibrations,  106-110,  412, 
424,  440  ;  of  Ether,  479,  509 

Travelling  of  Electric  Condition,  685  ; 
—  of  Molecules  of  Solids,  257  ;  —  of 
Wave-Form,  104,  111 

Trevelyan's  Rocker,  455 

Triangle  of  Accelerations,  68 ;  —  of 
Forces,  145  ;  —  of  Velocities,  63 

Triphase  Electromotors,  741 

Trombone,  422,  453 

Trough  and  Crest,  104 

True  Contact-effect,  612,  616,  624,  627, 
649 

Trumpet,  453 

Tubes,  Capillary,  315  ;  Elastic  — ,  320- 
323  ;  Rigid  — ,  309  ;  Geissler's  — s, 

656,  723;  Piezometer s,  295,  310, 

334  ;  Pilot's  — s,  319,  336  ;  Safety- 
—,340 

Tubes  of  Flow,  602  ;  —  of  Force,  197, 
583,  602 

Tuning-Fork  :  35,  48,  267,  335,  412,  414, 
419,  421,  429,  430,  442,  446,  447,  456, 
458,  471,  475,  477,  735 ;  Electromag- 
netic Interrupter  for  — ,  267,  447, 
735 

Twinkling  of  Stars,  550 

Twist  in  a  Magnet,  688 

Twisting  Moment,  158,  160,  166,  263, 
716,  723 

ULTRA-GASEOUS  Matter,  233,  252,  656 

Ultra-red  Rays,  486 

Ultra-violet  Rays,  482,  484,  486,  589 

Umbra,  547 

Uniaxial,  551,  694 

Uniform  Field,  198,  583,  608,  673 


780 


INDEX. 


Units :  of  Acceleration,  18  ;  Angle,  75  ; 
Area,  12  ;  Astronomical  Units,  202  ; 
Density,  220.  Electrical :  —  Electro- 
dynamic,  670 ;  Electromagnetic  or 
Magnetic  — ,  625,  634,  670,  684,  703, 
707,  709  ;  Magnetic  Field,  Intensity 
of,  677  ;  Magnetic  Force,  675  ;  Mag- 
netic Pole,  676.  Electrostatic:  — 
Capacity,  592,  603;  Conductivity, 
634;  Density,  579;  Difference  of 
Potential,  584,  585,  603  ;  Force,  578, 
602;  Inductive  Capacity,  597,  604; 
Intensity  of  Current,  633  ;  Quantity, 
578,  579,  709 ;  Resistance,  633  ;  Re- 
sistivity, 634.  Practical :  —  Capacity 
(Farad),  711  ;  Difference  of  Poten- 
tial (Volt),  587,  618,  711  ;  Intensity 
of  Current  (Ampere),  711 ;  Quantity 
(Coulomb),  638,  711 ;  Resistance 
(Ohm),  353,  634,  637,  640,  711 ;  Self- 
induction  (Henry),  711 ;  Elongation, 
260 ;  Force,  21 ;  Heat,  353,  404 ; 
Length,  10;  Light,  513;  Mass,  12; 
Astronomical,  202  ;  Space,  10  ;  Time, 
9  ;  Velocity,  14  ;  Volume,  11 ;  Work, 
41 

Universal  Gravitation,  203 

Universe,  Electricity  in  the,  =  0,  581  ; 
Energy  in  the  —  a  constant,  47 

Unsaturated  Vapour,  231 

Utility  of  Electromotor,  738 

U-tube,  294 

VACUUM,  235,  237,  360, 480  ;  Discharge 
through  — ,  656,  662  ;  —  as  an  In- 
sulator, 589  ;  Torricelli's  — ,  342,  344, 
349  ;  _  -Tubes,  656 

Valves,  346  ;  —  of  Heart,  346,  431 

Van  der  Waals's  Law,  375 

Vapour  (1)  in  presence  of  Liquid,  ready 
to  condense,  231 ;  (2)  saturated  or 
unsaturated,  231  ;  (3)  a  gas  con- 
densible  by  pressure  alone,  232; 
Boyle's  Law  in  — s,  390  ;  saturated 
— ,  231,  sp.  heat  of,  371 ;  unsaturated, 
231 

Vapour-Density,  measurement  of,  391 

Vapour- Friction,  electrification  on,  623 

Vapour-Pressure  of  Solution,  281,  387 

Variable  Period,  695-699,  721 

Variation,  Magnetic,  678 

Variations  in  Barometric  Pressure,  348  ; 
—in  Difference  of  Potential,  199,  737  ; 
—  of  Conductivity,  737  ;  —  in  the 
Earth's  Magnetic  Force,  679;  —  of 
Gravity,  22,  41,  205 ;  —  of  Stress, 
267 

Varley's  Condenser,  664 

Velocity,  14,  58,  59, 166  ;  Absolute,  17  ; 
Average  —  under  uniform  accelera- 
tion, 70  ;  mean,  17,  79  ;  Relative,  17  ; 
Uniform  — ,  15,  58  ;  Variable,  17,  58  ; 
Angular  — ,75,  166 ;  Change  of  — , 


18,  68 ;  in  curved  paths,  58  ;  Mini- 
mum angular,  164  ;  Resolution  of  — 
into  Components,  60,  63  ;  Tangential 
— ,  58,  167  ;  —  Conductance  (elec- 
trostatic) a  — ,  638  ;  Electrostatic- 
Electromagnetic  Ratio  a  — ,  708,  709  ; 
Magnetic  Force  possibly  a  — ,  746 ; 
Resistance  (electromagnetic),  a  — , 
710 

Velocities  :  Composition  of  two  — ,  60  ; 
of  more  than  two,  65 ;  Parallele- 
pipedon  of  — ,  67  ;  Parallelogram  of, 
61 ;  Polygon  of,  66,  67  ;  Skew-Poly- 
gon of,  68  ;  Triangle  of  — ,  63 

Velocity  of  Ether- Waves,  480,  510,  512, 
743  ;  in  metals,  637  ;  of  Propagation 
of  an  Electromagnetic  Disturbance, 
234,  635,  698,  743,  744;  Measure- 
ment of  do.,  708  ; Head,  310, 334  ; 

—  of  Irons,  658 ;  —  of  Molecules,  250, 
465,  610,  613  ;  —  of  Outflow,  302,  305, 
334  ;  —  of  Sound,  267,  461-465  ;  — 
of  a  Stream,  317-319  ;  —  of  Trans- 
mission of  Telegraph  Signals,  697  ;  — 
of  Wave- Motion,  105,  112  ;  in  a  Gas, 
268,  370,  464  ;  in  an  Elastic  Solid, 
267  ;  in  a  Liquid,  461 

Vena  Contracta,  304 

Venturi's  Water-Meters,  314 

Verniers,  28 

Vibrating  Body,  Loading  a,  445  , 

Vibrating  Fluid,  Pressure  in,  335 

Vibrations  :  Free  — ,  434^45,  479  ; 
Forced  — ,  445-449,  479  ;  —  of  Bells, 
444  ;  —  of  Condenser,  737  ;  of  Cords, 
133,  413;  Transverse  — ,  106-110, 
412,  424,  435-441,  479,  509  ;  Longi- 
tudinal, 110,  135,  441  ;  —  of  Discs, 

443,  479  ;  —  the  Result  of  Elasticity, 
151, 251,  266  ;  —  as  affecting  Electric 
Discharge,  599  ;  —  of  the  Ether,  479, 
508, 509,  513,  593,  612  ;  —  on  Impact, 
151,  251,  266;    of  Membranes,  137, 

444,  448  ;    of  Molecules,    247,    251, 
351,  479,  506  ;  —  as  affecting  Mag- 
netic   Permeability,    684,    687  ;     — 
photographed,    433  ;    —    of    Plates, 
443  ;   of  Reeds,  442,  443,  451,  736  ; 
Resolution  of—,  107, 110  ;  —  affected 
by  Rigidity,  415,  435;  —  of  Rods, 
135,   441,  443  ;    Stationary  — ,  134  ; 
Stopping  Component  — ,  135  ;  —  of 
Stretched  Strings,  413,  434-441,  446. 

Vibrations,  Method  of  (magnetometry), 

(580 

"  Vibrations  Simples,"  414 
Vibrator,  Herz's,  741 
Villari's  Critical  Value,  690 
Viola,  450 
Violin,  412,  415,  422,  425,  435, 436,  438, 

449,  450 
Violoncello,  450 
Virial,  249 


INDEX. 


781 


Virtual  Image,  525,  527,  535,  537  ;  — 
Focus,  527,  535 

Viscosity,  185,  186,  226,  227,  579,  703  ; 
Coefficient  of  — ,  226,  307,  316; 
Kinematical  Coefficient  of  — ,  227  ; 
—  in  Gases,  251,  404,  428,  477  ;  —  in 
Liquids,  254,  300,  307,  315;  —  in 
Liquid  Streams,  301,  307  ;  Magnetic 
— ,  716  ;  —  Resistances,  185  ;  —  in 
S.H.M.,  185;  —  of  Elastic  Solids, 
267  ;  of  a  Sounding  Body,  414,  415  ; 
Superficial  —  in  Liquids,  278  ;  —  in 
Gaseous  Streams,  331,  334 

Vis  viva,  517 

Vision,  Energy  in,  573 

Vitreous  (positive),  579,  746 

Voice,  475 

Volatilisation,  236  ;  of  carbon,  654, 655  ; 
of  Pt  and  Ir,  654  ;  of  Snow,  236,  390 

Volt,  587,  618,  647,  711,  715 

Voltage,  687,  609,  724,  732 

Volt-meter,  609,  645,  649 

Voltaic  :  see  Galvanic 

Voltaic  Balance,  617 

Voltameter,  661,  662 

Volta's  Pile,  617,  618,  623 

Volume,  12,  218;  Critical  — ,  233,  375  ; 

Density,  188  ;  Elasticity  of,  229, 

259  ;  Measurement  of,  33  ;  —  and 
Pressure  in  Gases,  230 

Vortex-atom,  Vortex-ring,  246,  745 

Vortices,  Magnetic,  234 

Vowel,  475 

WALKING,  6,  175,  181,  209,  212 

Water :  Conductivity  of  — ,  635  ;  Den- 
sity of,  13,  220,  224 ;  maximum 
density  (at  3°-9  C.),  13,  402;  Elec- 
trolysis of  — ,  658  ; Equivalent, 

405  ;  —  Level,  294  ;  —  Meters,  314  ; 
Viscosity  of,  308 

Watt,  42,  647 

Wave :  in  Air,  see  Sound  ;  in  Ether, 
(chap,  xv.)  ;  originated,  506,  741 ;  pro- 
pagated, 508,  742 ;  measured,  542, 
544,  550  ;  relation  to  Fringes,  138  ; 

Compression s  in  the  Ether,  510, 

744 ; Form,  Travelling  of,  104, 

111 ;  Velocity  of  Propagation  of  — , 
105, 112,  269,  370,  461,  464  ;  of  groups 
of  — s,  142  ; Front,  111,  112  ;  Di- 
rection of,  115 ;  Normal  to,  115 ;  — 
-Length,  104,  112  ;  Crest  and  Trough, 
104 ;  Loss  of  half  wave-length,  125, 
518,  545;  —  -Motion:  Energy  of, 
142,  414,  476 ;  Velocity  of,  in  Elastic 
gas,  267,  370  ;  liquid,  461, 462  ;  solid, 
267 

Waves :  bidimensional,  112  ;  concen- 
tric, 114;  flat,  115;  distorted,  116; 
linear,  104  ;  tridimensional,  114  ;  Ef- 
fect of  Screen  on  waves,  139 ;  Fre- 
quency of  — ,  112 ;  Interference  of 


— ,  137 ;  —  Propagated  along  Nor- 
mals, 113,  131,  139;  —  Reflected, 
117-124;  —  reflected  in  Elastic 
Tubes,  321;  —  Refracted,  124; 
Spherical,  128,  130,  514;  -Surface, 
557,  744  ;  —  Traversing  an  Aperture, 
116,  117,  131,  140 

Waves  of  Potential,  697  ;  —  of  Propa- 
gation of  Lines  of  Electric  Force, 
721 ;  —  of  Temperature,  409 ;  Elec- 
tromagnetic — ,  479,  480,  481,  498 

Webers,  715 

Weber's  measurement  of  v,  708;  his 
Theory  of  Induced  Magnetisation, 

Wedge,  174 

Weighing,  333 

Weight,  12,  21,  201,  219,  333;  atomic 
— ,  217  ;  —  measured,  37 

Welding,  257,  258  ;  Electric  — ,  653 

Well,  depth  of,  463 

Wheatstone's  Bridge,  645,  706 

Wheel  and  Axle,  171 

Wheels  of  Railway  Train,  182  ;  Fly 

89,  164,  168,  169,  667;  Friction , 

182  ;  Savart's  — ,  418 

Wheel  work,  172  ;  —  in  a  Clock,  34 

Whispering  Galleries,  460 

White  Light,  484,  485,  487 ; de- 
composed, 485,  530 ;  —  —  recom- 
pounded,  531 

Whitworth's  Measuring  Machine.  29, 
31 

Wickholder,  408 

Winch,  172 

Wind,  116, 348, 463  ;  Work  done  against 
— ,  40  :  Electrical  — ,  580 

Wire  Telephone,  456 

Work,  40  ;  Unit  of,  41 

Work  done,  7,  40  ;  Mean  Rate  of  doing 
— ,  42 ;  —  done  against  Attraction, 

192  ; in  Charging  a  Conductor, 

593,  599 ; in  producing  Com- 
pression, 260 ;  by  Ether- Waves,  670  ; 
by  Expanding  Substance,  54,  359, 
360,  394,  396 ;  in  producing  Exten- 
sion, 260  ;  in  Electrolysis,  662  ;  in 
moving  across  Equipotential  Surfaces, 
194,  685 ;  by  or  against  Force,  40  ; 
against  Friction,  181,  185  ;  by  the 
Heart,  320 ;  by  Heat,  359,  394 ;  dar- 
ing Overturning,  209;  in  moving  a 
Pendulum,  213  ;  by  Repulsion,  190  ; 
in  producing  Shear,  260  ;  in  keeping 
up  a  Stream  of  Liquid,  319 ;  in  pro- 
ducing Twist,  263;  Internal  —  in 
Gases,  359,  369-371,  375 

Worm-wheel,  29 

YELLOW  Spot  (in  eye),  573 
Young's  Experiment,  547 
Young's  Modulus,  261,  321,  441,  443, 
461 


782 


INDEX. 


ZERO  Potential,  192,  588 

Zero  Temperature :  Absolute,  364,  400 ; 

Centigrade,  402  ;  Fahrenheit,  386,  402 
Zero  the  reciprocal  of  Infinity,  191;   — 

of  Thermometers,  402  ;  —  rising,  403 


Zinc,  amalgamated,  356,  618,  064  (elec- 
trodes) ;  —  electropositive  to  Copper, 
negative  in  the  Battery,  611,  615, 
617,  643 


A  LABORATORY  MANUAL 


OF 


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